GEOMETRY - Julian High School



GEOMETRY

POINTS, LINES AND PLANES

Name: __________________________________ Date: ______________

Lesson 1.1: Points, Lines, and Planes

Objective: To understand the basic terms of geometry

|Geometric |Properties |Drawing |Symbol |Read as |

|Figure | | | | |

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|Point |has no size |A |A |point A |

| |indicates a definite | | | |

| |location | | | |

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| |is straight |A B |AB BA |line AB or BA |

| |has no thickness | | | |

|Line |extends indefinitely | | | |

| |in two opposite directions | |l |line l |

| | |l | | |

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|Segment |is part of a line | |AB or BA |segment AB or BA |

| |has two endpoints |A B | | |

| | | | | |

| |is part of a line | |AB |ray AB |

|Ray |has only one endpoint |A B | | |

| |extends indefinitely in one direction | | | |

| | | |BA |ray BA |

| | |A B | | |

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|Collinear points |points that lie on the same line | | |Points A, B, C are collinear |

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| | |A B C | | |

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| |is a flat surface |B | |plane ABC |

|Plane |has no thickness |C | | |

| |extends indefinitely in all directions|A | | |

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| | | | |plane x |

| | |[pic] | | |

|Coplanar points |points that lie on the same plane |W | | |

| | | | |Points X, Y, Z are coplanar |

GEOMETRY

Classwork 1.1: POINTS, LINES AND PLANES

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To draw collinear and noncollinear points

2) To identify collinear points and coplanar points

|1) Draw three noncollinear points J, K, L, and J. Then, using your straightedge, |3) State whether the points are collinear or noncollinear. |

|draw JK, KL and LJ. | |

| |●A |

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| |C D E F |

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| |●B |

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| |a) D, E, F ________________ |

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| |b) C, B, D ________________ |

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| |c) D, C, F ________________ |

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|H |d) A, C, B ________________ |

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|●G | |

|F | |

|E |4) True or False. |

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|D |_______________a) AB and BA are the same |

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| |_______________b) AB and BA are the same |

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| |_______________c) AB and BA are the same |

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| |_______________d) Any three points A, B, and C |

|a) Name three points that are collinear. |must lie in exactly one plane. |

|______, ______, ______ | |

| |5) State whether each is modeled by a point, line or plane. |

|b) Name four points that are coplanar. |a) a star in the sky _______________ |

|______, ______, ______ , ________ | |

| |b) an ice skating rink ______________ |

|c) Name three points that are noncollinear. | |

|______, ______, ______ |c) a telephone wire strung between two poles _______ |

GEOMETRY

MEASURING LINE SEGMENTS

Name: __________________________________ Date: ______________

Lesson 1.2: Measuring Line Segments

Objectives: To find the distance between two points on a number line.

| Distances can be determined on both a number line and in the |The distance between points A and B, written AB, is the absolute value of the |

|Coordinate Plane.   |difference between the coordinates of A and B. |

|A number line is a straight horizontal line on which each point represents a real | |

|number.  Integers are points marked at unit distance apart (....-3,-2,-1,0,1,2,3,...) |Thus, AB = │x2 – x1│ or BA = │x1 – x2│ |

|as shown below. | |

|[pic] |Note: When you use absolute value, the order in which you subtract the |

|Distance on a number line |coordinates does not matter. |

|Consider the line segment AB below: | |

|names of points |Thus, AB = BA. |

|A B | |

|x1 x2 |Example: |

| |[pic] |

|coordinates of points | |

| |Find AB, AC, AD, BC, and BD. |

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|The real number that corresponds to a point is the coordinate of the point. |Solution: |

| |AB = │-2 – (-3.5)│= 1.5 = BA |

|Here x1 is the coordinate of point A, | |

|x2 is the coordinate of point B. |AC = │1 – (-3.5)│ = 4.5 = CA |

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|Illustration: |BC = │1 – (-2)│ = 3 = CB |

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|A B |BD = │3 – (-2)│ = 5 = DB |

|-1 0 1 2 | |

| |Or count the units from one point to another point. |

|Then -1 is the coordinate of point A |---------------------------------------------------------------- |

|2 is the coordinate of point B |When three points lie on a line, you can say that one of them is between the |

| |other two. This concept applies to collinear points only. |

|The length or measure of AB, written AB (no bar over the letters), is the distance | |

|between A and B. Thus, the measure of a segment is the same as the distance between|In the figure below, point B is between points A and C, while point D is not |

|its two endpoints. Note that the distance between A and B is the same as the |between A and C. For B to be between A and C, all three points must be |

|distance between B and A. |collinear. |

| |D● |

| |C● |

|Recall: | |

|Postulate or Axioms = rules that are accepted without |B● |

|proof. |A● |

|Theorems = rules that are proved | |

| |Segment AB, written AB, consists of points A and B and all points between A |

|Segment Addition Postulate |and B. |

|If B is between A and C, then AB + BC = AC. |2) If B is between A and C, find the value of x and the measure of BC and AB. |

|If AB + BC = AC, then B is between A and C. |a) AB = 3x, BC = 5x, AC = 8 |

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|AC |AB + BC = AC |

| |3x + 5x = 8 |

|A B C |8x = 8 |

|● ● ● |x = 1 |

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|AB BC |BC = 5x = 5(1) = 5 |

| |AB = 3x = 3(1) = 3 |

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|Example: |b) AB = 6x - 5, BC = 2x + 3, AC = 30 |

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|1) Given that B is between A and C, find the missing measure. |AB + BC = AC |

|a) AB = 5, BC = ? AC = 7 |6x - 5 + 2x + 3 = 30 |

| |8x = 32 |

|AB + BC = AC |x = 4 |

|5 + BC = 7 | |

|BC = 2 |BC = 2x + 3 = 2(4) + 3 = 11 |

| |AB = 6x - 5 = 6(4) – 5 = 19 |

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|b) AB = ? BC = 18.9, AC = 23 | |

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|AB + BC = AC | |

|AB + 18.9 = 23 | |

|AB = 4.1 | |

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|c) AB = 21, BC = 4.3, AC = ? | |

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|AB + BC = AC | |

|21 + 4.3 = AC | |

|25.3 = AC | |

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GEOMETRY

Classwork 1.2: MEASURING LINE SEGMENTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the distance between two points on a number line.

|1) Refer to the number line below to find each measure. Show your work. |3) Given that E is between D and F, find the missing measure. Show your work. |

| |a) DE = ?, EF = 2.75 in, AC = 12 in |

|A B C D E | |

|● ● ● ● ● | |

|-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 | |

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|a) AC = b) BA = | |

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| |b) DE = 6.13 cm, EF = ?, DF = 10.25 |

|c) EB = d) DB = | |

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|d) DC = e) CA = | |

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|2) Given that B is between C and A, find the missing measure. Show your work. |4) If P is between Q and R, find the value of y and the measure of PQ and QR. Show|

|a) AB = 3.3cm, BC = 3.3 cm, AC = ? |your work. |

| |PQ = 2y, QR = 3y + 1, PR = 21 |

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|b) AB = ?, BC = 5.4 cm, AC = 10.25 | |

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|5) Find the value of the variable and ST if S is between R and T. | |

|a) RS = 7x, ST = 12x, RT = 38 |d) RS = 12, ST = 2x, RT = 34 |

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|b) RS = 2x, ST = 3x, RT = 28 | |

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| |e) RS = 16, ST = 2x, RT = 5x + 10 |

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|c) RS = 3y + 1, ST = 2y, RT = 21 | |

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| |f) RS = 4y - 1, ST = 2y - 1, RT = 5y |

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GEOMETRY

Classwork 1.3: MEASURING LINE SEGMENTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the distance between two points on a number line.

|1) Refer to the number line below to find each measure. Show your work. |3) If B is between A and C, find the value of x and the measure of AB and BC. Show|

| |your work. |

|A B C D E |a) AB = 4x, BC = 6x, AC = 40 |

|● ● ● ● ● | |

|-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 | |

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|a) AE = b) BD = | |

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|c) EC = d) EB = | |

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| |b) AB = 3x + 21, BC = 2x -6, AC = 50 |

|d) DA = e) DA = | |

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|2) Given that B is between A and C, find the missing measure. Show your work. | |

|a) AB = 10, BC = ?, AC = 25 | |

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| |c) AB = 4x - 3, BC = 6x - 4, AC = 113 |

|b) AB = ?, BC = 25.4, AC = 45 | |

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|c) AB = 41, BC = 10.3, AC = ? | |

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GEOMETRY

Classwork 1.4: MEASURING LINE SEGMENTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the distance between two points on a number line.

|1) Refer to the number line below to find each measure. Show your work. |3) Given that D is between E and F, find the missing measure. Show your work. |

| |Draw the line and find the formula. |

|E A B C D | |

|● ● ● ● ● |a) DE = ?, EF = 12.75 in, DF = 6 in |

|-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 | |

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|a) CD = b) BD = | |

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|c) EC = d) DB = | |

| |b) DE = 6.13 cm, EF = ?, DF = 10.25 |

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|d) DC = e) CA = | |

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|2) Given that C is between A and B, find the missing measure. Show your work. | |

|Draw the line and find the formula. | |

|a) AB = 10.3cm, BC = 5.6 cm, AC = ? |4) If Q is between P and R, find the value of y and the measure of PQ. Show your |

| |work. Draw the line and find the formula. |

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| |PQ = 3y, QR = 25, PR = 4y + 1 |

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|b) AB = ?, BC = 5.4 cm, AC = 10.25 | |

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|5) Find the value of the variable and SQ if Q is between S and T. Show your |6) Find the value of the variable and ON if O is between M and N. Show your |

|work. Draw the line and find the formula. |work. Draw the line and find the formula. |

| |a) MO = 12, ON = 2x, MN = 34 |

|a) SQ = 7x, QT = 7, ST = 2x + 10 | |

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|b) SQ = 35 - x, ST = 50, QT = 2x - 3 |b) ON = 3x, MO = 16, MN = 5x + 10 |

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|c) ST = 21, SQ = 3y + 1, QT = 2y |c) MO = 4y - 1, MN = 5y , ON = 2y - 3 |

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|d) QT = 5, ST = 2x + 5, SQ = 7x - 2 |d) MN = 12 – 3x, ON = 5x, MO = 4 |

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GEOMETRY

Classwork 1.5: MEASURING LINE SEGMENTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To find the distance between two points on a number line, using the Segment Addition

Postulate.

2) To use the Segment addition Postulate to solve real-world problems.

|Use the Segment Addition Postulate to solve the indicated distances. Draw the |2) Towns M, P and Q are located along a straight highway. Town Q is between towns|

|corresponding line segment with complete labels. Show your work. |M and P, and the distance from M to P is 60 miles. If QP is 6 miles more than twice|

| |MQ, find the distance QP and MQ. Represent each town as a point on a line segment, |

|1) If Q is between S and T, and |then use the segment addition postulate. Show your work. |

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|a) SQ = 7x, QT = 10, ST = 2x + 30, find x, SQ, and ST. | |

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|b) SQ = 2x – 3, ST = 50, QT = 35 – x, find x, SQ and QT. | |

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| |3) Cities S, Q and T are located along a straight highway. City Q is between |

| |cities S and T, and the distance from Q to T is 20 miles. If ST is 30 miles less |

| |than six times SQ, find the distance SQ and ST. Represent each town as a point on a|

| |line segment, then use the segment addition postulate. Show your work. |

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|c) ST = 21, SQ = 3y + 1, QT = 2y, find y, SQ and QT. | |

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|Use the Segment Addition Postulate to solve the indicated distances. Draw the |5) Towns LaGrange(L), Belmont(B) and Naperville(N) are located along a straight |

|corresponding line segment with complete labels. Show your work. |highway. Town Belmont is between towns LaGrange and Naperville, and the distance |

| |from LaGrange to Naperville is 56 miles. If LB is 5 miles more than twice BN, find |

|4) If O is between M and N, and |the distance BN and LB. Represent each town as a point on a line segment, then use |

|a) MO = 10, ON = 2x, MN = 34, find x and ON. |the segment addition postulate. Show your work. |

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|b) ON = 5x – 9, MO = 3, MN = 3x , find x, ON, and MN. | |

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| |6) Towns Hinsdale(H) , Lisle(L) and Naperville(N) are located along a straight |

| |highway. Town Lisle is between towns Hinsdale and Naperville, and the distance |

| |from Lisle to Naperville is 4 miles. If HN is 31 less than six times HL, find the |

| |distance HL and HN. Represent each town as a point on a line segment, then use the |

| |segment addition postulate. Show your work. |

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|c) MO = 4y - 2, MN = 5y , ON = 3y – 2, find y, MO, MN and ON. | |

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|d) MN = 12 – 3x, ON = 5x, MO = 4, find x, MN and ON. | |

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GEOMETRY

Classwork 1.6: MEASURING LINE SEGMENTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To find the distance between two points on a number line, using the Segment Addition

Postulate.

2) To use the Segment addition Postulate to solve real-world problems.

|Use the Segment Addition Postulate to solve the indicated distances. Draw the |2) Towns LaGrange(L), Belmont(B) and Naperville(N) are located along a straight |

|corresponding line segment with complete labels. Show your work. |highway. Town Belmont is between towns LaGrange and Naperville, and the distance |

| |from LaGrange to Naperville is 56 miles. If LB is 5 miles more than twice BN, find |

|1) If O is between M and N, and |the distance BN and LB. Represent each town as a point on a line segment, then use |

|a) MO = 15, ON = 3x, MN = 36, find x and ON. |the segment addition postulate. Show your work. |

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|b) ON = 6x – 1, MO = 8, MN = 8x , find x, ON, and MN. | |

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| |3) Towns Hinsdale(H) , Lisle(L) and Naperville(N) are located along a straight |

| |highway. Town Lisle is between towns Hinsdale and Naperville, and the distance |

| |from Lisle to Naperville is 4 miles. If HN is 31 less than six times HL, find the |

|c) MO = 5y - 3, MN = 7y , ON = 3y – 4, find y, MO, MN and ON. |distance HL and HN. Represent each town as a point on a line segment, then use the |

| |segment addition postulate. Show your work. |

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|d) MN = 12 – 5x, ON = 7x, MO = 6, find x, MN and ON. | |

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| |5) Towns M, P and Q are located along a straight highway. Town Q is between towns|

| |M and P, and the distance from M to P is 68 miles. If QP is 5 miles more than twice|

|4) If R is between P and T, and |MQ, find the distance QP and MQ. Represent each town as a point on a line segment, |

| |then use the segment addition postulate. Show your work. |

|a) PR = 8x, RT = 15, PT = 3x + 25, find x, PR, and PT. | |

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|b) PR = 2x – 3, PT = 50, RT = 35 – x, find x, PR and RT. | |

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| |6) Cities S, Q and T are located along a straight highway. City Q is between |

| |cities S and T, and the distance from Q to T is 25 miles. If ST is 35 miles less |

| |than six times SQ, find the distance SQ and ST. Represent each town as a point on a|

| |line segment, then use the segment addition postulate. Show your work. |

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|c) PT = 24, PR = 4y + 3, RT = 3y, find y, PR and RT. | |

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|d) RT = 5y, PR = 4y - 3, PT = 24 - y, find y, PR and RT. | |

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GEOMETRY

MIDPOINTS ON A NUMBER LINE AND IN THE COORDINATE PLANE

Name: __________________________________ Date: ______________

Lesson 1.4: Midpoints and Segment Congruence

Objective: To find the midpoint of a segment on a number line and in the coordinate plane.

| Sometimes you need to find the point that is exactly between two other points.| |

|For instance, you might need to find a line that bisects (divides into equal halves) a|w = [pic] |

|given line segment. This middle point is called the "midpoint". The concept doesn't | |

|come up often, but the Formula is quite simple and obvious, so you should remember it |b) R = 1, W = -3 |

|for later. |S W R |

|Definition of Midpoint: |-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 |

|The midpoint M of PQ is the point between P and Q such that PM = MQ. |w = [pic] -3 = [pic] |

| |Solving for s, |

|The Midpoint Formula |s = -7 |

|In One Dimension |----------------------------------------------------------------- |

|On a number line, the number halfway between x1 and x2 is | |

|[pic]  |In Two Dimensions |

|Example 1: |Suppose you are given two points in the plane (x1, y1) and (x2, y2), and asked |

|Find the midpoint between -1 and 4. |to find the point halfway between them. The coordinates of this midpoint will |

|Use the formula. The midpoint is |be: |

|(-1 + 4)/2 |[pic] |

|= 3/2 or 1.5. |An easy way to think about this is that the x-coordinate of the midpoint is the |

|[pic] |average of the x-coordinates of the two points, and likewise with the |

|Example 2: W, R, and S are points on a number line, and W is the midpoint of the |y-coordinate. |

|line segment RS. For each pair of coordinates given, find the coordinate of the third |Example 3: |

|point. |Find the midpoint between (-2, 5) and (7, 7). |

|a) R = 4, S = -6 |Use the formula. The coordinates of the midpoint are: |

|S W R |[pic] |

|-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 |Simplify. |

|[pic] |[pic] |

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| |Example 5: |

|Example 4: |If Y is the midpoint of the line segment XZ, XY = 2a + 11, and YZ = 4a –|

|Find the coordinates of point Q if L(4, -6) is the midpoint of the segment NQ and|5, find the value of a and the measure of the line segment XZ. |

|the coordinates of N are (8, -9). | |

| |Solution: Y is the midpoint of XZ, so XY = YZ. |

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| |Z |

| |4a - 5 |

|0 | |

| |2a + 11 Y |

|Q | |

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| |Write an equation and solve for a. |

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|N |XY = YZ |

| |2a + 11 = 4a – 5 |

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|[pic] = (4, -6) |-2a = -16 |

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|[pic] = 4 [pic] = -6 |a = 8 |

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|[pic] = 4 [pic] = -6 | |

| |Now use the value of a to find XZ. |

|8 + x2 = 8 -9 + y2 = -12 | |

| |XZ = XY + YZ |

|x2 = 0 y2 = -3 | |

| |= (2a + 11) + (4a – 5) |

|The coordinates of Q are (0, -3). | |

|---------------------------------------------------------------------- |= 6a + 6 |

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| |= 6(8) + 6 |

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| |= 54 |

| |----------------------------------------------------------------- |

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GEOMETRY

Classwork 1.9: MIDPOINTS ON A NUMBER LINE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the midpoint of a segment on a number line.

| Find the midpoint between the indicated points on a number line. Draw each line | In each of the following, draw the indicated line segment, and the corresponding |

|segment and solve. |coordinates. |

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|Between -2 and 4. |If the midpoint of the segment PQ is 5 and the coordinate of P is -1, find the |

| |coordinate of Q. |

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|Between -4 and 2. | |

| |If the midpoint of the segment AB is -5 and the coordinate of A is -8, find the |

| |coordinate of B. |

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|Between 5 and 8. | |

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| |If the midpoint of the segment RS is 2.5 and the coordinate of R is -1.5, find the|

| |coordinate of S. |

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|Between -4.5 and 0. | |

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| |If the midpoint of the segment KL is 8 and the coordinate of K is 3, find the |

| |coordinate of L. |

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|Between 2 and 8.5. | |

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GEOMETRY

Classwork 1.10: MIDPOINTS AND SEGMENT CONGRUENCE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the midpoint of a segment on a number line and in the coordinate plane

|Use the figure below for Problems 1 - 7 |In the figure below, [pic] bisects [pic] at C, and [pic] bisects [pic] at B. |

| |For each of the following, find the value of x and the measure of the indicated |

| |segment. |

|A | |

| |A F● |

|●B | |

|G C H I E J |● B |

|● ● ● ● ● ● |E |

|-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 |C |

| | |

|●F | |

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| |D |

|Find the coordinates of the midpoint of each segment. |8) AB = 3x + 6, BC = 2x + 14; [pic] |

|1) GI [pic] = | |

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|2) CE = | |

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|3) HJ = | |

| |9) AC = 5x - 8, CD = 16 - 3x; [pic] |

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|Determine whether each statement is true or false. If false, state why. | |

|4) _________ | |

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|5) [pic] bisects [pic]. ______ | |

| |10) AD = 5x + 2, BC = 7 - 2x; [pic] |

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|6) AE bisects [pic]._______ | |

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|7) E is the midpoint of [pic].______ | |

GEOMETRY

Classwork 1.11: MIDPOINTS ON A NUMBER LINE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objective: To find the midpoint of a segment on a number line.

|In each of the following problems, draw the line segment with complete labels. |2) T is the midpoint of [pic]. Find x, PT, TQ and PQ, if |

| | |

|C is the midpoint of [pic]. Find x, AC, CB and AB, if |PT = 5x + 3 and TQ = 7x - 9 |

|AC = 2x + 1 and CB = 3x – 4. | |

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|CB = 6x + 2 and AC = 8x. |TQ = 3x + 4 and PT = 4x - 6 |

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|AC = 5x – 3 and CB = 3x + 4. |PT = 7x - 24 and TQ = 6x - 2 |

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|3) M is the midpoint of [pic]. Find y, LM, MN and LN, if | |

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|a) LM = 12 – 5y and MN = 7y. | |

| |Applications: |

| |4) Highways and the mile markers along their sides suggest a number line. You can |

| |find the distance between mile markers in the same way that you find distance on a |

| |number line. |

| |Michael sees mile marker 237 when he enters the highway and mile marker 159 when he |

| |exits. How far did he travel? |

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|b) MN = 2y – 4 and LM = 35 - y. | |

| |5) Use the following figure to describe the statement as true or false. |

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| |A B C D E |

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| |-8 -6 0 1 3 7 |

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| |AB = CD _____________________ |

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| |BD < CD _____________________ |

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| |AC + BD = AD __________________ |

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|c) LM = 5y and MN = 27 - y. |AC + CD = AD _________________ |

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| |Use the following figure to answer problems 6 - 8. |

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| |W A S Q B |

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| |-8 0 2 4 8 |

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| |6) Find the midpoint of [pic]. |

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|d) MN = 5y - 4 and LM = 27 -3y. | |

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| |7) What is the coordinate of the midpoint of [pic]? |

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| |8) What is the coordinate of the midpoint of [pic]? |

GEOMETRY

Classwork 1.12: MEASURING LINE SEGMENTS AND MIDPOINTS

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To find the distance between two points on a number line, using the Segment Addition

Postulate.

2) To use the Segment addition Postulate to solve real-world problems.

3) To find the midpoint of a segment on a number line.

|Use the Segment Addition Postulate to solve the indicated distances. Draw the |Find the midpoint between the indicated points on a number line. Draw each line |

|corresponding line segment with complete labels. Show your work. |segment and solve. Show your work. |

|1) If O is between M and N, and |Between -5 and -1. |

|a) MO = 23, ON = 5x, MN = 42, find x and ON. | |

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| |Between 4 and 9. |

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|b) ON = 7x – 5, MO = 18, MN = 12x , find x, ON, and MN. |Between -1 and 4. |

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| |In each of the following, draw the indicated line segment, and the corresponding |

| |coordinates. Show your work. |

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| |If the midpoint of the segment PQ is 1.5 and the coordinate of P is -2, find the |

| |coordinate of Q. |

|c) MO = 7y - 5, MN = 5y , ON = 13y – 10, find y, MO, MN and ON. | |

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| |If the midpoint of the segment AB is -3.4 and the coordinate of A is -6.5, find the|

| |coordinate of B. |

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|d) MN = 15 – 7x, ON = 9x, MO = 3, find x, MN and ON. | |

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| |If the midpoint of the segment RS is 12.5 and the coordinate of R is 8.4, find the|

| |coordinate of S. |

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|3) M is the midpoint of [pic]. Find y, LM, MN and LN, if | |

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|a) LM = 12 – 15y and MN = 7y. |4) Towns Kenosha(K), Waukegan(W) and LakeBluff(L) are located along a straight |

| |highway. Waukegan is between Kenosha and LakeBluff, and the distance from Kenosha |

| |to LakeBluff is 64.5 miles. If WL is 6 miles more than twice KW, find the distance |

| |KW and WL. Represent each town as a point on a line segment, then use the segment |

| |addition postulate. Show your work. |

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|b) MN = 5y – 13 and LM = 35 - 3y. | |

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| |5) Towns Glencoe(G), Winnetka(W) and Ravenswood(R) are located along a straight |

| |highway. Winnetka is between Glencoe and Ravenswood, and the distance from |

| |Winnetka to Ravenswood is 40 miles. If GR is 18 less than thirty times GW, find |

|c) LM = 6y and MN = 37 - 4y. |the distance GW and GR. Represent each town as a point on a line segment, then use |

| |the segment addition postulate. Show your work. |

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|d) MN = 15y - 9 and LM = 29 - 4y. | |

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|e) LM = 28 - 6y and MN = 37 - 14y. | |

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Name: __________________________________ Date: ______________

Lesson 1.13: Exploring Angles

Objectives: 1) To identify angles and classify angles

2) To use the Angle Addition Postulate to find the measures of angles, and

3) To identify and use congruent angles and the bisector of an angle.

|ANGLE - An angle is formed by two rays which begin at the same point (if the |[pic] |

|two rays lie on the same line, then it is called a straight angle).   | |

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|There are several ways of naming angles: |EXTERIOR & INTERIORS OF ANGLES - An angle separates a plane into THREE distinct regions,|

|a) a capital letter at its vertex ([pic] P); |the interior of the angle, the exterior of the angle, and the angle itself.  (The Blue |

|b) a small letter within the angle; |is the interior angle, and the Yellow is the exterior angle.) |

|c) a number within the angle, or |  |

|d) by three capital letters, the middle letter is the vertex, the other two |  |

|are points on each ray ([pic] SPQ or [pic]QPS).  We will often name angles| |

|by this last approach. |Angle Classification:   Angles are measured in degrees.  A circle has 360 degrees, and |

|SIDES OF THE ANGLE - The two rays that form the angle are called the sides of |of course half of a circle is 180 degrees. |

|the angle.  (Side PS & Side PQ) |1) An ACUTE ANGLE is an angle whose measure is between 0 and 90 degrees. |

|VERTEX - The common point for both rays is called the vertex.  (Point P) |[pic] |

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| |The following angles are all obtuse. |

|Degrees: Measuring Angles | |

|We measure the size of an angle using degrees. |[pic] |

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|Example: Here are some examples of angles and their degree measurements. | |

| |4) A STRAIGHT ANGLE is one whose measure is EXACTLY 180 DEGREES.  A straight angle is |

| |made up of two opposite rays.   Another important fact is that a straight angle forms a |

| |straight line.  This information will be used very frequently throughout the year. |

| |[pic] |

|The following angles are all acute angles. | |

|[pic] | |

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|A RIGHT ANGLE is an angle whose measure is EXACTLY 90 DEGRRES. Right angles |5) A REFLEX ANGLE is one whose measure is GREATER THAN 180 AND LESS THAN 360 DEGREES. |

|are denoted by a small square in its interior. |[pic] |

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|[pic] |The following angles are all reflex. |

|Two lines or line segments that meet at a right angle are said to be | |

|perpendicular. Note that any two right angles are supplementary angles (a | |

|right angle is its own angle supplement). | |

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|3) An OBTUSE ANGLE is one whose measure is greater than 90 and less than 180 | |

|degrees. | |

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|[pic] | |

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| |Angle Bisector |

| |An angle bisector is a ray that divides an angle into two equal angles. |

| |The blue ray on the right is the angle bisector of the angle on the left. |

|6) CONGRUENT ANGLES - Two angles that have the same measure are called |[pic] |

|Congruent Angles.  Equal measure angles are labeled as shown in the diagram. |The red ray on the right is the angle bisector of the angle on the left. |

|[pic] |[pic] |

|7) Complementary Angles -Two angles are called complementary angles if the |[pic] |

|sum of their degree measurements equals 90 degrees. One of the complementary | |

|angles is said to be the complement of the other. |ANGLE ADDITION POSTULATE |

|These two angles are complementary. | |

|[pic] |If R is in the interior of [pic]PQS, then |

|Note that these two angles can be "pasted" together to form a right angle! |m[pic]PQR + m[pic]RQS = m[pic]PQS. |

|8) Supplementary Angles | |

|Two angles are called supplementary angles if the sum of their degree |If m[pic]PQR + m[pic]RQS = m[pic]PQS, then R is in the interior of [pic]PQS. |

|measurements equals 180 degrees. One of the supplementary angles is said to be| |

|the supplement of the other. |[pic] |

|These two angles are supplementary. | |

|[pic][pic] | |

|Note that these two angles can be "pasted" together to form a straight line! |In the special case where [pic]PQS is a straight angle or in other words a straight line|

|[pic] |then [pic]PQR + [pic]RQS = 180. |

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|Example: In the figure below, BA and BC are opposite | |

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|rays, and BE bisects [pic]ABD. | |

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|●D | |

|E● | |

|A ● | |

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|B | |

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|C ● | |

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|a) If m[pic]ABE = 6x + 2 and m[pic]DBE = 8x – 14, find m[pic]ABE. | |

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|From the figure, m[pic]ABE = m[pic]DBE | |

|Thus, 6x + 2 = 8x – 14 | |

|Solving for x: | |

|x = 8 | |

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|So, m[pic]ABE = 6x + 2 | |

|= 6(8) + 2 = 50 | |

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|b) Given that m[pic]ABD = 2y and m[pic]DBC = 6y – 12, find m[pic]DBC. | |

|m[pic]ABD + m[pic]DBC = 180 | |

|2y + 6y – 12 = 180 | |

|Solving for y: y = 24 | |

|Thus, | |

|m[pic]DBC = 6y – 12 | |

|= 6(24) – 12 | |

|= 132 | |

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GEOMETRY

Classwork 1.14: USING THE ANGLE ADDITION POSTULATE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To find the measures of angles.

2) To use the Angle Addition Postulate to find the measures of angles.

|A) Use the following figure to solve the following problems 1 - 3. Show your work.|B) Use the following figure to solve the following problems 4 - 6. Show your work.|

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|C |R |

|M |P |

|2 | |

|1 | |

|A B | |

| |Q S |

|1) m[pic]1 = 350, m[pic]2 = 460, find m[pic]ABC. | |

| |4) m[pic]PQR = 750, m[pic]RQS = 480, find m[pic]PQS. |

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|2) m[pic]2 = 350, m[pic]ABC = 590, find m[pic]1. | |

| |5) m[pic]RQS = 530, m[pic]PQS = 1230, find m[pic]PQR. |

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|3) m[pic]1 = 230, m[pic]ABC = 83.50, find m[pic]2. | |

| |6) m[pic]PQR = 920, m[pic]PQS = 134.60, find m[pic]RQS. |

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|C) Use the following figure to solve problems 7 – 11. Show your work. | |

| |D) Use the following figure to solve the following problems 12 -15 . Show your |

|C |work. |

|M | |

|2 |R |

|1 |P |

|A B | |

|7) m[pic]1 = 450, m[pic]2 = 420, find m[pic]ABC. | |

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| |Q S |

| |12) m[pic]PQR = 77.50, m[pic]RQS = 490, find m[pic]PQS. |

|8) m[pic]1 = 370, m[pic]ABC = 800, find m[pic]2. | |

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| |13) m[pic]RQS = 630, m[pic]PQS = 1320, find m[pic]PQR. |

|9) m[pic]2 = 460, m[pic]ABC = 520, find m[pic]1. | |

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|10) m[pic]1 = 29.60, m[pic]2 = 340, find m[pic]ABC. |14) m[pic]PQR = 950, m[pic]PQS = 143.50, find m[pic]RQS. |

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|11) m[pic]2 = 44.350, m[pic]ABC = 600, find m[pic]1. | |

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| |15) m[pic]1 = 360, m[pic]ABC = 83.50, find m[pic]2. |

GEOMETRY

Classwork 1.15: USING THE ANGLE ADDITION POSTULATE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To find the measures of angles.

2) To use the Angle Addition Postulate to find the measures of angles.

|Use the following figure to solve problems 1 – 5. Show your work. |4) m[pic]2 = 4x - 10, m[pic]1 = x + 20, m[pic]GHT = 800, find m[pic]1 |

|G H |and m[pic]2. |

|1 2 | |

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|B | |

|T | |

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|1) m[pic]1 = 3x + 5, m[pic]2 = 2x + 35, m[pic]GHT = 800, find m[pic]1 | |

|and m[pic]2. | |

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| |5) m[pic]2 = 4x + 20, m[pic]1 = 32 – 3x, m[pic]GHT = 620, find m[pic]1 |

| |and m[pic]2. |

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|2) m[pic]2 = x + 28, m[pic]1 = 2x + 2, m[pic]GHT = 540, find m[pic]1 | |

|and m[pic]2. |Use the following figure to solve problem 6 . Show your work. |

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| |R |

| |S |

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| |2 |

| |1 Q |

| |T 3 |

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| |P |

|3) m[pic]1 = 2x - 4, m[pic]2 = 5x + 2, m[pic]GHT = 770, find m[pic]1 | |

|and m[pic]2. | |

| |6) m[pic]2 = 180, m[pic]1 = 370, find m[pic]TQR . |

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|Use the following figure to solve problems 7 – 11. Show your work. |Use the following figure to solve problems 11 – 14 . Show your work. |

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|C | |

|M |R |

|2 |S |

|1 | |

|A B |2 |

| |1 Q |

|7) m[pic]1 = 290, m[pic]2 = 360, find m[pic]ABC. |T 3 |

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| |P |

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| |12) m[pic]2 = 280, m[pic]1 = 4x - 7, m[pic]TQR = 85, find m[pic]1 . |

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|8) m[pic]1 = 370, m[pic]ABC = 780, find m[pic]2. | |

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| |13) m[pic]1 = 3x + 5, m[pic]3 = 2x - 5, m[pic]SQP = 6x - 4, |

|9) m[pic]2 = 460, m[pic]ABC = 620, find m[pic]1. |find m[pic]1, m[pic]3, and m[pic]SQP. |

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|10) m[pic]1 = 2x + 6, | |

|m[pic]2 = 3x + 45, m[pic]ABC = 860, find m[pic]1 and m[pic]2. | |

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| |14) m[pic]3 = 5x - 3, m[pic]1 = 2x + 5, m[pic]SQP = 10x - 4, |

| |find m[pic]1, m[pic]3, and m[pic]SQP. |

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|11) m[pic]2 = 4x + 20, | |

|m[pic]1 = 32 – 3x, m[pic]CBA = 620, find m[pic]1 and m[pic]2. | |

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GEOMETRY

Classwork 1.16: CLASSIFICATION OF ANGLES AND THE ANGLE ADDITION POSTULATE

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To classify angles by their measures

2) To apply the Angle Addition Property

College Readiness Standards:

Exhibit some knowledge of basic angle properties (strand 20 – 23)

Use several angle properties to find an unknown angle measure (strand 24 – 27)

|1. What is the measure of each of the following | Use the following figure to solve problems 3 – 8. |6) m[pic]1 = 2x – 5, m[pic]2 = 3x + 4, |

|angles? |Show your work. |m[pic]ABC = 89, find m[pic]1 and m[pic]2. |

|a) right | | |

| |C | |

|b) straight |M | |

| |2 | |

|c) acute |1 | |

| |A B |7) m[pic]2 = 25 - 4x, m[pic]ABC = 590, m[pic]1 = |

| |3) m[pic]1 = 450, m[pic]2 = 420, find |10x + 7, find m[pic]1 and m[pic]2. |

|d) obtuse |m[pic]ABC. | |

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|e) reflex | | |

| |4) m[pic]1 = 370, m[pic]ABC = 800, find m[pic]2. | |

|2. Name each angle in four ways. | |8) m[pic]1 = 26 – 2x, |

|a) | |m[pic]ABC = 73.40, m[pic]2 = 5x + 12, find m[pic]1 and|

| | |m[pic]2. |

|G |5) m[pic]1 = 4x + 6, m[pic]ABC = 520, m[pic]2 = 3x| |

|1 |+ 4, find m[pic]1 and m[pic]2. | |

|T R | | |

|_____ _______ _______ _______ | | |

| | |16) m[pic]1 = 2x + 6, m[pic]3 = 4x + 3, m[pic]SQP =|

| | |78, find m[pic]1 and m[pic]2. |

|b) S |12) m[pic]2 = 4x - 10, | |

| |m[pic]1 = x + 19.5, m[pic]GHT = 8x - 10, find | |

|2 |m[pic]1, m[pic]2 and m[pic]GHT. | |

|W P | | |

|_____ _______ _______ _______ | |17) m[pic]TQR = 6x - 3, |

| | |m[pic]2 = 23 -2x, m[pic]1 = 5x + 4, find m[pic]1, |

| | |m[pic]2 and m[pic]TQR. |

|D |13) m[pic]2 = 4x + 20, | |

| |m[pic]1 = 32 – 3x, m[pic]GHT = 6x + 2, find m[pic]1, | |

| |m[pic]2 and m[pic]GHT. | |

|3 | | |

|S Y | |18) m[pic]SQP = 9x + 15, |

| | |m[pic]3 = 33 – 2x, m[pic]1 = 9x – 5, find m[pic]1, |

|______ ______ _______ _______ | |m[pic]2 and m[pic]SQP. |

|Use the following figure to solve problems 9 – 13. |Use the following figure to solve | |

|Show your work. |problems 14 – 19 . Show your | |

|G H |work. | |

|1 2 | | |

| |R |19) m[pic]PQR = 19x + 15, |

|B |S |m[pic]3 = 33 – 2x, m[pic]1 = 9x – 5, m[pic]2 =|

|T |2 |7x – 3, find m[pic]1, m[pic]2, m[pic]3 and |

| |1 |m[pic]PQR. |

| |3 Q | |

|9) m[pic]1 = 3x + 2, |T | |

|m[pic]2 = 2x + 4, m[pic]GHT = 4x + 10, find m[pic]1, |P | |

|m[pic]2 and m[pic]GHT. | | |

| |14) m[pic]2 = 280, m[pic]1 = 450, find m[pic]TQR | |

| |. | |

| | | |

| | | |

|10) m[pic]2 = x + 28, |15) m[pic]3 = 420, m[pic]1 = 370, find m[pic]PQS.| |

|m[pic]1 = 2x + 2, m[pic]GHT = 5x - 4, find m[pic]1, | | |

|m[pic]2 and m[pic]GHT. | | |

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|11) m[pic]1 = 3x - 5, | | |

|m[pic]2 = 5x + 2, m[pic]GHT = 7x + 3, find m[pic]1, | | |

|m[pic]2 and m[pic]GHT. | | |

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GEOMETRY

ANGLE RELATIONSHIPS

Name: __________________________________ Date: ______________

Lesson 1.17: Angle Relationships

Objective: To identify and use adjacent, vertical, complementary, supplementary, and linear pairs of angles,

and perpendicular lines.

|1) Adjacent Angles --- two angles in the same plane that have a common vertex|3) Complementary Angles --- are two angles whose sum is 900. |

|and a common side, but no common interior points. | |

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|A |350 550 |

| |A B |

|C | |

|B 1 |m[pic] + m[pic] = 350 + 550 = 900 |

|2 D | |

|3 |[pic] and [pic] are complementary angles. |

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|E | |

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|[pic]1 and [pic]2 are adjacent angles. | |

|[pic]2 and [pic]3 are adjacent angles. |1 |

|[pic]1 and [pic]3 are NOT adjacent angles. |2 |

|[pic]1 and [pic]ABD are NOT adjacent angles. | |

| |[pic] and [pic] are complementary angles. |

|2) Vertical angles --- are two nonadjacent angles formed by two intersecting | |

|lines. |Each angle is the complement of the other. |

|X | |

|W |4) Supplementary angles are two angles whose sum is 1800. |

|1 | |

|2 3 | |

|Y | |

|4 |1230 570 |

|Z V |C D |

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|[pic]1 (or [pic]WYX) and [pic]4 (or [pic]ZYV) are vertical angles. |m[pic]C + m[pic]D = 1230 + 570 = 1800 |

|[pic]2 and [pic]3 are vertical angles. | |

|[pic]1 and [pic]2 are NOT vertical angles. |[pic]C and [pic]D are supplementary angles. |

|[pic]2 and [pic]4 are NOT vertical angles. | |

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|Vertical angles are congruent, that is, they have equal measure. | |

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|[pic]1 [pic] [pic]4, which means m[pic]1 = m[pic]4 ; |1 2 |

|[pic]2 [pic] [pic]3, which means m[pic]2 = m[pic]3 ; |straight angle |

| | |

| |[pic]1 and [pic]2 are supplementary angles. |

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| |Each angle is the supplement of the other. |

|5) Linear Pair of angles--- A pair of adjacent angles formed by intersecting | |

|lines. Adjacent angles whose noncommon sides are opposite rays. |Since [pic]GIJ and [pic]JIH are a linear pair, |

| |m[pic]GIJ + m[pic]JIH = 180 |

|X |124 + m[pic]JIH = 180 |

|W |m[pic]JIH = 56 |

|1 |------------------------------------------------------------------------ |

|2 3 |Example 2: Given [pic]. |

|Y | |

|4 |G |

|Z V |D |

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| |2 |

|[pic]1 and [pic]2 are a linear pair. |1 |

|[pic]2 and [pic]4 are a linear pair. |W P |

|[pic]3 and [pic]4 are a linear pair. | |

|[pic]1 and [pic]3 are a linear pair. |m[pic]1 = 4x + 10, |

| |m[pic]2 = 2x + 20, find m[pic]1 and m[pic]2. |

|Linear pairs of angles are supplementary, that is, the sum of the measures of| |

|the angles in a linear pair is 1800. | |

|m[pic]1 + m[pic]2 = 1800 |----------------------------------------------------------------------- |

|m[pic]2 + m[pic]4 = 1800 |Example 3: The measure of an angle is 60 more than the measure of its complement. Find |

|m[pic]3 + m[pic]4 = 1800 |the measure of each angle. |

|m[pic]1 + m[pic]3 = 1800 | |

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|--------------------------------------------------------------- | |

|Example 1: In the figure below, GH and JK intersect at I. Find the value | |

|of x and m[pic]JIH. | |

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|J |------------------------------------------------------------------------ |

|G |Example 4: |

|16x - 20 | |

|I | |

|13x + 7 | |

|H | |

|K |2 1 |

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| |m[pic]1 = 4x + 10, |

|[pic]GIJ and [pic]KIH are vertical angles, so |m[pic]2 = 2x + 20, find m[pic]1 and m[pic]2. |

| | |

|m[pic]GIJ = m[pic]KIH |------------------------------------------------------------------------ |

|16x – 20 = 13x + 7 |Example 5: |

|Solving for x: | |

|x = 9 |3 |

| |1 2 |

|m[pic]GIJ = 16x – 20 |4 |

|= 16(9) – 20 | |

|= 124 | |

| |m[pic]1 = 3x – 30, |

| |m[pic]2 = x + 20, find m[pic]1, m[pic]2, m[pic]3 and m[pic]4. |

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GEOMETRY

Classwork 1.18: COMPLEMENTARY ANGLES AND SUPPLEMENTARY ANGLES (PART 2)

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To solve problems about complementary angles

2) To solve problems about supplementary angles

College Readiness Standards:

Exhibit some knowledge of basic angle properties (strand 20 – 23)

Use several angle properties to find an unknown angle measure (strand 24 – 27)

Solve multistep geometry problems that involve integrating concepts, planning, visualization,

and/or making connections with other content areas.

|Use the following figure to solve problems 1 – 4. Given|3) m[pic]1 = 15x + 25, |6) m[pic]2 = 65 – 3x, |

|[pic]. |m[pic]2 = 5x + 5, find m[pic]1 and m[pic]2. |m[pic]1 = 8x + 15, find m[pic]1 and m[pic]2. |

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|F W | | |

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|1) m[pic]1 = 2x - 5, | | |

|m[pic]2 = 4x - 1, find m[pic]1 and m[pic]2. |4) m[pic]1 = 25 - x, |7) m[pic]2 = x - 30, |

| |m[pic]2 = 6x + 5, find m[pic]1 and m[pic]2. |m[pic]1 = 4x + 50, find m[pic]1 and m[pic]2. |

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| |Use the following figure to solve problems 5 – 12. Show| |

|2) m[pic]1 = 3x - 15, |your work. | |

|m[pic]2 = 2x - 5, find m[pic]1 and m[pic]2. | |8) m[pic]2 = 65 + 3x, |

| | |m[pic]1 = 9x - 5, find m[pic]1 and m[pic]2. |

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| |1 2 | |

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| |5) m[pic]1 = 4x + 13, | |

| |m[pic]2 = 3x + 20, find m[pic]1 and m[pic]2. | |

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|Applications. Show your work. Draw the figure and |13) The measure of an angle is 25 more than 5 times the|17) The measure of an angle is 56 more than the |

|solve. |measure of its supplement. Find the measure of each |measure of its complement. Find the measure of each |

|9) The measure of an angle is 65 more than the measure|angle. |angle. |

|of its complement. Find the measure of each angle. | | |

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| |14) The measure of an angle is [pic] the measure of |18) The measure of an angle is 27 more than [pic] the |

|10) The measure of an angle is 35 less than 4 times |its supplement. Find the measure of each angle. |measure of its supplement. Find the measure of each |

|the measure of its complement. Find the measure of each| |angle. |

|angle. | | |

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| |15) The measure of an angle is 9 less than [pic] the |19) The measure of an angle is [pic] the measure of |

| |measure of its supplement. Find the measure of each |its complement. Find the measure of each angle. |

| |angle. | |

|11) The measure of an angle is 125 more than the | | |

|measure of its supplement. Find the measure of each | | |

|angle. | | |

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| |16) The measure of an angle is 45 more than [pic] the |20) The measure of an angle is [pic] the measure of |

|12) The measure of an angle is 30 less than 5 times |measure of its complement. Find the measure of each |its supplement. Find the measure of each angle. |

|the measure of its supplement. Find the measure of each|angle. | |

|angle. | | |

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GEOMETRY

Classwork 1.19: VERTICAL ANGLES

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To identify vertical angles

2) To apply the Vertical Angles Property

College Readiness Standards:

Exhibit some knowledge of basic angle properties (strand 20 – 23)

Use several angle properties to find an unknown angle measure (strand 24 – 27)

Solve multistep geometry problems that involve integrating concepts, planning, visualization,

and/or making connections with other content areas.

|Use the following figure to solve problems 1 – 3. |5) m[pic]3 = 3y – 20, |8) m[pic]1 = 7x – 4, |

| |m[pic]4 = y + 40, find m[pic]3 and m[pic]4. |m[pic]2 = 3x + 12, find m[pic]1, m[pic]2, m[pic]3 and|

| | |m[pic]4. |

|1340 3 | | |

|1 2 | | |

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|1) m[pic]1 = ? | | |

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|2) m[pic]2 = ? |6) m[pic]1 = 4x – 18, | |

| |m[pic]2 = 2x + 20, find m[pic]1, m[pic]2, m[pic]3, |9) m[pic]1 = 8x + 12, |

| |and m[pic]4. |m[pic]2 = 3x + 32, find m[pic]1 m[pic]2, m[pic]3 and |

| | |m[pic]4. |

|3) m[pic]3 = ? | | |

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|Use the following figure to solve problems 4 – 11. | | |

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|3 | | |

|1 2 | | |

|4 |7) m[pic]2 = 6x – 10, | |

| |m[pic]1 = x + 30, find m[pic]1 and m[pic]2. |10) m[pic]3 = 2y – 30, |

| | |m[pic]4 = y + 50, find m[pic]3, m[pic]4, m[pic]1, |

|4) m[pic]1 = 3x – 20, | |and m[pic]2. |

|m[pic]2 = x + 10, find m[pic]1 and m[pic]2. | | |

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|11) m[pic]3 = 7x + 18, | | |

|m[pic]4 = 5x + 48, find m[pic]3, m[pic]4, m[pic]2 and |12) | |

|m[pic]1 | | |

| |A |13) |

| |B | |

| |(4x+50) (2x+60) | |

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| |C |1 |

| | |4 3 |

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| |Find the value of x and m[pic]ABC. | |

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| | |m[pic]1 = 7x + 20, m[pic]2 = 5x + 10, |

| | |find m[pic]1, m[pic]2, m[pic]3, and m[pic]4. |

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GEOMETRY

Classwork 1.20: CLASSIFICATION OF ANGLES, ANGLE ADDITION PROPERTY, COMPLEMENTARY, SUPPLEMENTARY ANGLES AND VERTICAL ANGLES

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

Objectives: 1) To classify angles by their measures

2) To apply the Angle Addition Property

3) To solve problems about complementary angles

4) To solve problems about supplementary angles

5) To identify vertical angles

6) To apply the Vertical Angles Property

College Readiness Standards:

Exhibit some knowledge of basic angle properties (strand 20 – 23)

Use several angle properties to find an unknown angle measure (strand 24 – 27)

Solve multistep geometry problems that involve integrating concepts, planning, visualization,

and/or making connections with other content areas.

|Refer to the figure below for Problems 1 – 6. |5) If m[pic]PMO = 55 and m[pic]OMN = 65, what is |8) Find the measure of two complementary angles, |

| |the measure of [pic]PMN? |[pic]A and [pic]B, if m[pic]A = 7x + 4 and |

|I Q | |m[pic]B = 4x+9. |

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|P O | | |

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|J M N |6) Is [pic]P a valid name for one of the angles? | |

|R |Explain. | |

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|K L | | |

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|1) Name two angles that have N as a vertex. | | |

| |7) [pic]N is a complement of [pic]M, m[pic]N = 8x – 6|9) The measure of an angle is 44 more than the measure|

| |and [pic]M = 14x + 8. Find the values of x, [pic]N, |of its supplement. Find the measures of the angles. |

|2) Name five angles that have O as the vertex. |and [pic]M. | |

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|3) Name four angles that have MN | | |

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|4) Name three angles that appear to be acute. | | |

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|10) Suppose [pic]T and [pic]U are complementary. Find | | |

|m[pic]T and m[pic]U if m[pic]T = 16x – 9 and | |14) |

|m[pic]U = 4x + 3. |12) m[pic]2 = 228 – 3x, m[pic]3 = x. Find m[pic]2, | |

| |m[pic]3, m[pic]1, and m[pic]4. | |

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| | |7 8 9 |

| | |12 10 11 |

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| | |m[pic]7 = x + 20, m[pic]8 = x + 40, m[pic]9 = x + 30.|

| | |Find m[pic]7, m[pic]8, m[pic]9, m[pic]10, |

|Refer to the figure below for Problems 11 – 13. | |m[pic]11, and m[pic]12. |

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|3 4 | | |

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| |13) m[pic]3 = 2x - 4, | |

|11) m[pic]1 = 2x – 5, m[pic]2 = x – 4. Find m[pic]1,|m[pic]4 = 2x + 4. Find m[pic]3, m[pic]4, m[pic]1, | |

|m[pic]2, m[pic]3, and m[pic]4. |and m[pic]2. | |

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GEOMETRY

Classwork 1.21: SEGMENT ADDITION POSTULATE, DISTANCE ON A NUMBER LINE, MIDPOINTS, ANGLE ADDITION POSTULATE, COMPLEMENTARY ANGLES, SUPPLEMENTARY ANGLES, AND VERTICAL ANGLES

Name: ________________________________ Period: ____ Date: ___________ Score:

Remarks: Parent’s Signature: _______________________

Tel: ___________________________________

|For problems 1 – 2, use the Segment Addition Postulate |3) Towns Hinsdale (H), Lisle (L) and Naperville (N) |6) C is the midpoint of [pic]. Find AC, CB, and AB |

|to solve the indicated distances. Draw the |are located along a straight highway. Lisle is between |if AC = 8x - 6, CB = 2x + 9. |

|corresponding line segment with complete labels. Show |Hinsdale and Naperville, and the distance from Lisle to| |

|your work. |Naperville is 12 miles more than HL. If HN is 43 miles| |

|1) If S is between R and T, and RS = 3x - 5, ST = |less than six times HL, find HL, LN and HN. Represent | |

|2x + 3, RT = 73, find x, RS and ST. |each town as a point on a line segment, then use the | |

| |Segment Addition Postulate. | |

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| | |Use the following figure to solve problems 7 - 8. Show|

| | |your work. |

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| | |1 2 |

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| |In problems 4 - 5, draw the indicated line segment, and|B |

| |the corresponding coordinates. |T |

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| |4) If the midpoint of the segment ST is -5 and the | |

| |coordinate of T is -3, find the coordinate of S. |7) m[pic]1 = 2x - 5, m[pic]2 = 420, m[pic]GHT = 4x -|

|2) If S is between R and T, and RT = 9y + 14, RS =| |5, find m[pic]1 and m[pic] GHT. |

|4y + 8, ST = 8y + 5, , find y, RS, ST | | |

|and RT. | | |

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| |5) If the midpoint of the segment JK is 4 and the | |

| |coordinate of K is -5, find the coordinate of J. | |

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| | |8) m[pic]1 = 5x - 2, m[pic]2 = 2x + 4, m[pic]GHT = 9x|

| | |- 5, find m[pic]1, m[pic]2 and m[pic] GHT. |

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| |Use the following figure to solve problems 11 – 12. | |

| |Show your work. Given [pic]. | |

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|Use the following figure to solve problems 9 – 10 . |D | |

|Show your work. | |14) m[pic]1 = 35 – 3x, |

| |2 |m[pic]2 = 15x + 25, find m[pic]1 and m[pic]2. |

| |1 | |

|R |W P | |

|S | | |

| |11) m[pic]1 = 5x + 10, | |

|2 |m[pic]2 = 35 - x, find m[pic]1 and m[pic]2. | |

|1 Q | | |

|T 3 | | |

| | |15) The measure of an angle is 28.8 less than [pic] |

|P | |times the measure of its complement. Find the measure |

| | |of each angle. |

|9) m[pic]TQR = 7x - 2, | | |

|m[pic]SQR = 2x + 8, | | |

|m[pic]TQS = 3x + 14, find x, | | |

|m[pic] TQR, m[pic] SQR , and | | |

|m[pic] TQS . |12) m[pic]1 = 15x - 12, | |

| |m[pic]2 = 3x + 12, find m[pic]1 and m[pic]2. | |

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| | |16) The measure of an angle is 70 more than [pic]the |

| | |measure of its supplement. Find the measure of each |

| | |angle. |

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| |Use the following figure to solve problems 13 – 14. | |

| |Show your work. | |

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|10) m[pic]PQS = 7x + 10, | |Use the following figure to solve problem 17. |

|m[pic]PQT = 4x - 2, |1 2 | |

|m[pic]TQS = 2x + 14, find x, | | |

|m[pic] PQS, m[pic] PQT , and |13) m[pic]1 = 12y - 10, |3 |

|m[pic] TQS . |m[pic]2 = 4y + 22, find m[pic]1 and m[pic]2. |1 2 |

| | |4 |

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| | |17) m[pic]3 = 5y – 40, |

| | |m[pic]4 = 2y + 50, find m[pic]3, m[pic]4, m[pic]1, |

| | |and m[pic]2. |

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x

X

Y

Z

1830

2250

2700 3490

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