Chapter 1: Basics of Geometry



Unit 1: Essentials of Geometry

Lesson 1.1: Points, Lines, and Planes

Objective

Understand and use the basic undefined terms and defined terms in geometry such as point, line, plane, collinear, coplanar.

USING UNDEFINED TERMS AND DEFINTIONS

Point Line Plane

________________ ______________________ _____________________

Name Names Names

Example 1

a. Name three points that are collinear: ___________________________

b. Name four points that are coplanar: ____________________________

c. Name three points that are not collinear: ________________________

Line

Segment Ray Opposite Rays

Example 2

Give another name for [pic] _______________

Name all the rays with endpoint J. ______________________

Which of those rays are opposite rays? ___________________

INTERSECTIONS

The intersection of two lines: _____________________

The intersection of two planes: ____________________

Example 3

Sketch a plane and a line Sketch a plane and a line that Sketch a plane and a line

that is in the plane. does not intersect the plane. that intersects at one point.

Example 4

Name the intersection of [pic]and line k. ________________

Name the intersection of plane A and plane B. ________________

Name the intersection of plane A and line k. _________________

Example 5

You are given an equation of a line and a point. Use substitution to determine whether the point is on the line.

y = x – 4; A (5, 1) _______________________

y = x + 1; A (1, 0) _______________________

Example 6

Graph the inequality on a number line. Tell whether the graph is a segment, a ray or rays, a point, or a line.

x < 3 _______________________

-7 < x < 4 _______________________

Unit 1: Essentials of Geometry

Lesson 1.2: Use Segments and Congruence

Objective

Find the distance between two points using the Ruler Postulate and Segment Addition Postulate

Use the Distance Formula to find the distance between two points in the coordinate plane.

Understand and apply the definition of congruence (congruent segments).

USING SEGMENT POSTULATES

Ruler Postulate

Example 1

Measure the length of the segment to the nearest millimeter.

[pic]

When three points are collinear, one point is ____________________ the other two.

For instance, point _____________ is between points A and C.

Segment Addition Postulate

Example 2

Use the map to find the distance between Lubbock and St. Louis.

__________________.

Example 3

Use the diagram to find [pic]. GH = ____________

CONGRUENT SEGMENTS

Example 4:

Plot J(-3, 4), K(2, 4), L(1, 3), and M(1, -2) in a

coordinate plane. Then determine whether

[pic]and [pic]are congruent.

Example 5:

Use the number line to find the indicated distance.

VW = ___________ XY = ___________ XZ = ___________ VX= ___________

Unit 1: Essentials of Geometry

Lesson 1.3 Apply Pythagorean Theorem

Lesson 7.1 from Geometry Textbook

Objectives

Use the Pythagorean Theorem to find the length of the unknown side of a right triangle and the area of the triangle.

Determine which numbers form a Pythagorean Triple

RIGHT TRIANGLES Pythagorean Theorem

_____________________________

Example 1

Find the length of the sides of each of the given triangles.

x = ___________ x = ___________

Example 2

Find the area of the isosceles triangle with side A = __________________

lengths 10 meters, 13 meters, and 13 meters.

HINT: A = ½(base of triangle)(height of triangle).

Example 3

A = __________________

PYTHAGOREAN TRIPLES

__________________________________________

Example 3

Complete the table of Pythagorean Triples and their multiples.

| |3, 4, 5 |5, 12, 13 |8, 15, 17 |7, 24, 25 |

|Multiple of 2 |6, 8, 10 | | |14, 28, 50 |

| | |15, 36, 39 | | |

| | | |40, 75, 85 | |

|Multiple of x | | | | |

Example 4

Ladder A 20 foot ladder is resting against the side of a house. The base of the ladder is 4 feet away from the house. Approximately how high above the ground does the ladder touch the house?

Example 5

Real Estate An investor owns a triangular plot of land as shown in the diagram.

Find the perimeter of the plot of land.

One acre of land is equivalent to 43,560 square feet.

How many acres are in this plot of land? Round to

two decimal places.

The investor is planning on selling the land. The

market rate in this area is $5000 per acre. How

much should the investor ask for the land?

Unit 1: Essentials of Geometry

[pic]Lesson 1.4 Using the Midpoint and Distance Formulas

Lesson 1.3 from Geometry Textbook

Objectives

Use midpoint and distance formula to find the distance between two points and their midpoint.

MIDPOINTS AND SEGMENT BISECTORS

Example 1

The figure shows a gate with diagonal braces.

[pic]bisects[pic]at Q. If PQ is 22.6, find PN.

PN = ___________________

Example 2

Point M is the midpoint of [pic]. Find x and the length of [pic].

VM = _________________________

COORDINATE PLANE

The Midpoint Formula

________________________________

Example 3

Find the coordinates of the midpoint M. Find the coordinates of the endpoint K.

M = ___________________ K = ____________________

Example 4

Find the length of the segment and then find the coordinate of the midpoint of the segment.

Length = ____________ Midpoint _____________

DISTANCE FORMULA

The Distance Formula

If A(x1, y1) and B(x2, y2) are points in the coordinate plane,

then the distance between A and B is

___________________________________________

Example 5

Find the length of [pic]

RS = ______________

Example 6

The coordinates of two segments are given. Find each segment length. Tell whether the segments are congruent.

AB = ________________ CD = ___________________

[pic]___________________

Unit 1: Essentials of Geometry

Lesson 1.5: Measure and Classify Angles

Lesson 1.4 from Geometry Textbook

Objective

Find the measure of an angle using different postulates such as the Protractor Postulate and Angle Addition Postulate.

Classify angles as acute, right, obtuse, and straight.

Use a protractor to construct and find the measure of an angle.

USING ANGLE POSTULATES

The angle that has sides [pic]and [pic]is denoted _______________________.

The point A is the ____________ of the angle.

Example 1

Name the angles in the figure.

____________________________________________

Protractor Postulate

Words: ____________________________________

Symbols: __________________________________

CLASSIFYING ANGLES

________________ ________________ _______________ ________________

Example 2

Use the diagram to find the measure of the indicated angle. Then classify the angle.

a) [pic] b) [pic]

_______________ ________________

_______________ ________________

c) [pic] d) [pic]

________________ ________________

_______________ ________________

Angle Addition Postulate

Example 3

Given that [pic], find [pic]and [pic].

[pic]= _______________ [pic]= ________________

CLASSIFYING ANGLES

Angles that have the same measure are called _____________________.

MEASURES ARE EQUAL ANGLES ARE CONGRUENT

______________________ ______________________

Example 4

The figure shows angles formed by the legs of an ironing board. Identify the

congruent angles. If [pic], what is [pic]

______________________________ ________________

Angle Bisectors

[pic]is the ______________________ of [pic].

[pic]

Example 5

In the diagram at the right, [pic]bisects [pic]and [pic] Find [pic]

[pic]

Unit 1: Basics of Geometry

Lesson 1.6 Describe Angle Pair Relationships

Lesson 1.5 from Geometry Textbook

Objective

Classify pairs of angles as vertical, supplementary, complementary, and a linear pair.

Apply understanding of angle pair relationship to find the measures of given angles.

COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Example 1

In the figure, name a pair of complementary angles

and supplementary angles, and a pair of adjacent angles.

Example 2

Given that ................
................

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