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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