Unit 1

Geometry Essentials

Section 1

Points, Lines, and Planes

Exploration

Undefined Terms

Point:

Line:

Plane:

Collinear Points:

Coplanar Points:

Example 1.1

(a) Give two other names for and plane R.

(b) Name three points that are collinear. Name four points that are coplanar.

Example 1.2

Use the diagram from example 1.1. Give two other names for . Name a point that is not coplanar with points Q, S, and T.

Line segment/Segment:

Endpoints:

Ray:

Opposite Rays:

Example 1.3

(a) Give another name for .

(b) Name all rays with endpoint J. Which of these rays are opposite rays?

Example 1.4

(a) Give another name for .

(b) Are and the same ray? Are and the same ray? Explain.

Intersection:

With a Partner

1. A plane and a line that is in the plane.

2. A plane and a line that does not intersect the plane.

3. A plane and a line that intersects the plane at a single point.

4. Two planes that intersect in a line.

5. Two different lines that intersect a plane at the same point.

Example 1.5

Name the intersection of and line k.

Name the intersection of plane A and plane B.

Name the intersection of line k and plane A.

Example 1.6

The diagram shows a molecule of sulfur hexafluoride, the most potent greenhouse gas in the world. Name two different planes that contain line r.

Example 1.7

Use the diagram that shows a molecule of phosphorus pentachloride.

Name two different planes that contain line s.

Name three different planes that contain point K.

Name two different planes that contain .

Section 2

Measuring and Constructing Segments

Postulate/Axiom:

Theorem:

Construction:

Construct a copy:

Congruent segments: line segments that have the same ________________

Example 2.1

Plot J(-3,4), K(2,4), L(1,3), and M(1,-2) in a coordinate plane. Then determine whether and are congruent.

Example 2.2

Plot A(-2,4), B(3,4), C(0,2), and D(0,-2) in a coordinate plane. Then determine whether and are congruent.

Between:

____ is between ___ and ___

Is B between A and C?

Example 2.3

(a) Find DF.

(b) Find GH.

Example 2.4

Use the Segment Addition Postulate to find XZ.

In the diagram, WY=30. Can you use the Segment Addition Postulate to find the distance between points W and Z? Explain your reasoning.

Use the diagram to find KL.

Example 2.5

The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri.

Section 1.3

Using Midpoint and Distance Formulas

Exploration

Work with a partner. Use centimeter graph paper.

a. Graph , where the points A and B are as shown.

b. Explain how to bisect , that is, to divide into two congruent line segments. Then bisect and use the result to find the midpoint M of .

c. What are the coordinates of the midpoint M?

d. Compare the x-coordinates of A, B, and M. Compare the y-coordinates of A, B, and M. How are the coordinates of the midpoint M related to the coordinates of A and B?

Midpoint:

Segment Bisector:

Example 3.1

In the skateboard design, bisects at point T, and XT=39.9 cm. Find XY.

Example 3.2

Identify the segment bisector of . Then find PQ.

Questions (Answers)

Example 3.3

Point M is the midpoint of . Find the length of .

Example 3.4

Identify the segment bisector of . Then find MQ.

Identify the segment bisector of . Then find RS.

Construction

Midpoint Formula

Example 3.5

(a) The endpoints of are R(1,-3) and S(4,2). Find the coordinates of the midpoint M.

(b) The midpoint of is M(2,1). One endpoint is J(1,4). Find the coordinates of endpoint K.

With a Partner

(a) The endpoints of are A(1,2) and B(7,8). Find the coordinates of the midpoint M.

(b) The endpoints of are C(-4,3) and D(-6,5). Find the coordinates of the midpoint M.

(c) The midpoint of is M(2,4). One endpoint is T(1,1). Find the coordinates of endpoint U.

(d) The midpoint of is M(-1.-2). One endpoint is W(4,4). Find the coordinates of endpoint V.

The Distance Formula

Example 3.6

Your school is 4 miles east and 1 mile south of your apartment. A recycling center, where your class is going on a field trip, is 2 miles east and 3 miles north of your apartment. Estimate the distance between the recycling center and your school.

Example 3.7

What is the length of RS with endpoints R(2, 3) and S(4, -1)?

On Your Own

Find the distance between P(1, 2) and Q(5, 4). Round to the nearest tenth.

Challenge Problem

The distance between A(4, a) and B(1, 6) is 5 units. Find the value of a.

Section 4

Perimeter and Area in the Coordinate Plane

Exploration

Work with a partner.

a. On a piece of centimeter graph paper, draw quadrilateral ABCD in a coordinate plane. Label the points A(1, 4), B(−3, 1), C(0, −3), and D(4, 0).

b. Find the perimeter of quadrilateral ABCD.

c. Are adjacent sides of quadrilateral ABCD perpendicular to each other? How can you tell?

d. What is the definition of a square? Is quadrilateral ABCD a square? Justify your answer. Find the area of quadrilateral ABCD.

Part 2

a. Partition quadrilateral ABCD into four right triangles and one square, as shown. Find the coordinates of the vertices for the five smaller polygons.

b. Find the areas of the five smaller polygons.

Area of Triangle BPA:

Area of Triangle AQD:

Area of Triangle DRC:

Area of Triangle CSB:

Area of Square PQRS:

c. Is the sum of the areas of the five smaller polygons equal to the area of quadrilateral ABCD? Justify your answer.

Polygons

Convex:

Concave:

Number of Sides

Type of Polygon

3

4

5

6

7

8

9

10

12

n

Example 4.1

Classify each polygon by the number of sides. Tell whether it is convex or concave.

Example 4.2

Example 4.3

Find the perimeter of △ABC with vertices A(−2, 3), B(3, −3), and C(−2, −3).

Example 4.4

Find the perimeter of the polygon with the given vertices.

3. D(−3, 2), E(4, 2), F(4, −3)

4. G(−3, 2), H(2, 2), J(−1, −3)

5. K(−1, 1), L(4, 1), M(2, −2), N(−3, −2)

6. Q(−4, −1), R(1, 4), S(4, 1), T(−1, −4)

Example 4.5

Find the area of △DEF with vertices D(1, 3), E(4, −3), and F(−4, −3).

Example 4.6

Find the area of the polygon with the given vertices.

7. G(2, 2), H(3, −1), J(−2, −1)

8. N(−1, 1), P(2, 1), Q(2, −2), R(−1, −2)

9. F(−2, 3), G(1, 3), H(1, −1), J(−2, −1)

10. K(−3, 3), L(3, 3), M(3, −1), N(−3, −1)

Example 4.7

You are building a shed in your backyard. The diagram shows the four vertices of the shed. Each unit in the coordinate plane represents 1 foot. Find the area of the floor of the shed.

Section 1.5

Measuring and Constructing Angles

Exploration

Work with a partner. Find the degree measure of each of the following angles. Classify each angle as acute, right, or obtuse.

a. ∠AOB

b. ∠AOC

c. ∠BOC

d. ∠BOE

e. ∠COE

f. ∠COD

g. ∠BOD

h. ∠AOE

Angle: Consists of two _______________ with the same endpoint

Sides of the angle: the ________________

Vertex: the ________________

Interior:

Exterior:

Example 5.1

A lighthouse keeper measures the angles formed by the lighthouse at point M and three boats. Name three angles shown in the diagram.

Example 5.2

Write three names for each angle.

Acute Angle:

Right Angle:

Obtuse Angle:

Straight Angle:

Example 5.3

Find the measure of each angle.

Then classify each angle.

a. ∠GHK

b. ∠JHL

c. ∠LHK

Construction

Congruent Angles: two angles with the same ________________ ________________

Example 5.4

a. Identify the congruent angles labeled in the quilt design.

b. m∠ADC = 140°. What is m∠EFG?

Example 5.5

Given that m∠LKN = 145°, find m∠LKM and m∠MKN.

Example 5.6

Find the indicated angle measures.

8. Given that ∠KLM is a straight angle, find m∠KLN and m∠NLM.

9. Given that ∠EFG is a right angle, find m∠EFH and m∠HFG.

Angle Bisector: a ____________ that divides an angle into two angles that are _____________

Example 5.7

bisects ∠PQR, and m∠PQS = 24°. Find m∠PQR.

Construction

Example 5.8

10. Angle MNP is a straight angle, and bisects ∠MNP. Draw ∠MNP and . Use arcs to mark the congruent angles in your diagram. Find the angle measures of these congruent angles.

Section 1.6

Describing Pairs of Angles

Exploration

Work with a partner. The five-pointed star has a regular pentagon at its center.

a. What do you notice about the following angle pairs?

x° and y°

y° and z°

x° and z°

b. Find the values of the indicated variables. Do not use a protractor to measure the angles.

x=

y=

z=

w=

v =

Explain how you obtained each answer.

Complementary Angles:

Supplementary Angles:

Adjacent Angles:

Example 6.1

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Example 6.2

a. ∠1 is a complement of ∠2, and m∠1 = 62°. Find m∠2.

b. ∠3 is a supplement of ∠4, and m∠4 = 47°. Find m∠3.

Example 6.3

In Exercises 1 and 2, use the figure.

1. Name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

2. Are ∠KGH and ∠LKG adjacent angles? Are ∠FGK and ∠FGH adjacent angles? Explain.

3. ∠1 is a complement of ∠2, and m∠2 = 5°. Find m∠1.

4. ∠3 is a supplement of ∠4, and m∠3 = 148°. Find m∠4.

Example 6.4

When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground.

Linear Pair:

Vertical Angles:

Example 6.5

Identify all the linear pairs and all the vertical angles in the figure.

Example 6.6

Two angles form a linear pair. The measure of one angle is five times the measure of the other angle. Find the measure of each angle.

Example 6.7

Do any of the numbered angles in the diagram form a linear pair? Which angles are vertical angles? Explain.

The measure of an angle is twice the measure of its complement. Find the measure of each angle.

Tell whether the statements are always, sometimes, or never true.

1. An obtuse angle has a complement.

2. A straight angle has a complement.

3. An angle has a supplement.

4. The complement of an acute angle is an acute angle.

5. The supplement of an acute angle is an obtuse angle.

Example 6.8

Example 6.9