Geometry Geometry Honors T.E.A.M.S. Geometry Honors ...

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Geometry

Geometry Honors

T.E.A.M.S. Geometry Honors

Summer Assignment

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Dear Parents and Students:

All students entering Geometry or Geometry Honors are required to complete this assignment.

This assignment is a review of essential topics to strengthen math skills for the upcoming

school year.

If you need assistance with any of the topics included in this assignment, we strongly

recommend that you to use the following resource: http://www.khanacademy.org/.

If you would like additional practice with any topic in this assignment visit: http://www.math-

drills.com.

Below are the POLICIES of the summer assignment:

The summer assignment is due the first day of class. On the first day of class,

teachers will collect the summer assignment. Any student who does not have the

assignment will be given one by the teacher. Late projects will lose 10 points each

day.

Summer assignments will be graded as a quiz. This quiz grade will consist of 20%

completion and 80% accuracy. Completion is defined as having all work shown in the

space provided to receive full credit, and a parent/guardian signature.

Any student who registers as a new attendee of Teaneck High School after August

15th will have one extra week to complete the summer assignment.

Summer assignments are available on the district website and available in the THS

guidance office.

HAVE A GREAT SUMMER!

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An Introduction to the Basics of Geometry

Directions: Read through the definitions and examples given in each section, then complete the practice questions,

found on pages 20 to 26. Those pages will be collected by your Geometry Teacher on the first day of school.

Section 1: Points, Lines and Planes

Undefined term: words that do not have formal definitions, but there is an agreement about what they mean. In

Geometry, the words point, line and plane are undefined terms.

Undefined Term Meaning Example/Picture and symbols

Point A point has no dimension but has location. A dot is used to represent a point.

Line

A line has one dimension. It is represented by a line with two arrowheads, showing that it extends in two directions without end. Through any two points there is exactly one line. You can use any two points on a line to name it, or it can be named by a lowercase letter written by the line.

Plane

A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. You can use three points that are not on the same line to name a plane, or you can use a capital letter (without a point next to it) to name a plane.

Collinear points: points that lie on the same line. Coplanar points: points that lie in the same plane.

Example 1: Naming Points, Lines and Planes

a. Give two other names for 𝑃𝑄 ⃡ and plane R.

Answer: Other names for 𝑃𝑄 ⃡ are 𝑄𝑃 ⃡ and line n. Other names for plane R are

plane SVT and plane PTV.

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

Answer: Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so

they are coplanar.

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Defined Terms: Segment and Ray

The definitions below use line AB (written as 𝐴𝐵 ⃡ ) and points A and B.

Defined Term Definition Example/Picture and Symbols

Segment

The line segment AB, or segment AB (written as 𝐴𝐵̅̅ ̅̅ ) consists

of the endpoints A and B and all points on 𝐴𝐵 ⃡ that are between A and B. The endpoints are like stop and start points. Unlike lines, segments do not continue on forever in both directions and they can be measured. Note that 𝐴𝐵̅̅ ̅̅ can also be called 𝐵𝐴̅̅ ̅̅ .

𝐴𝐵̅̅ ̅̅ (read “segment AB”)

Ray

The ray AB (written as 𝐴𝐵 ) consists of the endpoint A and all

points on 𝐴𝐵 ⃡ that lie on the same side of A as B. In other words, rays have a starting point (called an endpoint) and continue in the direction of the other point.

Note that 𝐴𝐵 and 𝐵𝐴 are two different rays because they are going in different directions.

Top: 𝐴𝐵 (read “ray AB”)

Bottom: 𝐵𝐴 (read “ray BA”)

Opposite Rays

If point C lies on 𝐴𝐵 ⃡ between A and B, then 𝐶𝐴 and 𝐶𝐵 are opposite rays. They have the same point but go in opposite directions to form a line.

𝐶𝐴 and 𝐶𝐵 are opposite rays.

Intersection

Two or more geometric figures intersect when they have one or more points in common. The intersection of the figures is the set of all points they have in common.

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Example 2: Naming Segments, Rays and Opposite Rays

a. Give another name for 𝐺𝐻̅̅ ̅̅ .

Answer: Another name for 𝐺𝐻̅̅ ̅̅ is 𝐻𝐺̅̅ ̅̅ .

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

Answer: The rays with endpoint 𝐽 are 𝐽𝐸 , 𝐽𝐺 , 𝐽𝐹 , and 𝐽𝐻 . The pairs of opposite rays with endpoint 𝐽 are 𝐽𝐸 and

𝐽𝐹 , and 𝐽𝐺 and 𝐽𝐻 .

Practice Questions for Section 1 can be found on Page 20.

Section 2: Measuring Segments

In Geometry, a rule that is accepted without proof is called a postulate or an axiom. A rule that can be proved is called a

theorem.

The Ruler Postulate: The points on a line can be matched one to one with the real numbers.

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

The distance

between points A and B, written as AB (notice there is no symbol above the 2 letters), is

the absolute value of the difference of the coordinates of A and B.

Congruent Segments: Line segments that have the same length are called congruent

segments. You can say “the length of 𝐴𝐵̅̅ ̅̅ is equal to the length of 𝐶𝐷̅̅ ̅̅ ,” or you can say “𝐴𝐵̅̅ ̅̅

is congruent to 𝐶𝐷̅̅ ̅̅ .” The symbol ≅ means “is congruent to.”

In the diagram above, of 𝐴𝐵̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ have tick marks on them, indicating 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ . When there is more than one pair

of congruent segments, use multiple tick marks.

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When three points are collinear, you can say that one

point is between the other two.

Segment Addition Postulate

If B is between A and C, then AB + BC=AC.

If AB+BC=AC, then B is between A and C.

Example 1: Comparing Segments for Congruence

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

congruent.

Answer:

Plot the points, as shown.

To find the length of a horizontal segment, find the absolute value of the

difference of the x-coordinates of the endpoints.

𝐽𝐾 = |−3 − 2| = 5, Ruler Postulate

To find the length of a vertical segment, find the absolute value of the

difference of the y-coordinates of the endpoints.

𝐿𝑀 = |3 − (−2)| = 5, Ruler Postulate

𝐽𝐾 = 𝐿𝑀. So, 𝐽𝐾̅̅ ̅ ≅ 𝐿𝑀̅̅ ̅̅ .

Example 2: Using the Segment Addition Postulate

a. Find 𝐷𝐹.

Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 𝐷𝐹.

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b. Find 𝐺𝐻.

Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 𝐺𝐻.

Example 3: Using the Segment Addition Postulate

The cities shown on the map lie

approximately in a straight line. Find the

distance from Tulsa, Oklahoma, to St.

Louis, Missouri.

Answer:

1. Understand the Problem. You are given the distance from Lubbock to St. Louis and the distance from

Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis.

2. Make a Plan. Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis.

3. Solve the Problem. Use the Segment Addition Postulate to write an equation. Then solve the equation

to find 𝑇𝑆.

So, the distance from Tulsa to St. Louis is 361 miles.

Practice Questions for Section 2 can be found on Page 21.

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Section 3: Using Midpoint and Distance Formulas

Midpoints and Segment Bisectors

The midpoint of a segment is the point that divides the segment into two congruent segments.

A segment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint. A midpoint or

a segment bisector bisects a segment. (“Bi” means two, “sect” means sections).

Example 1: Finding Segment Lengths

In the skateboard design, 𝑉𝑊̅̅ ̅̅ ̅ bisects 𝑋𝑌̅̅ ̅̅ at point T, and 𝑋𝑇 = 39.9 cm.

Find 𝑋𝑌.

Answer:

Point 𝑇 is the midpoint of 𝑋𝑌̅̅ ̅̅ , so 𝑋𝑇 = 𝑇𝑌 = 39.9 cm.

𝑋𝑌 = 𝑋𝑇 + 𝑇𝑌 Segment Addition Postulate 𝑋𝑌 = 39.9 + 39.9 Substitute

𝑋𝑌 = 79.8 Add

So, 𝑋𝑌 = 79.8 cm

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Example 2: Using Algebra with Segment Lengths

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

Answer:

1. Write and solve an equation. Use the fact that 𝑉𝑀̅̅̅̅̅ = 𝑀𝑊̅̅ ̅̅ ̅̅ .

𝑉𝑀 = 𝑀𝑊 Write the equation

4𝑥 − 1 = 3𝑥 + 3 Substitute

𝑥 − 1 = 3 Subtract 3𝑥 from both sides

𝑥 = 4 Add 1 to each side

2. Evaluate 𝑉𝑀 = 4𝑥 − 1 when 𝑥 = 4

𝑉𝑀 = 4(4) − 1 = 15

So the length of 𝑉𝑀̅̅̅̅̅ is 15 units.

Using the Midpoint Formula

The coordinates of the midpoint of a segment are the averages of the

x-coordinates and of the y-coordinates of the endpoints.

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

midpoint M of AB has coordinates

Example 3: Using the Midpoint Formula

a. The endpoints of 𝑅𝑆̅̅̅̅ are R(1, −3) and S(4, 2). Find the

coordinates of the midpoint M.

Answer:

Use the midpoint formula: 𝑀 (1+4

2,−3+2

2) = 𝑀 (

5

2,−1

2)

So the coordinates of 𝑀 are (5

2, −

1

2).

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

Answer:

Let (x, y) be the coordinates of endpoint 𝐾. Use the Midpoint Formula.

Step 1: Find x Step 2: Find y 1 + 𝑥

2= 2

4 + 𝑦

2= 1

1 + 𝑥 = 4 4 + 𝑦 = 2 𝑥 = 3 𝑦 = −2

The coordinates of endpoint 𝐾 are (3, -2).

Using the Distance Formula

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If A(𝑥1, 𝑦1), and B(𝑥2, 𝑦2) are points in a coordinate plane, then the distance between A and B is

𝐴𝐵 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

Example 4: Using the Distance Formula

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.

Answer:

You can model the situation using a coordinate plane with your apartment at the origin (0, 0). The coordinates

of the recycling center and the school are 𝑅(2, 3) and 𝑆(4, −1), respectively. Use the Distance Formula. Let

(𝑥1, 𝑦1) = (2, 3) and (𝑥2, 𝑦2) = (4, −1).

𝑅𝑆 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Distance Formula

𝑅𝑆 = √(4 − 2)2 + (−1 − 3)2 Substitute

𝑅𝑆 = √(2)2 + (−4)2 Subtract

𝑅𝑆 = √4 + 16 Evaluate Powers

𝑅𝑆 = √20 Add

𝑅𝑆 ≈ 4.5 Use Calculator

So, the distance between your school and the recycling center is about 4.5 miles.

Practice Questions for Section 3 can be found on Page 22.

Section 4: Perimeter and Area in the Coordinate Plane

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Polygons

In geometry, a figure that lies in a plane is called a plane figure. Recall that a

polygon is a closed plane figure formed by three or more line segments called

sides. Each side intersects exactly two sides, one at each vertex, so that no two

sides with a common vertex are collinear. You can name a polygon by listing

the vertices in consecutive order.

A polygon is convex when no line that contains a side of

the polygon contains a point in the interior of the

polygon. A polygon that is not convex is concave.

*One way to determine that a polygon is convex is to

imagine turning it on all sides and pouring waters over it

each time. If the water will roll off the polygon no

matter which side it is sitting on, then it is convex.

Polygon Types

Number of sides

Type of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Undecagon

12 Dodecagon

N n-gon (example – 23 sides 23-gon)

Example 1: Classifying Polygons

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

a.

Answer: The polygon has four sides. So, it is a

quadrilateral. The polygon is concave.

b.

Answer: The polygon has six sides. So, it is a

hexagon. The polygon is convex.

Finding Perimeter and Area in the Coordinate Plane

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You can use the formulas given below and the Distance Formula to find the perimeters and areas of polygons in the

coordinate plane.

Example 2: Finding Perimeter in the Coordinate Plane

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

Answer:

Step 1 Draw the triangle in a coordinate plane. Then find the length of each side.

Side 𝑨𝑩̅̅ ̅̅

𝐴𝐵 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Distance Formula

𝐴𝐵 = √(3 − (−2))2 + (−3 − 3)2 Substitute

𝐴𝐵 = √(5)2 + (−6)2 Subtract

𝐴𝐵 = √25 + 36 Evaluate powers

𝐴𝐵 = √61 Add

𝐴𝐵 ≈ 7.81 Use Calculator Side 𝑩𝑪̅̅ ̅̅

𝐵𝐶 = ∣ −2 − 3 ∣ = 5 Ruler Postulate Side 𝑨𝑪̅̅ ̅̅

𝐴𝐶 = ∣ 3 − (− 3) ∣ = 6 Ruler Postulate

Step 2 Find the sum of the side lengths.

𝐴𝐵 + 𝐵𝐶 + 𝐶𝐴 ≈ 7.81 + 5 + 6 = 18.81

So, the perimeter of △ 𝐴𝐵𝐶 is about 18.81 units.

Example 3: Finding Area in the Coordinate Plane

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

Answer:

Step 1 Draw the triangle in a coordinate plane by plotting the vertices and

connecting them.

Step 2 Find the lengths of the base and height.

Base

The base is 𝐹𝐸̅̅ ̅̅ , which is a horizontal segment so we can use the

Ruler Postulate:

𝐹𝐸 = |−4 − 4| = 8

Height

The height is the distance from point 𝐷 to 𝐹𝐸̅̅ ̅̅ . By counting grid lines, you can determine the height is 6

units.

Step 3 Substitute the values for the base and height into the formula for the area of a triangle.

𝐴 =1

2𝑏ℎ

𝐴 =1

28 ∙ 6

𝐴 = 24

So, the area of △ 𝐷𝐸𝐹 is 24 square units.

Practice Questions for Section 4 can be found on Page 23.

Section 5 – Angles and their Measures

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Naming Angles

An angle is a set of points consisting of two different rays that have the same endpoint, called the vertex. The rays are

the sides of the angle.

You can name an angle in several different ways.

Use its vertex, such as ∠A.

Use a point on each ray and the vertex, such as ∠BAC or ∠CAB.

(notice the vertex is always in the middle when naming this way).

Use a number, such as ∠1.

The region that contains all the points between the sides of the angle is the interior of

the angle. The region that contains all the points outside the angle is the exterior of the

angle.

Example 1: Naming Angles

A lighthouse keeper measures the angles formed by

the lighthouse at point M and three boats. Name

three angles shown in the diagram.

Answer:

∠JMK or ∠KMJ

∠KML or ∠LMK

∠JML or ∠LMJ

*Common Error: When a point is the vertex of more than one angle, you cannot use the vertex alone to name

the angle.

Measuring and Classifying Angles

A protractor helps you approximate the measure of an angle. The measure is usually given in degrees.

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Protractor Postulate

Consider 𝑂𝐵 ⃡ and a point 𝐴 on one side of 𝑂𝐵 ⃡ . The rays of the form 𝑂𝐴 can be

matched one to one with the real numbers from 0 to 180.

The measure of ∠AOB, which can be written as 𝑚∠𝐴𝑂𝐵, is equal to the absolute

value of the difference between the real numbers matched with 𝑂𝐴 and 𝑂𝐵 on a

protractor.

In the diagram, 𝑚∠𝐴𝑂𝐵 = 140°, because 𝑂𝐴 passes through the 40°/140° line and 𝑂𝐵 passes through the 180°/0° line.

Using the outer numbers |180 – 40| = 140° Using the inner numbers |0 – 140|=140°.

Types of Angles

Acute angle Right angle Obtuse angle Straight angle

(a small square drawn at the vertex of an angle symbolizes a right angle)

Measures greater than 0° and less than 90°

Measures 90° Measures greater than 90° and less than 180°

Measures 180°

Example 2: Measuring and Classifying Angles

Find the degree measure of each of the following

angles. Classify each angle as acute, right, or

obtuse.

a. ∠AOB

𝑂𝐴 lines up with 0° on the inner scale, and

𝑂𝐵 passes through 35° on the inner scale,

so 𝑚∠𝐴𝑂𝐵 = 35°. It is an acute angle.

b. ∠BOE

𝑂𝐵 lines up with 35° on the inner scale,

and 𝑂𝐸 passes through 145° on the inner

scale, so 𝑚∠𝐵𝑂𝐸 = |35 − 145|° = 110°.

It is an obtuse angle.

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Congruent angles are two angles with equal measures. If 𝑚∠𝐴𝐵𝐶 =

𝑚∠𝐷𝐸𝐹, then ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹. (Angles are congruent when their measures are

equal.

An angle bisector is a ray between two sides of an angle the creates two

congruent angles.

Example 3: Identifying Congruent Angles

Use the diagram to answer the questions.

a. Identify the angles congruent to ∠ADG.

Because ∠𝐵𝐸𝐻 and ∠𝐶𝐹𝐼 have matching arcs, ∠𝐴𝐷𝐺 ≅

∠𝐵𝐸𝐻 ≅ ∠𝐶𝐹𝐼.

b. Identify the angles congruent to ∠DAG.

Because ∠𝐷𝐴𝐺, ∠𝐴𝐺𝐷, ∠𝐸𝐵𝐻, ∠𝐸𝐻𝐵, ∠𝐹𝐶𝐼 and ∠𝐹𝐼𝐶 have

matching arcs, so

∠𝐷𝐴𝐺 ≅ ∠𝐴𝐺𝐷 ≅ ∠𝐸𝐵𝐻 ≅ ∠𝐸𝐻𝐵 ≅ ∠𝐹𝐶𝐼 ≅ ∠𝐹𝐼𝐶.

Angle Addition Postulate

Words

If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST.

Symbols If P is in the interior of ∠RST, then m∠RST = m∠RSP + m∠PST.

If 𝑆𝑉 bisects ∠𝑅𝑆𝑇 , then

∠𝑅𝑆𝑉 ≅ ∠𝑉𝑆𝑇

Reading: In diagrams, matching

arcs indicate congruent angles.

When there is more than one pair

of congruent angles, use multiple

arcs.

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Example 4: Using the Angle Addition Postulate to Find Angle Measures

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

Step 1 Write and solve an equation to find the value of x.

𝑚∠𝐿𝐾𝑁 = 𝑚∠𝐿𝐾𝑀 + 𝑚∠𝑀𝐾𝑁 Angle Addition Postulate 145° = (2𝑥 + 10)° + (4𝑥 − 3)° Substitute

145 = 6𝑥 + 7 Simplify (Combine Like Terms) 138 = 6𝑥 Subtract 23 = 𝑥 Divide

Step 2 Evaluate the given expressions when x = 23.

𝑚∠𝐿𝐾𝑀 = (2𝑥 + 10)° = (2 ⋅ 23 + 10)° = 56° 𝑚∠𝑀𝐾𝑁 = (4𝑥 − 3)° = (4 ⋅ 23 − 3)° = 89°

So, 𝑚∠𝐿𝐾𝑀 = 56° and 𝑚∠𝑀𝐾𝑁 = 89°.

Example 5: Using a Bisector to Find Angle Measures

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

Step 1 Draw a diagram.

Step 2 Because 𝑄𝑆 bisects ∠𝑃𝑄𝑅, 𝑚∠𝑃𝑄𝑆 = 𝑚∠𝑅𝑄𝑆. So, 𝑚∠𝑅𝑄𝑆 = 24°.

Use the Angle Addition Postulate to find 𝑚∠𝑃𝑄𝑅.

𝑚∠𝑃𝑄𝑅 = 𝑚∠𝑃𝑄𝑆 + 𝑚∠𝑅𝑄𝑆 Angle Addition Postulate 𝑚∠𝑃𝑄𝑅 = 24° + 24° Substitute Angle

Measures 𝑚∠𝑃𝑄𝑅 = 48° Add

So, 𝑚∠𝑃𝑄𝑅 = 48°.

Practice Questions for Section 5 can be found on Page 25.

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Section 6 – Describing Pairs of Angles

Complementary and Supplementary Angles

Complementary angles

Two positive angles whose measures have a sum of 90°. Each angle is the complement of the other.

Supplementary angles

Two positive angles whose measures have a sum of 180°. Each angle is the supplement of the other.

Adjacent Angles

Angles can be adjacent angles or nonadjacent angles. Adjacent angles are two angles that share a common vertex and

side, but have no common interior points.

Example 1: Identifying Pairs of Angles

In the figure, name a pair of complementary angles, a pair of

supplementary angles, and a pair of adjacent angles.

Answer:

Because 37° + 53° = 90°, ∠BAC and ∠RST are complementary

angles.

Because 127° + 53° = 180°, ∠CAD and ∠RST are supplementary angles.

Because ∠BAC and ∠CAD share a common vertex and side, they are adjacent angles.

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Example 2: Finding Angle Measures

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

Answer: If ∠1 is a complement of ∠2, then

𝑚∠1 + m∠2 = 90 Definition of Complementary Angles

62 + m∠2 = 90 Substitute

m∠2 = 28° Subtract

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

Answer: If ∠3 is a complement of ∠4, then

𝑚∠3 + m∠4 = 180 Definition of Supplementary Angles

𝑚∠3 + 47 = 180 Substitute

m∠3 = 133° Subtract

Linear Pairs and Vertical Angles

Two adjacent angles are a linear pair when their noncommon sides are opposite rays. The angles in a linear pair are supplementary angles.

∠1 and ∠2 are a linear pair.

Two angles are vertical angles when their sides form two pairs of opposite rays.

∠3 and ∠6 are vertical angles. ∠4 and ∠5 are vertical angles.

Example 4: Identify Angle Pairs

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

Answer:

To find vertical angles, look for angles formed by intersecting lines.

∠1 and ∠5 are vertical angles.

To find linear pairs, look for adjacent angles whose noncommon sides

are opposite rays.

∠1 and ∠4 are a linear pair. ∠4 and ∠5 are also a linear pair.

Practice Questions for Section 6 can be found on Page 26.

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Name: _________________________________ Teacher:____________________

Practice Questions

Section 1 Questions – Points. Line and Planes:

In exercises 1 – 4, use the diagram at right.

1. Give two other names for 𝐶𝐷 ⃡

2. Give another name for plane M.

3. Name three points that are collinear. Then name a fourth point that is not collinear with these three points.

4. Name a point that is not coplanar with points A, C, E.

In exercises 5 – 7, use the diagram at right.

5. What are two other names for 𝑃𝑄 ⃡ ?

6. What is another name for 𝑅𝑆̅̅̅̅ ?

7. Name all rays with endpoint T. Which of these rays are opposite rays?

In Exercises 9 and 10, sketch the figure described.

8. 𝐴𝐵̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅

9. line k in plane M.

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Section 2 Questions – Measuring Segments:

In Exercises 1–3, plot the points in the coordinate plane. Then determine whether 𝐴𝐵̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ are congruent.

1. A(-5, 5), B(-2, 5)

C(2, -4), D(-1, -4)

2. A(4, 0), B(4, 3)

C(-4, -4), D((-4, 1)

3. A(-1, 5), B(5, 5)

C(1, 3), D(1, -3)

In exercises 4 – 6, find VW.

4.

5.

6.

7. A bookstore and a movie theater are 6 kilometers apart along the same street. A florist is located between the

bookstore and the theater on the same street. The florist is 2.5 kilometers from the theater. How far is the

florist from the bookstore? (Hint: draw a line segment and label 3 points with each location)

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Section 3 Questions – Using Midpoint and Distance Formulas:

In Exercises 1–3, identify the segment bisector of 𝐴𝐵̅̅ ̅̅ . Then find AB.

1.

2.

3.

In Exercises 4-6, identify the segment bisector of 𝐸𝐹̅̅ ̅̅ . Then find EF.

4.

5.

6.

In Exercises 7–9, the endpoints of 𝑃𝑄̅̅ ̅̅ are given. Find the coordinates of the midpoint M and the length of 𝑃𝑄̅̅ ̅̅ .

7. P(-4, 3) and Q(0, 5)

8. P(-2, 7) and Q(10, -3)

9. P(3, -15) and Q(9, -3)

In Exercises 10–12, the midpoint M and one endpoint of JK are given. Find the coordinates of the other endpoint.

10. J(7, 2) and M(1, -2)

11. J(5, -2) and M(0, -1)

12. J(2, 16) and M(−9

2, 7)

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Section 4 Questions – Perimeter and Area in the Coordinate Plane:

In Exercises 1–4, classify the polygon by the number of sides. Tell whether it is convex or concave.

1.

2.

3.

4.

In Exercises 5–8, find the perimeter and area of the polygon with the given vertices.

5. X(2, 4), Y(0, -2), Z(2, -2)

6. P(1, 3), Q(1, 1), R(-4, 2)

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7. J(-4, 1), K(-4, -2), L(6, -2), M(6, 1)

8. D(5, -3), E(5, -6), F(2, -6), G(2, -3)

In Exercises 9–14, use the diagram.

9. Find the perimeter of ΔABD.

10. Find the perimeter of ΔBCD.

11. Find the perimeter of quadrilateral ABCD.

12. Find the area of ΔABD.

13. Find the area of ΔBCD.

14. Find the area of quadrilateral ABCD.

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Section 5 Questions – Measuring Angles:

In Exercises 1–3, name three different angles in the diagram.

1.

2.

3.

In Exercises 4–9, find the indicated angle measure(s).

4. Find 𝑚∠𝐽𝐾𝐿.

5. 𝑚∠𝑅𝑆𝑈 = 91°. Find 𝑚∠𝑅𝑆𝑇.

6. ∠𝑈𝑊𝑋 is a straight angle. Find

𝑚∠𝑈𝑊𝑉 𝑎𝑛𝑑 𝑚∠𝑋𝑊𝑉.

7. Find 𝑚∠𝐶𝐴𝐷 and 𝑚∠𝐵𝐴𝐷.

8. 𝐸𝐺 bisects ∠𝐷𝐸𝐹. Find 𝑚∠𝐷𝐸𝐺 and 𝑚∠𝐺𝐸𝐹.

9. 𝑄𝑅 bisects ∠𝑃𝑄𝑆. Find 𝑚∠𝑃𝑄𝑅 and 𝑚∠𝑃𝑄𝑆.

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Section 6 Questions – Describing Pairs of Angles:

In Exercises 1 and 2, use the figure.

1. Name the pair(s) of adjacent complementary angles.

2. Name the pair(s) of nonadjacent supplementary angles.

In Exercises 3 and 4, find the angle measure.

3. ∠A is a complement of ∠B and m∠=36 .° Find m∠B.

4. ∠C is a supplement of ∠D and m∠D=117 .° Find m∠C.

In Exercises 5 and 6, find the measure of each angle.

5.

6.

In Exercises 7–9, use the figure.

7. Identify the linear pair(s) that include ∠1.

8. Identify each vertical angle pair.

9. Are ∠6 and ∠7 a linear pair? Explain.