exterior angles of polygons activity what’s your …tristanbates.wikispaces.com/file/view/geometry...

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Unit 2 • Congruence, Triangles, and Quadrilaterals 101

My Notes

ACTIVITY

2.2Exterior Angles of PolygonsWhat’s Your Angle?SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Graphic Organizer, Interactive Word Wall, Think/Pair/Share, Use Manipulatives

Dee Zine is a landscape architect. Her business motto is Creating Unique Spaces for Your Unique Places. She has commissioned you to determine exterior angle measures for various polygon types since her customers have homes and structures in many di! erent shapes.

1. Your teacher has provided your group with a page containing polygons. One is a triangle, one a quadrilateral, one a pentagon and one a hexagon. In your groups, measure each exterior angle (one at each vertex of the polygon). Also determine the sum of the exterior angles for each polygon and record the sums in the table provided.

Polygon Number of Sides Calculations

Sum of the Measures of the Exterior Angles

Triangle 3Quadrilateral 4Pentagon 5HexagonHeptagonOctagonNonagonDecagonDodecagonn-gon

2. Describe any patterns you notice in the table values.

ACADEMIC VOCABULARY

An exterior angle of a polygon is formed by extending a side of the polygon. The vertex of the exterior angle is the vertex of the polygon. The sides of the angle are determined by a side of the polygon and the extension of the adjacent side at the vertex.

∠TAL is an exterior angle of polygon PENTA

N

LA

E

P

T

Because P, A, and L are collinear and T is not on ! " # PL , ∠TAP and ∠TAL form a linear pair.

101-104_SB_Geom_2-2_SE.indd 101101-104_SB_Geom_2-2_SE.indd 101 1/21/10 4:16:56 PM1/21/10 4:16:56 PM

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102 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

ACTIVITY 2.2continued

SUGGESTED LEARNING STRATEGIES: Close Reading, Summarize/Paraphrase/Retell, Activating Prior Knowledge, Notetaking, Think/Pair/Share, Quickwrite, Self/Peer Revision, Look for a Pattern

3. Compare your sums to the results of other groups in your class. Are the results always the same? Explain.

4. Complete the Exterior Angle ! eorem.

If a polygon has n sides, the sum of the measures of its exterior angles is .

TRY THESE

Write your answers in the My Notes space. Show your work.

a. Determine the sum of the exterior angles of a 15-sided polygon. Explain.

Determine the value of x in each " gure. Explain.

b. c.

d.

Items c and d illustrate how the measure of an exterior angle of a triangle is related to the interior angles.

5. In Items c and d, compare the measure of the exterior angle to the sum of the two non-adjacent interior angles (sometimes called the remote interior angles). What do you notice?

6. Complete this theorem.

! e measure of an exterior angle of a triangle is the sum of the two non-adjacent interior angles of the triangle.

You will prove the theorem you wrote in Item 5 in Exercise 6 of Check Your Understanding.

90°

105° 105°

90°

x60°

110°x

75°

40°x

What’s Your Angle?What’s Your Angle?Exterior Angles of Polygons


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Unit 2 • Congruence, Triangles, and Quadrilaterals 103

ACTIVITY 2.2continued

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING


1. What is the sum of the measures of the exterior angles of an octagon? Explain.

2. What is the measure of each exterior angle of an equiangular octagon? Explain.

3. If the sum of the measures of the interior angles of a polygon is 720°, how many sides does it have?

a. 5 b. 6

c. 8 d. Not enough information

4. If the sum of the measures of the exterior angles of a polygon is 360°, how many sides does it have?

a. 3 b. 4

c. 6 d. Not enough information

5. Supply the reasons in the proof of the Exterior Angle ! eorem.

Given: a triangle as shown

Prove: m∠1 + m∠4 + m∠6 = 360°

Statements Reasons1. ∠1 and ∠2 form a linear pair. ∠3 and ∠4 form a linear pair. ∠5 and ∠6 form a linear pair.

1. De" nition of linear pair

2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. ∠5 and ∠6 are supplementary.

2. Angles that form a linear pair are supplementary.

3. m∠1 + m∠2 = 180° m∠3 + m∠4 = 180° m∠5 + m∠6 = 180°

3.

4. m∠1 + m∠2 + m∠3 + m∠4 + m∠5 + m∠6 = 540°

4.

5. m∠1 + m∠4 + m∠6 + m∠2 + m∠3 + m∠5 = 540°

5.

6. m∠2 + m∠3 + m∠5 = 180° 6. 7. m∠1 + m∠4 + m∠6 = 360° 7.

1 2

3

56

4

Exterior Angles of Polygons What’s Your Angle?What’s Your Angle?


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104 SpringBoard® Mathematics with Meaning™ Geometry

Exterior Angles of Polygons ACTIVITY 2.2continued What’s Your Angle?What’s Your Angle?


CHECK YOUR UNDERSTANDING (continued)

6. Provide the missing Statements and Reasons in this proof of the Triangle Exterior Angle ! eorem:

Given: a triangle as shown

Prove: m∠4 = m∠1 + m∠2

Statements Reasons1. ∠3 and ∠4 form a linear pair. 1. 2. ∠3 and ∠4 are supplementary. 2. ! e angles that form a linear pair are supplementary.3. 3. De" nition of supplementary angles4. 4. ! e sum of the measures of the angles in a triangle is 180°.5. m∠3 + m∠4 =

_____ + ______ + _____5. Substitution property

6. 6.

7. Determine the value of x in this " gure. ! en calculate m∠A and m∠TNG.Show the process.

8. Without using a protractor, determinethese angle measures.

a. m∠JKC b. m∠CJK c. m∠IJK

d. m∠IHL e. m∠CAB f. m∠BMG

g. m∠MLB h. m∠HDK i. m∠DJI

j. m∠DIJ k. m∠HLM l. m∠JGL

m. m∠HEL n. m∠EBL o. m∠ACG

1

2

34

TB

C

A

6x°2x°

T

A N

G

64°

G

C

D

A

H

I

J

E

L

BM

K70°

130°

25°

70°

50°

10°55°


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