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1. 2.

3.

(For help, go to Lesson 1-6 and 3-4.)

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

Find the measure of each angle of quadrilateral ABCD.

Check Skills You’ll Need

Solutions

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

1. m DAB = 32 + 45 = 77; m B = 65; m BCD = 70 + 61 = 131; m D = 87

2. m DAC = m ACD = m D and m CAB = m B = m BCA; by the Triangle

Angle-Sum Theorem, the sum of the measures of the angles is 180,

so each angle measures , or 60. So, m DAB = 60 + 60 = 120,

m B = 60, m BCD = 60 + 60 = 120, and m D = 60.

3. By the Triangle Angle-Sum Theorem m A + 55 + 55 = 180, so m A = 70. m ABC = 55 + 30 = 85; by the Triangle Angle-Sum Theorem, m C + 30 + 25 = 180, so m C = 125; m ADC = 55 + 25 = 80

180 3

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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1. A triangle with a 90° angle has sides that are 3 cm, 4 cm, and 5 cm long. Classify the triangle by its sides and angles.

Use the diagram for Exercises 2–6.

2. Find m 3 if m 2 = 70 and m 4 = 42.

3. Find m 5 if m 2 = 76 and m 3 = 90.

4. Find x if m 1 = 4x, m 3 = 2x + 28, and m 4 = 32.

5. Find x if m 2 = 10x, m 3 = 5x + 40, and m 4 = 3x – 4.

6. Find m 3 if m 1 = 125 and m 5 = 160.

GEOMETRY LESSON 3-4GEOMETRY LESSON 3-4

scalene right triangle

68

166

30

8

105

Parallel Lines and the Triangle Angle-Sum TheoremParallel Lines and the Triangle Angle-Sum Theorem

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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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A polygon is a closed plane figure with at least three sides that are line segments. The sides intersect only at their endpoints, and no adjacent sides are collinear.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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To name a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction.

Two names for this polygon are ABCDE and CBAED.

vertices:

sides: , , , ,AB BC CD DE EAA, B, C, D, E

angles: , , , ,A B C D E

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex.

In this textbook, a polygon is convex unless stated otherwise.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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You can name a polygon by the number of its sides. The table shows the names of some common polygons.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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All the sides are congruent in an equilateral polygon.

All the angles are congruent in an equiangular polygon.

A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

A regular polygon is always convex.

Name the polygon. Then identify its vertices, sides,

and angles.

The polygon can be named clockwise or counterclockwise, starting at any vertex.

Possible names are ABCDE and EDCBA.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

Its vertices are A, B, C, D, and E.

Its angles are named by the vertices, A (or EAB or BAE), B (or ABC or CBA), C (or BCD or DCB), D (or CDE or EDC), and E (or DEA or AED).

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE.

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Quick Check

Naming Polygons

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

Starting with any side, count the number of sides clockwise around the figure. Because the polygon has 12 sides, it is a dodecagon.

Classify the polygon below by its sides. Identify it as convex or

concave.

Think of the polygon as a star. If you draw a diagonal connecting two points of the star that are next to each other,that diagonal lies outside the polygon, so the dodecagon is concave.

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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Quick Check

Classifying Polygons

A decagon has 10 sides, so n = 10.

Sum = (n – 2)(180) Polygon Angle-Sum Theorem

= (10 – 2)(180) Substitute 10 for n.

= 8 • 180 Simplify.

= 1440

Find the sum of the measures of the angles of a decagon.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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Quick Check

Finding a Polygon Angle Sum

m X + m Y + m Z + m W = (4 – 2)(180) Polygon Angle-Sum Theorem

m X + m Y + 90 + 100 = 360 Substitute.

m X + m Y + 190 = 360 Simplify.

m X + m Y = 170 Subtract 190 from each side.

2m X = 170 Simplify.

m X = 85 Divide each side by 2.

m X + m X = 170 Substitute m X for m Y.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

The figure has 4 sides, so n = 4.

Find m X in quadrilateral XYZW.

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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Quick CheckUsing the Polygon Angle-Sum Theorem

Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures.

Sample: The hexagon is regular, so all its angles are congruent.

An exterior angle is the supplement of a polygon’s angle because they are adjacent angles that form a straight angle.

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

A regular hexagon is inscribed in a rectangle. Explain how you

know that all the angles labeled 1 have equal measures.

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

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Quick Check

1. 2.

3. Find the sum of the measures of the angles in an octagon.

4. A pentagon has two right angles, a 100° angle and a 120° angle. What is the measure of its fifth angle?

5. Find m ABC.

6. XBC is an exterior angle at vertex B. Find m XBC.

quadrilateral ABCD;

AB, BC, CD, DA

not a polygon becausetwo sides intersect at a point other than endpoints

GEOMETRY LESSON 3-5GEOMETRY LESSON 3-5

1080

140

144

36

The Polygon Angle-Sum TheoremsThe Polygon Angle-Sum Theorems

For Exercises 1 and 2, if the figure is a polygon, name it by its vertices and identify its sides. If the figure is not a polygon, explain why not.

ABCDEFGHIJ is a regular decagon.

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