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(For help, go to Lesson 1-6 and 3-4.) GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems Find the measure of each angle of quadrilateral ABCD. Check Skills Youll Need

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Solutions GEOMETRY LESSON m DAB = = 77; m B = 65; m BCD = = 131; m D = 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 = = 120, m B = 60, m BCD = = 120, and m D = By the Triangle Angle-Sum Theorem m A = 180, so m A = 70. m ABC = = 85; by the Triangle Angle-Sum Theorem, m C = 180, so m C = 125; m ADC = = The Polygon Angle-Sum Theorems 3-5

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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 = Find m 5 if m 2 = 76 and m 3 = Find x if m 1 = 4x, m 3 = 2x + 28, and m 4 = Find x if m 2 = 10x, m 3 = 5x + 40, and m 4 = 3x – Find m 3 if m 1 = 125 and m 5 = 160. GEOMETRY LESSON 3-4 scalene right triangle Parallel Lines and the Triangle Angle-Sum Theorem 3-4

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 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.

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 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.

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 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: A, B, C, D, E angles:

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 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.

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 You can name a polygon by the number of its sides. The table shows the names of some common polygons.

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5

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GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 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.

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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 3-5 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 Theorems Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE. 3-5 Quick Check Naming Polygons

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GEOMETRY 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 Theorems 3-5 Quick Check Classifying Polygons

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A decagon has 10 sides, so n = 10. Sum = (n – 2)(180) Polygon Angle-Sum Theorem = (10 – 2)(180) Substitute 10 for n. = Simplify. = 1440 Find the sum of the measures of the angles of a decagon. GEOMETRY LESSON 3-5 The Polygon Angle-Sum Theorems 3-5 Quick Check Finding a Polygon Angle Sum

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m X + m Y + m Z + m W = (4 – 2)(180) Polygon Angle-Sum Theorem m X + m Y = 360 Substitute. m X + m Y = 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 3-5 The figure has 4 sides, so n = 4. Find m X in quadrilateral XYZW. The Polygon Angle-Sum Theorems 3-5 Quick Check Using the Polygon Angle-Sum Theorem

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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 polygons angle because they are adjacent angles that form a straight angle. GEOMETRY 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 Theorems 3-5 Quick Check

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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 because two sides intersect at a point other than endpoints GEOMETRY LESSON The 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. 3-5

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