The Polygon Angle-Sum Theorems

The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 (For help, go to Lesson 1-6 and 3-4.) Find the measure of each angle of quadrilateral ABCD. 1.2. 3. Check Skills You’ll Need 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 Theorems GEOMETRY LESSON 3-5 Solutions 3-5

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. Parallel Lines and the Triangle Angle-Sum Theorem GEOMETRY LESSON 3-4 scalene right triangle 68 166 30 8 105 3-4

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

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

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

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

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

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

The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. A regular polygon is always convex. 3-5

Its vertices are A, B, C, D, and E. Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE. 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 GEOMETRY LESSON 3-5 Naming Polygons 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. Quick Check 3-5

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 GEOMETRY LESSON 3-5 Classifying Polygons Classify the polygon below by its sides. Identify it as convex or concave. Starting with any side, count the number of sides clockwise around the figure. Because the polygon has 12 sides, it is a dodecagon. Quick Check 3-5

Find the sum of the measures of the angles of a decagon. The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 Finding a Polygon Angle Sum 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 Quick Check 3-5

Find m X in quadrilateral XYZW. 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. m X + m X = 170 Substitute m X for m Y. 2m X = 170 Simplify. m X = 85 Divide each side by 2. The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 Using the Polygon Angle-Sum Theorem Quick Check The figure has 4 sides, so n = 4. 3-5

A regular hexagon is inscribed in a rectangle. Explain how you know that all the angles labeled 1 have equal measures. Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures. The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 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. Quick Check 3-5

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 The Polygon Angle-Sum Theorems GEOMETRY LESSON 3-5 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. not a polygon because two sides intersect at a point other than endpoints 1080 140 ABCDEFGHIJ is a regular decagon. 144 36 3-5

The Polygon Angle-Sum Theorems