Quadrilaterals 8 8.1 Find Angle Measures in Polygons 8.2 Use Properties of Parallelograms 8.3 Show that a Quadrilateral is a Parallelogram 8.4 Properties of Rhombuses, Rectangles, and Squares 8.5 Use Properties of Trapezoids and Kites 8.6 Identify Special Quadrilaterals Before In previous chapters, you learned the following skills, which you’ll use in Chapter 8: identifying angle pairs, using the Triangle Sum Theorem, and using parallel lines. Prerequisite Skills VOCABULARY CHECK Copy and complete the statement. ∠ 1 2 1. 1 and ? are vertical angles. 3 4 2. ∠ 3 and ? are consecutive interior angles. 5 6 3. ∠ 7 and ? are corresponding angles. 7 8 4. ∠ 5 and ? are alternate interior angles. SKILLS AND ALGEBRA CHECK 5. In nABC, m∠ A 5 x8, m∠ B 5 3x8, and m∠ C 5 (4x 2 12)8. Find the measures of the three angles. (Review p. 217 for 8.1.) Find the measure of the indicated angle. (Review p. 154 for 8.2–8.5.) 6. If m∠ 3 5 1058, then m∠ 2 5 ? . 1 7. If m∠ 1 5 988, then m∠ 3 5 ? . 2 8. If m∠ 4 5 828, then m∠ 1 5 ? . 3 4 9. If m∠ 2 5 1028, then m∠ 4 5 ? . 1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN 504 Now In Chapter 8, you will apply the big ideas listed below and reviewed in the Chapter Summary on page 559. You will also use the key vocabulary listed below. Big Ideas 1 Using angle relationships in polygons 2 Using properties of parallelograms 3 Classifying quadrilaterals by their properties KEY VOCABULARY • diagonal, p. 507 • square, p. 533 • midsegment of a • parallelogram, p. 515 • trapezoid, p. 542 trapezoid, p. 544 • rhombus, p. 533 bases, base angles, legs • kite, p. 545 • rectangle, p. 533 • isosceles trapezoid, p. 543 Why? You can use properties of quadrilaterals and other polygons to find side lengths and angle measures. Geometry The animation illustrated below for Example 4 on page 545 helps you answer this question: How can classifying a quadrilateral help you draw conclusions about its sides and angles? % $ & M& M$ $|& ' 3TART #HECK!NSWER Many real-world kites are shaped like Use properties of quadrilaterals to write geometric kites. an equation about the angle measures. Geometry at classzone.com Other animations for Chapter 8: pages 509, 519, 527, 535, 551, and 553 505 InvestigatingInvestiggggating Geometry ACTIVITYACTIVITY Use before Lesson 8.1 8.1 Investigate Angle Sums in Polygons MATERIALS • straightedge • ruler QUESTION What is the sum of the measures of the interior angles of a convex n-gon? Recall from page 43 that an n-gon is a polygon with n sides and n vertices. EXPLORE Find sums of interior angle measures STEP 1 Draw polygons Use a straightedge to draw convex polygons with three sides, four sides, five sides, and six sides. An example is shown. STEP 2 Draw diagonals In each polygon, draw all the diagonals from one vertex. A diagonal is a segment that joins two nonconsecutive vertices. Notice that the diagonals divide the polygon into triangles. STEP 3 Make a table Copy the table below. By the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is 1808. Use this theorem to complete the table. Polygon Number of sides Number of triangles Sum of measures of interior angles Triangle 3 1 1 p 18085 1808 Quadrilateral ? ? 2 p 18085 3608 Pentagon ? ? ? Hexagon ? ? ? DRAW CONCLUSIONS Use your observations to complete these exercises 1. Look for a pattern in the last column of the table. What is the sum of the measures of the interior angles of a convex heptagon? a convex octagon? Explain your reasoning. 2. Write an expression for the sum of the measures of the interior angles of a convex n-gon. 3. Measure the side lengths in the hexagon you drew. Compare the lengths with those in hexagons drawn by other students. Do the side lengths affect the sum of the interior angle measures of a hexagon? Explain. 506 Chapter 8 Quadrilaterals Find Angle Measures 8.1 in Polygons Before You classified polygons. Now You will find angle measures in polygons. Why? So you can describe a baseball park, as in Exs. 28–29. Key Vocabulary In a polygon, two vertices that are endpoints C of the same side are called consecutive vertices. • diagonal B D • interior angle, A diagonal of a polygon is a segment that joins p. 218 two nonconsecutive vertices. Polygon} ABCDE} diagonals • exterior angle, has two diagonals from vertex B, BD and BE . AE p. 218 As you can see, the diagonals from one vertex form triangles. In the Activity on page 506, you used these triangles to find the sum of the interior angle measures of a polygon. Your results support the following theorem and corollary. THEOREMS For Your Notebook THEOREM 8.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles 2 2 p 8 3 of a convex n-gon is (n 2) 180 . 1 4 m∠ 1 1 m∠ 2 1 . 1 m∠ n 5 (n 2 2) p 1808 56 n 5 Proof: Ex. 33, p. 512 (for pentagons) 6 COROLLARY TO THEOREM 8.1 Interior Angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is 3608. Proof: Ex. 34, p. 512 EXAMPLE 1 Find the sum of angle measures in a polygon Find the sum of the measures of the interior angles of a convex octagon. Solution An octagon has 8 sides. Use the Polygon Interior Angles Theorem. (n 20 2) p 18 85 (8 02 2) p 18 8 Substitute 8 for n. 5 6 p 1808 Subtract. 5 10808 Multiply. c The sum of the measures of the interior angles of an octagon is 10808. 8.1 Find Angle Measures in Polygons 507 EXAMPLE 2 Find the number of sides of a polygon The sum of the measures of the interior angles of a convex polygon is 9008. Classify the polygon by the number of sides. Solution Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides. (n 2 2) p 18085 9008 Polygon Interior Angles Theorem n 2 2 5 5 Divide each side by 1808. n 5 7 Add 2 to each side. c The polygon has 7 sides. It is a heptagon. ✓ GUIDED PRACTICE for Examples 1 and 2 1. The coin shown is in the shape of a regular 11-gon. Find the sum of the measures of the interior angles. 2. The sum of the measures of the interior angles of a convex polygon is 14408. Classify the polygon by the number of sides. EXAMPLE 3 Find an unknown interior angle measure ALGEBRA Find the value of x in 1088 1218 the diagram shown. x8 598 Solution The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation. x81 10881 12181 5985 3608 Corollary to Theorem 8.1 x 1 288 5 360 Combine like terms. x 5 72 Subtract 288 from each side. c The value of x is 72. ✓ GUIDED PRACTICE for Example 3 3. Use the diagram at the right. Find m∠ S and P m∠ T. P 1568 R 938 858 4. The measures of three of the interior angles of a quadrilateral are 898, 1108, and 468. Find the measure of the fourth interior angle. T S 508 Chapter 8 Quadrilaterals EXTERIOR ANGLES Unlike the sum of the interior angle measures of a convex polygon, the sum of the exterior angle measures does not depend on the number of sides of the polygon. The diagrams below suggest that the sum of the measures of the exterior angles, one at each vertex, of a pentagon is 3608. In general, this sum is 3608 for any convex polygon. 3608 VISUALIZE IT 2 1 2 1 A circle contains two 1 5 straight angles. So, 2 4 81 8 5 5 3 there are 180 180 , 3 3 8 or 360 , in a circle. 4 4 1808 STEP 1 Shade one exterior STEP 2 Cut out the STEP 3 Arrange the exterior 1808 angle at each vertex. exterior angles. angles to form 3608. (FPNFUSZ at classzone.com THEOREM For Your Notebook THEOREM 8.2 Polygon Exterior Angles Theorem The sum of the measures of the exterior angles of a 2 3 convex polygon, one angle at each vertex, is 3608. 1 4 m∠ 1 1 m∠ 2 1 . 1 m∠ n 5 3608 5 Proof: Ex. 35, p. 512 n 5 5 ★ EXAMPLE 4 Standardized Test Practice What is the value of x in the diagram shown? 898 A 67 B 68 2x8 8 C D 67 ELIMINATE CHOICES 91 136 x8 You can quickly eliminate choice D. If x were equal to 136, then Solution the sum of only two of the angle measures Use the Polygon Exterior Angles Theorem to write and solve an equation. (x8 and 2x8) would be x81 2x81 8981 6785 3608 Polygon Exterior Angles Theorem greater than 3608. 3x 1 156 5 360 Combine like terms. x5 68 Solve for x. c The correct answer is B. ABCD ✓ GUIDED PRACTICE for Example 4 5. A convex hexagon has exterior angles with measures 348, 498, 588, 678, and 758. What is the measure of an exterior angle at the sixth vertex? 8.1 Find Angle Measures in Polygons 509 EXAMPLE 5 Find angle measures in regular polygons READ VOCABULARY TRAMPOLINE The trampoline shown is Recall that a dodecagon shaped like a regular dodecagon.