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Page 1 of 6 4.3 Goal Use properties of isosceles and equilateral triangles. Isosceles and Equilateral Triangles Geo-Activity Properties of Isosceles Triangles 1 Fold a sheet of paper in half. ● Key Words • legs of an isosceles triangle 2 Unfold and label the angles ● Use a straightedge to draw a line from the fold to the bottom edge. Cut along the line to form an isosceles triangle. as shown. Use a protractor to measure aH and aK. What do you notice? • base of an isosceles triangle • base angles J H K 3 Repeat Steps 1 and 2 for different isosceles triangles. What can ● you say about aH and aK in the different triangles? Student Help VOCABULARY TIP Isos- means “equal,” and -sceles means “leg.” So, isosceles means equal legs. The Geo-Activity shows that two angles of an isosceles triangle are always congruent. These angles are opposite the congruent sides. leg The congruent sides of an isosceles triangle are called legs . leg base angles The other side is called the base . base The two angles at the base of the triangle are called the base angles . Isosceles Triangle THEOREM 4.3 Base Angles Theorem Words A If two sides of a triangle are congruent, then the angles opposite them are congruent. Symbols &* c AC If AB &*, then aC c aB. 4.3 B C Isosceles and Equilateral Triangles 185 Page 2 of 6 EXAMPLE 1 Use the Base Angles Theorem Find the measure of aL. M Solution Angle L is a base angle of an isosceles triangle. From the Base Angles Theorem, aL and aN have the same measure. ANSWER ? © The measure of aL is 528. 528 L N Rock and Roll Hall of Fame, Cleveland, Ohio THEOREM 4.4 Converse of the Base Angles Theorem Words A If two angles of a triangle are congruent, then the sides opposite them are congruent. Symbols If aB c aC, then AC &* c AB &*. Visualize It! EXAMPLE Base angles don’t have to be on the bottom of an isosceles triangle. Find the value of x. 2 B Use the Converse of the Base Angles Theorem F Solution C C 12 By the Converse of the Base Angles Theorem, the legs have the same length. D B DE 5 DF A x 1 3 5 12 x59 ANSWER x13 Converse of the Base Angles Theorem Substitute x 1 3 for DE and 12 for DF. Subtract 3 from each side. © The value of x is 9. Use Isosceles Triangle Theorems Find the value of y. 1. 2. 3. y 508 186 Chapter 4 Triangle Relationships y8 16 9 y14 E Page 3 of 6 Student Help THEOREMS 4.5 and 4.6 LOOK BACK 4.5 Equilateral Theorem For the definition of equilateral triangle, see p. 173. Words Symbols If AB &* c AC &* c BC &*, then aA c aB c aC. 4.6 Equiangular Theorem Words B If a triangle is equilateral, then it is equiangular. A C If a triangle is equiangular, then it is equilateral. Symbols If aB c aC c aA, then AB &* c AC &* c BC &*. Constructing an Equilateral Triangle You can construct an equilateral triangle using a straightedge and compass. &*. Draw 1 Draw AB ● 2 Draw an arc with ● an arc with center A that passes through B. center B that passes through A. 3 The intersection of ● the arcs is point C. TABC is equilateral. C 60° 60° A B A B 60° A B By the Triangle Sum Theorem, the measures of the three congruent angles in an equilateral triangle must add up to 1808. So, each angle in an equilateral triangle measures 608. EXAMPLE 3 Find the Side Length of an Equiangular Triangle Find the length of each side of the equiangular triangle. R Solution The angle marks show that TQRT is equiangular. So, TQRT is also equilateral. 3x 5 2x 1 10 x 5 10 3(10) 5 30 ANSWER Sides of an equilateral T are congruent. 3x P 2x 1 10 T Subtract 2x from each side. Substitute 10 for x. © Each side of TQRT is 30. 4.3 Isosceles and Equilateral Triangles 187 Page 4 of 6 4.3 Exercises Guided Practice Vocabulary Check Skill Check 1. What is the difference between equilateral and equiangular? Tell which sides and angles of the triangle are congruent. M 2. R 3. L N U 4. T S W V Find the value of x. Tell what theorem(s) you used. 5. 6. x cm 8.8 cm x8 508 Practice and Applications Extra Practice Finding Measures Find the value of x. Tell what theorem(s) you used. See p. 681. B 7. 8. H x8 688 J 9. E x8 F 8 A 55 x8 C G D Using Algebra Find the value of x. 10. 12 11. 11 12. 13 6x x14 x Homework Help Example 1: Exs. 7–9, 14, 15, 17–19, 27, 28 Example 2: Exs. 10–13 Example 3: Exs. 20–25 188 Chapter 4 13. (5x 1 7)8 14. 7x 1 5 Triangle Relationships 15. 3x8 19 528 4x8 Page 5 of 6 16. Someone in your class tells you that all equilateral triangles are isosceles triangles. Do you agree? Use theorems or definitions to support your answer. Using Algebra Find the measure of aA. IStudent Help ICLASSZONE.COM You be the Judge 17. C A HOMEWORK HELP Extra help with problem solving in Exs. 17–19 is at classzone.com 18. x8 A 19. B 2x 8 508 x8 x8 B B x8 308 A C C Using Algebra Find the value of y. 20. 2y 21. 11 22. 2y 1 5 10 y15 y 23. 4y 2 3 y 24. 25. 8y 2 10 3y 5 Sports 4y 1 2 5y 2 14 26. Challenge In the diagram at the right, TXYZ is Y equilateral and the following pairs of segments &* and LK &*; ZY &* and LJ &; XZ &* and JK &. are parallel: XY Describe a plan for showing that TJKL must be equilateral. J X K L Z Rock Climbing In one type of rock climbing, climbers tie themselves to a rope that is supported by anchors. The diagram shows a red and a blue anchor in a horizontal slit in a rock face. ROCK CLIMBING The climber is using a method of rock climbing called top roping. If the climber slips, the anchors catch the fall. Application Links CLASSZONE.COM 27. If the red anchor is longer than the blue anchor, are the base angles congruent? 28. If a climber adjusts the anchors so they are the same length, do you think that the base angles will be congruent? Why or why not? 4.3 Isosceles and Equilateral Triangles 189 Page 6 of 6 Z Y X Tiles In Exercises 29–31, use the diagram at the left. In the diagram, &** c WX &*** c YX &* c ZX &*. VX 29. Copy the diagram. Use what you know about side lengths to mark your diagram. W V 30. Explain why aXWV c aXVW. 31. Name four isosceles triangles. 32. Technology Use geometry software to complete the steps. B 1 Construct circle A. ● 2 Draw points B and C on the circle. ● 3 Connect the points to form TABC. ● C A Is TABC isosceles? Measure the sides of the triangle to check your answer. Standardized Test Practice Multiple Choice In Exercises 33 and 34, use the diagram below. 33. What is the measure of aEFD? A X C X 558 B 658 X D 1808 X 1258 E 34. What is the measure of aDEF ? F X H X Mixed Review 508 G 708 X J 1808 X 1258 1258 F G D &( is the angle bisector. Find maDBC and maABC. Angle Bisectors BE (Lesson 2.2) 35. 36. E A E A E A C D 428 37. C 568 B B C 758 B D Vertical Angles Find the value of the variable. (Lesson 2.4) 38. 39. 558 40. (x 2 8)8 (x 1 20)8 818 428 (2x 1 1)8 Algebra Skills Evaluating Square Roots Evaluate. (Skills Review, p. 668) 41. Ï4 w9 w 190 Chapter 4 Triangle Relationships 42. Ï1 w2 w1 w 43. Ï1 w 44. Ï4 w0 w0 w D