Isosceles Triangles. leg. base

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1 6 4 What ou ll Learn ou ll learn to identif and use properties of isosceles triangles. Isosceles riangles ecall from Lesson 5 that an isosceles triangle has at least two congruent sides. he congruent sides are called legs. he side opposite the vertex angle is called the base. In an isosceles triangle, there are two base angles, the vertices where the base intersects the congruent sides. Wh It s Important vertex angle dvertising Isosceles triangles can be found in business logos. ee xercise 7. leg base angle leg base base angle ou can use a I 83/84 lus graphing calculator to draw an isosceles triangle and stud its properties. Graphing alculator utorial ee pp tep raw a circle using the ircle tool on the 2 menu. Label the center of the circle. tep 2 Use the riangle tool on the 2 menu to draw a triangle that has point as one vertex and its other two vertices on the circle. Label these vertices and. tep 3 Use the Hide/how tool on menu 5 to hide the circle. ress the L ke to quit the 7 menu. he figure that remains on the screen is isosceles triangle. r hese. ell how ou can use the measurement tools on 5 to check that is isosceles. Use our method to be sure it works. 2. Use the ngle tool on 5 to measure and. What is the relationship between and? 3. Use the ngle isector tool on 3 to bisect. Use the Intersection oint tool on 2 to mark the point where the angle bisector intersects Label the point of intersection. What is point in relation to side? 246 hapter 6 ore bout riangles

2 4. Use the ngle tool on 5 to find the measures of and. 5. Use the istance & Length tool on 5 to measure and What is the relationship between the lengths of and? 6. Is part of the perpendicular bisector of? xplain. he results ou found in the activit are expressed in the following theorems. heorem Words 6 2 If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Isosceles riangle heorem 6 3 xample odels mbols If, then. he median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle. If and, then and. ind the value of each variable in isosceles triangle if G is an angle bisector. irst, find the value of x. ince is an isosceles triangle,. o, x 49. G ow find the value of. heorem 6 3, G o, our urn or each triangle, find the values of the variables. a b. 50 O 70 Lesson 6 4 Isosceles riangles 247

3 uppose ou draw two congruent acute angles on two pieces of patt paper and then rotate one of the angles so that one pair of ras overlaps and the other pair intersects. X What kind of triangle is formed? What is true about angles and? What is true about the sides opposite angles and? Is the converse of heorem 6 2 true? heorem 6 4 onverse of Isosceles riangle heorem Words: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. odel: xample lgebra Link 2 mbols: If, then In, and m 48. ind m,, and. 4x 48 irst, find m. ou know that m 48. ince, m 48. lgebra eview olving ulti-tep quations, p. 723 m m m m m m m 84 6x 5 ngle um heorem eplace m and m with 48. dd. ubtract 96 from each side. implif. ext, find. ince, heorem 6 4 states that 6x 5 4x 6x 5 6x 4x 6x 5 2x 5 2 2x x efinition of ongruent egments eplace with 4x and with 6x 5. ubtract 6x from each side. implif. ivide each side b 2. implif. replacing x with 2.5, ou find that 4(2.5) or 0 and 6(2.5) 5 or hapter 6 ore bout riangles

4 quiangular: Lesson 5 2; quilateral: Lesson 5 In hapter 5, the terms equiangular and equilateral were defined. Using heorem 6 4, we can now establish that equiangular triangles are equilateral. is equiangular. ince m m m, heorem 6 4 implies that. heorem 6 5 triangle is equilateral if and onl if it is equiangular. heck for Understanding ommunicating athematics. raw an isosceles triangle. Label it with base hen state four facts about the triangle. 2. xplain wh equilateral triangles are also equiangular and wh equiangular triangles are also equilateral. Guided ractice or each triangle, find the values of the variables. xample xample 2 V 4. x xercises W 6 5. lgebra In, and m 37. ind m, Q, and Q. 37 3x 2 ractice 45 U Q 2x 3 or each triangle, find the values of the variables Homework Help or xercises 6-4, 7, 8 ee xamples 5, I 80 xtra ractice 8 59 O G. U 47 5 ee page 737. x J 68 K H x V 9 47 W Lesson 6 4 Isosceles riangles 249

5 2. In, If m 35, what is the value of x? 3. ind the value of if 4. In, ind m. 35 xercises 2 4 pplications and roblem olving 5. lgebra In, If m 5x 7 and m 4x 2, find m and m. (4x 2) 6. lgebra In,, m 70, 3x, and 7x 7. ind m,, and. 70 7x 7 (5x 7) 3x 7. dvertising business logo is shown. a. What kind of triangle does the logo contain? b. If the measure of angle is 0, what are the measures of the two base angles of that triangle? 8. ritical hinking ind the measures of the angles of an isosceles triangle such that, when an angle bisector is drawn, two more isosceles triangles are formed. ixed eview 9. In JK, JQ bisects KJ. If m KJ = 32, what is m? (Lesson 6 3) J K 20. In, ame a perpendicular bisector. (Lesson 6 2) 2. Graph and label point H at ( 4, 3) on a coordinate plane. (Lesson 2 4) tandardized est ractice 22. hort esponse arcus used 37 feet of fencing to enclose his triangular garden. What is the length of each side of the garden? (Lesson 6) Q X xercise 20 r 7 r 2 L r hort esponse Write a sequence in which each term is 7 less than the previous term. (Lesson ) 250 hapter 6 ore bout riangles