Notes 4 7 Isosceles and Equilateral Triangles ABC

Notes 4. 7 Isosceles and Equilateral Triangles. △ABC is an isosceles triangle. legs -- the congruent sides of an isosceles triangle. _ B base -- the third side (not congruent) of an isosceles triangle. vertex angle -- the angle formed by the legs of the triangle. A _ C base angles -- the angles in an isosceles triangle that are adjacted to the base.

Notes 4. 7 Isosceles and Equilateral Triangles. ∠C AC

Notes 4. 7 Isosceles and Equilateral Triangles. equiangular equilateral

EXAMPLE 1 In DEF, DE Apply the Base Angles Theorem DF. Name two congruent angles. SOLUTION DE DF , so by the Base Angles Theorem, E F.

GUIDED PRACTICE for Example 1 Copy and complete the statement. 1. If HG HK , then SOLUTION HGK HKG ? ? .

GUIDED PRACTICE for Example 1 Copy and complete the statement. 2. If KHJ KJH, then ? ? . SOLUTION If KHJ KJH, then , KH KJ

EXAMPLE 2 Find measures in a triangle Find the measures of P, Q, and R. The diagram shows that PQR is equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQR is equiangular. So, m P = m Q = m R. 3(m m P) = 180 P = 60 o Triangle Sum Theorem o Divide each side by 3. ANSWER The measures of P, Q, and R are all 60°.

GUIDED PRACTICE 3. for Example 2 Find ST in the triangle at the right. SOLUTION STU is equilateral, then its is equiangular ANSWER Thus ST = 5 ( Base angle theorem )

GUIDED PRACTICE 4. for Example 2 Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain. SOLUTION No; it is not possible for an equilateral triangle to have angle measure other then 60°. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60° because all pairs of angles could be considered base of an isosceles triangle

EXAMPLE 3 Use isosceles and equilateral triangles ALGEBRA Find the values of x and y in the diagram. SOLUTION STEP 1 STEP 2 Find the value of y. Because KLN is equiangular, it is also equilateral and KN Therefore, y = 4. KL. Find the value of x. Because LNM LMN, LN LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.

EXAMPLE 3 Use isosceles and equilateral triangles LN = LM Definition of congruent segments 4=x+1 Substitute 4 for LN and x + 1 for LM. 3=x Subtract 1 from each side.

EXAMPLE 4 Solve a multi-step problem Lifeguard Tower In the lifeguard tower, PS and QPS PQR. QR a. What congruence postulate can you use to prove that QPS PQR? b. Explain why c. Show that PQT is isosceles. PTS QTR.

EXAMPLE 4 Solve a multi-step problem SOLUTION a. Draw and label QPS and PQR so that they do not overlap. You can see that PQ QP , PS QR , and QPS PQR. So, by the SAS Congruence Postulate, QPS PQR. b. From part (a), you know that 1 2 because corresp. parts of are. By the Converse of the Base Angles Theorem, PT QT , and PQT is isosceles.

EXAMPLE 4 c. Solve a multi-step problem You know that PS QR , and 3 4 because corresp. parts of are. Also, PTS QTR by the Vertical Angles Congruence Theorem. So, PTS QTR by the AAS Congruence Theorem.

GUIDED PRACTICE 5. for Examples 3 and 4 Find the values of x and y in the diagram. SOLUTION y° = 120° x° = 60°

GUIDED PRACTICE 6. for Examples 3 and 4 Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that PTS QTR SOLUTION QPS PQR. Can be shown by segment addition postulate i. e a. QT + TS = QS and PT + TR = PR

GUIDED PRACTICE Since PT for Examples 3 and 4 QT from part (b) and TS TR from part (c) then, QS PR PQ PQ Reflexive Property and PS QR Given ANSWER Therefore Postulate QPS PQR. By SSS Congruence