4.7 Isosceles and Equilateral Triangles

1. 4.7 Isosceles and Equilateral Triangles Objective: You will use theorems about isosceles and equilateral triangles.

2. Base Angle Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent.

3. Converse of Base Angles Theorem • If two angles of a triangle are congruent, then the sides opposite them are congruent.

4. Corollary • Corollary to the Base Angle Theorem- if a triangle is equilateral, then it is equiangular. • Corollary to the Converse of Base Angle Theorem- If a triangle is equiangular, then it is equilateral.

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

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

7. Copy and complete the statement. If KHJKJH, then ?? . If KHJKJH, then ?? . If KHJKJH, then , KH KJ for Example 1 GUIDED PRACTICE SOLUTION

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

9. Find STin the triangle at the right. ( Base angle theorem ) ThusST = 5 ANSWER STUis equilateral, then its is equiangular for Example 2 GUIDED PRACTICE SOLUTION

10. Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain. for Example 2 GUIDED PRACTICE 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

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

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

13. In the lifeguard tower, PS QRand QPS PQR. What congruence postulate can you use to prove that QPS PQR? Explain why PQTis isosceles. Show thatPTS QTR. EXAMPLE 4 Solve a multi-step problem Lifeguard Tower

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

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

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

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

18. QS PR Reflexive Property and PQ PQ Given PS QR ANSWER ThereforeQPS PQR.BySSS Congruence Postulate TS TR for Examples 3 and 4 GUIDED PRACTICE SincePT QTfrom part (b) and from part (c) then,