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4.8 Use Isosceles and Equilateral Triangles

4.8 Use Isosceles and Equilateral Triangles. You will use theorems about isosceles and equilateral triangles. Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles?.

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4.8 Use Isosceles and Equilateral Triangles

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  1. 4.8 Use Isosceles and Equilateral Triangles • You will use theorems about isosceles and equilateral triangles. • Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles? You will learn how to answer this question by learning the Base Angles Theorem and its converse.

  2. 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

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

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

  5. 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 Can an equilateral triangle have an angle of 61?

  6. 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

  7. 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

  8. 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 Explain how you could find m ∠ M.

  9. 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.

  10. 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

  11. 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

  12. 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.

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

  14. 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

  15. 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,

  16. 1. ANSWER 8 Daily Homework Quiz Find the value of x.

  17. 2. ANSWER 3 Daily Homework Quiz Find the value of x.

  18. If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles? 3. ANSWER 34°, 34° Daily Homework Quiz

  19. Find the perimeter of triangle. 4. ANSWER 66 cm Daily Homework Quiz

  20. You will use theorems about isosceles and equilateral triangles. • Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles? • Angles opposite congruent sides of a triangle are congruent and conversely. • If a triangle is equilateral, then it is equiangular and conversely. If two sides of a triangle are congruent, then the angles opposite them are congruent. The converse is also true.

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