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Solving for x and y in Equilateral and Isosceles Triangles

This lesson focuses on finding the values of x and y in geometrical diagrams involving equilateral and isosceles triangles. First, learn to determine y using the properties of equiangular triangles, leading to y = 4. Next, find x using isosceles triangle properties, knowing two sides are equal. Guided practice examples illustrate multi-step problem-solving techniques, congruence postulates, and angle relationships, helping students understand geometric principles and their applications in solving problems step-by-step.

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Solving for x and y in Equilateral and Isosceles Triangles

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  1. 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

  2. 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.

  3. 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

  4. 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

  5. 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.

  6. Find the values of x and yin the diagram. We name the triangle as ABC and CBD CBD is equilateral triangle which guarantees the angle measure 60° therefore x° = 60° for Examples 3 and 4 GUIDED PRACTICE SOLUTION y° = 180° – x° y° = 180° – 60° y° = 120° x° = 180° – y° = 60°

  7. 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

  8. b. c. 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 and from part then,

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