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Isosceles and Equilateral Triangles

Learn about isosceles and equilateral triangles, their properties, angles, and congruence theorems in this detailed guide. Explore concepts with examples and solve related problems to master these triangle types.

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Isosceles and Equilateral Triangles

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  1. Isosceles and Equilateral Triangles Concept 25

  2. Has exactly three congruent sides Vertex Angle the angle formed by the legs. Leg Leg the 2 congruent sides of an isosceles triangle. Base Angle 2 angles adjacent to the base. the 3rd side of an isosceles triangle Base

  3. ___ BCA is opposite BA and A is opposite BC, so BCA  A. ___ 1. Name two unmarked congruent angles. Answer: BCAand A

  4. ___ BC is opposite D and BD is opposite BCD, so BC  BD. ___ ___ ___ ___ Answer: BC BD 2. Name two unmarked congruent segments.

  5. 3. Which statement correctly names two congruent angles? A.PJM PMJ B.JMK JKM C.KJP JKP D.PML PLK

  6. A.JP PL B.PM PJ C.JK MK D.PM PK 4. Which statement correctly names two congruent segments?

  7. Since QP = QR, QP  QR. By the Isosceles Triangle Theorem, base angles P and R are congruent, so mP = mR . Use the Triangle Sum Theorem to write and solve an equation to find mR. 5. Find mR. Triangle Sum Theorem mQ = 60, mP = mR Simplify. Subtract 60 from each side. Answer:mR = 60 Divide each side by 2.

  8. 6. Find PR. Since all three angles measure 60, the triangle is equiangular. Because an equiangular triangle is also equilateral, QP = QR = PR. Since QP = 5, PR = 5 by substitution. Answer:PR = 5 cm

  9. A. Find mT. A. 30° B. 45° C. 60° D. 65°

  10. B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7

  11. 7. Find the value of each variable. mDFE = 60 4x – 8 = 60 4x = 68 x = 17 The triangle is equilateral, so all the sides are congruent, and the lengths of all of the sides are equal. DF = FE 6y + 3 = 8y – 5 3 = 2y – 5 8 = 2y 4 = y

  12. 8. Find the value of each variable. A.x = 20, y = 8 B.x = 20, y = 7 C.x = 30, y = 8 D.x = 30, y = 7

  13. CPCTC Concept 26

  14. Corresponding Parts of Congruent Triangles are Congruent

  15. Use the diagram to answer the following. • What triangle appears to be congruent • to PAS? • What triangle appears to be congruent • to PAR? RLP • If and , what additional information would you need to prove PSL RSA? • If LPA ARL and PL≅AR, what additional information would you need to prove LPA ARL? SAS SSS ASA AAS SAS

  16. Use the diagram to answer the following questions. • To prove PSLPSA, which triangles must you prove to be congruent? • To prove≅, which triangles must you prove to be congruent? LPS APS APS LRS LPS ARS LPS APS LRS ARS

  17. Use the marked diagrams to state the method used to prove the triangles are congruent. Give the congruence statement, then name the additional corresponding parts that could then be concluded to be congruent. Missing Info/Why: Symmetric Prop. Triangle Congruence/Why: SSS CPCTC:

  18. Use the marked diagrams to state the method used to prove the triangles are congruent. Give the congruence statement, then name the additional corresponding parts that could then be concluded to be congruent. Missing Info/Why: Vertical Angles Triangle Congruence/Why: ASA CPCTC:

  19. Given: and  S R Prove:  S R Vertical Angles ASA CPCTC

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