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Understanding Isosceles, Equilateral, and Right Triangles

This chapter explores key theorems related to isosceles, equilateral, and right triangles including the Isosceles Triangle Theorem which states that the angles opposite equal sides are congruent. It also covers the properties of equilateral triangles, where each angle measures 60°. Students will engage with sample problems to find missing angle measures, understand congruence in triangles through HL Theorem, and apply concepts through various exercises. Essential for grasping foundational geometry, this lesson is designed for effective learning.

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Understanding Isosceles, Equilateral, and Right Triangles

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  1. Isosceles, Equilateral, and Right Triangles Chapter 4.6

  2. Isosceles Triangle Theorem Isosceles   The 2 Base s are  • Base angles are the angles opposite the equal sides.

  3. B A C If AB  BC, then A  C Isosceles Triangle Theorem

  4. B A C If A  C then AB  BC Isosceles Triangle Theorem

  5. B A C Sample Problem Solve for the variables • mA = 32° • mB = (4y)° • mC = (6x +2)° 32 + 32 + 4y = 180 4y + 64 = 180 4y = 116 y = 29 6x + 2 = 32 6x = 30 x = 5

  6. Find the Measure of a Missing Angle 120o 30o 30o 75o 30o 75o 180o – 120o = 60o 180o – 30o = 150o Lesson 6 Ex2

  7. A • B • C • D A. 25 B. 35 C. 50 D. 130 Lesson 6 CYP2

  8. A. B. C. D. A. Which statement correctly names two congruent angles? • A • B • C • D Lesson 6 CYP3

  9. A. B. C. D. B. Which statement correctly names two congruent segments? • A • B • C • D Lesson 6 CYP3

  10. Equilateral Triangle Theorem Equilateral   Equiangular Each angle = 60o !!!

  11. Use Properties of Equilateral Triangles Linear pair Thm. Substitution Subtraction Answer: 105 Lesson 6 Ex4

  12. A • B • C • D A.x = 15 B.x = 30 C.x = 60 D.x = 90 Lesson 6 CYP4

  13. A • B • C • D A. 30 B. 60 C. 90 D. 120 Lesson 6 CYP4

  14. Don’t be an ASS!!! • Angle Side Side does not work!!! • (Neither does ASS backward!) • It can not distinguish between the two different triangles shown below. However, if the angle is a right angle, then they are no longer called sides. They are called…

  15. Hypotenuse-Leg   Theorem • If the hypotenuse and one leg of a right triangle are congruent to the corresponding parts in another right triangle, then the triangles are congruent.

  16. Y B X Z A C ABC  XYZ Why?HL   Theorem

  17. Z X Y M Prove XMZ  YMZ Step Reason Given Given mZMX = mZMY = 90o Def of  lines Reflexive ZMX  ZMY HL   Thm

  18. AB  XY • BC  YZ • CA  ZX Corresponding Parts of Congruent Triangles are Congruent • Given ΔABC  ΔXYZ • You can state that: • A  X • B  Y • C  Z

  19. Suppose you know that ABD  CDB by SAS   Thm. Which additional pairs of sides and angles can be found congruent using Corr. Parts of  s are ?

  20. Complete the following two-column proof. Proof: Statements Reasons 1. Given 1. 2. 2. Isosceles Δ Theorem 3. 3. Given 4. 4. Def. of midpoint Lesson 6 CYP1

  21. Statements Reasons 4. 4. Def. of midpoint ? 5. ______ 6. 6. ? 5. ΔABCΔADC Complete the following two-column proof. • A • B • C • D Proof: SAS   Thm. Corr. Parts of s are  Lesson 6 CYP1

  22. Homework Video C Ch 4-6 • pg 248 1 – 10, 14 – 27, 32, 33, 37 – 39, & 48 Reminder! Midpoint Formula:

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