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Learn the 45°-45°-90° and 30°-60°-90° triangle theorems with examples on finding lengths of hypotenuses and legs. Solve real-world problems using properties of these triangles.

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 8–2) NGSSS Then/Now Theorem 8.8: 45°-45°-90° Triangle Theorem Example 1: Find the Hypotenuse Length in a 45°-45°-90° Triangle Example 2: Find the Leg Lengths in a 45°-45°-90° Triangle Theorem 8.9: 30°-60°-90° Triangle Theorem Example 3: Find Lengths in a 30°-60°-90° Triangle Example 4: Real-World Example: Use Properties of Special Right Triangles Lesson Menu

  3. A B C D A.5 B. C. D.10.5 Find x. 5-Minute Check 1

  4. A B C D A. B. C.45 D.51 Find x. 5-Minute Check 2

  5. A B C D Determine whether ΔQRS with vertices Q(2, –3), R(0, –1), and S(4, –1) is a right triangle. If so, identify the right angle. A. yes; S B. yes; Q C. yes; R D. no 5-Minute Check 3

  6. A B C D Determine whether the set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right.16, 30, 33 A. yes, acute B. yes, obtuse C. yes, right D. no 5-Minute Check 4

  7. A B C D Determine whether the set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. A. yes, acute B. yes, obtuse C. yes, right D. no 5-Minute Check 5

  8. A B C D A.25, 20, 15 B.4, 7 , 8 C.0.7, 2.4, 2.5 D.36, 48, 62 1 __ 2 Which of the following are the lengths of an acute triangle? 5-Minute Check 6

  9. MA.912.G.5.3Use special right triangles to solve problems. MA.912.G.5.4 Solve real-world problems involving right triangles. Also addresses MA.912.G.5.1. NGSSS

  10. You used properties of isosceles and equilateral triangles. (Lesson 4–6) • Use the properties of 45°-45°-90° triangles. • Use the properties of 30°-60°-90° triangles. Then/Now

  11. Concept

  12. Find the Hypotenuse Length in a 45°-45°-90° Triangle A. Find x. The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle. Example 1

  13. 45°-45°-90° Triangle Theorem Substitution Find the Hypotenuse Length in a 45°-45°-90° Triangle Example 1

  14. Find the Hypotenuse Length in a 45°-45°-90° Triangle B. Find x. The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem. Example 1

  15. 45°-45°-90° Triangle Theorem Substitution Find the Hypotenuse Length in a 45°-45°-90° Triangle x = 12 Answer:x = 12 Example 1

  16. A B C D A.3.5 B.7 C. D. A. Find x. Example 1

  17. A B C D A. B. C.16 D.32 B. Find x. Example 1

  18. The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle. Find the Leg Lengths in a 45°-45°-90° Triangle Find a. 45°-45°-90° Triangle Theorem Substitution Example 2

  19. Divide each side by Find the Leg Lengths in a 45°-45°-90° Triangle Rationalize the denominator. Multiply. Divide. Example 2

  20. A B C D A. B.3 C. D. Find b. Example 2

  21. Concept

  22. Find Lengths in a 30°-60°-90° Triangle Find x and y. The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle. Example 3

  23. Find Lengths in a 30°-60°-90° Triangle Find the length of the longer side. 30°-60°-90° Triangle Theorem Substitution Simplify. Example 3

  24. Answer:x = 4, Find Lengths in a 30°-60°-90° Triangle Find the length of hypotenuse. 30°-60°-90° Triangle Theorem Substitution Simplify. Example 3

  25. A B C D A.4 in. B.8 in. C. D.12 in. Find BC. Example 3

  26. Use Properties of Special Right Triangles QUILTINGA quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle? Example 4

  27. Use Properties of Special Right Triangles Understand You know that the length of the side of the square equals 3 inches. You need to find the length of the side and hypotenuse of one isosceles right triangle. Plan Find the length of one side of the isosceles right triangle, and use the 45°-45°-90° Triangle Theorem to find the hypotenuse. Example 4

  28. Use Properties of Special Right Triangles Solve Divide the length of the side of the square by 2 to find the length of the side of one triangle. 3 ÷ 2 = 1.5 So the side length is 1.5 inches. 45°-45°-90° Triangle Theorem Substitution Example 4

  29. Answer: The side length is 1.5 inches and the hypotenuse is ? ? 2.25 + 2.25 = 4.5 Use Properties of Special Right Triangles Check Use the Pythagorean Theorem to check the dimensions of the triangle. 4.5 = 4.5  Example 4

  30. A B C D A. B.10 C.5 D. BOOKENDS Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends. Example 4

  31. End of the Lesson

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