1. Bell Work: Use the difference of two squares theorem to write the answers to the following equation. w = 14 2

2. Answer: ±√14

3. Lesson 97: Angles and Triangles, Pythagorean Theorem, Pythagorean Triples

4. Two intersecting lines form four angles. Here are shown two intersecting lines and the four angles they formed.

5. An angle is formed by two half lines or rays that are in the same plane and that have a common endpoint.

6. To begin a quick review of angle measures, we remember that if two straight lines intersect and are perpendicular to one another, we define the measure of each of the four angles created to be 90 degrees.

7. It can be proved, by using geometry, that the sum of the interior angles of any triangle is 180°. We show three triangles and notes that the sum of the three interior angles in each triangle is 180°.

8. Any triangle that contains a right angle is called a right triangle, and the side of the triangle that is opposite the right angle is always the longest side. We call this side of a right triangle the hypotenuse. The other two sides are called legs, or simply sides.

10. It can be shown that the square drawn on the hypotenuse of a right triangle has the same area as the sum of the areas of the squares drawn on the other two sides. This idea is known as the Pythagorean theorem.

12. The general algebraic expression of the Pythagorean Theorem is a + b = c Where c is the length of the hypotenuse and a and b represent the lengths of the other two sides (legs). 2 2 2

13. Example: Given the triangle with the lengths of the sides as shown, use the Pythagorean theorem to find a. 5 4 a

14. Answer: a = 3

15. Example: Find side m. 12 m 8

16. Answer: m = 4√13

17. Example: Find k. k √61 5

18. Answer: k = 6

19. Pythagorean Triples: It is useful to commit to memory the lengths of the sides of certain right triangles. We show some of these right triangles below.

20. Note that all the sides of the right triangles show are integers. The triplets of numbers describing the lengths of the three sides of right triangles whose sides are integer lengths are called Pythagorean triples.

21. It is useful to know that any multiple of a Pythagorean triple is also a Pythagorean triple. For example, a 3-4-5 right triangle, 6-8- 10 right triangle, and 9-12-15 right triangle are all Pythagorean triples.

22. Example: Recall the appropriate Pythagorean triple to find the unknown length in each of the following right triangles. a c 8 5 17 8 10 12 b

23. Answer: a = 13 b = 15 c = 6

24. HW: Lesson 97 #1-30