1. Mrs. Samuelian CAHSEE Prep CONGRUENT FIGURES & PYTHAGOREAN THEOREM

2. Congruent Figures 7MG 3.4 Students will identify congruent figures and match corresponding sides and angles. Vocabulary: Angle- A figure formed by two rays with a common endpoint-the common endpoint is called the vertex of the angle Congruent Figures- Two or more figures that are the same shape and size

3. Introduction When two or more shapes are alike, they are called congruent shapes. Two or more shapes are congruent if they can be placed one on top of each other and all the points match. This means that all the matching sides have the same length and all matching angles have the same measure.

4. Understanding the Symbols Notice the tick mark on the top of each of the two triangles. This mark means that the segments AC and DF are congruent. ------- The symbol ≅ is used to show two objects are congruent. Therefore, AC ≅ DF. -------

5. Look for clues to help you match up sides and angles. Tick marks and angle marks are a great way to indicate side and angle congruence. Polygons with sides that have the same number of tick marks means that the sides are congruent. Angles with an equal number of tick marks indicate congruent angles.

6. Identifying Corresponding Parts of Congruent Figures The sides and angles that match and have the same measure are called corresponding parts. The two shapes below are congruent triangles. Side AB corresponds to side XY, since they have the same length. -- -----

7. Pythagorean Theorem 7MG 3.3 Students will use the Pythagorean theorem to find the length of the hypotenuse or to find the length of the missing leg of a right triangle. Vocabulary Hypotenuse: The side opposite the right angle in a right triangle (the longest side) Pythagorean theorem: The sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse Right triangle: A triangle with one 90 degree angle Square root: One of the two equal factors of a number

8. Introduction If you know the lengths of both legs of a right triangle, you can find the hypotenuse length using the Pythagorean theorem, a 2 + b 2 = c 2 Also, if you know the hypotenuse length and the length of one of the legs, then you can find the missing leg length!

9. Square Roots √1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5 √36 = 6 √49 = 7 √64 = 8 √81 = 9 √100 = 10 √121 = 11 √144 = 12 √169 = 13 Since 1 2 = 1 Since 2 2 = 4 Since 3 2 = 9 Since 4 2 = 16 Since 5 2 = 25 Since 6 2 = 36 Since 7 2 = 49 Since 8 2 = 64 Since 9 2 = 81 Since 10 2 = 100 Since 11 2 = 121 Since 12 2 = 144 Since 13 2 = 169

10. What information are you given? What information are you looking for? Write the Pythagorean theorem, filling in the information that you know. Square each of the known values. Solve this equation algebraically to find the unknown side length. 3 cm 5 cm ?

11. a 2 + b 2 = c 2 a 2 + 3 2 = 5 2 a 2 + 9 = 25 a 2 = 16 √a 2 = √16 a = 4 3 cm 5 cm ?