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Congruence vs. Similarity Polygon Congruence Polygon Similarity Triangle Congruence Shortcuts Triangle Similarity Shortcuts 1. Need to prove that corresponding angles are congruent SSS SAS ASA AAS HL AA~ SAS~ SSS~ 2. Need to prove that corresponding sides are congruent 2. Need to prove that corresponding sides are proportional

Let’s practice our new knowledge of similarity shortcuts. Determine whether the triangles are similar. If so, explain why then write a similarity statement and name the postulate or theorem you used. If not, explain. Yes, they are similar because LM || OP (given) which creates congruent corresponding angles: <L and <O are congruent as are <M and <P. Therefore: LMN ~ OPN AA~ Yes, they are similar because AB || CD (given) which creates congruent alternate interior angles: <A and <D are congruent as are <B and <C. Therefore: ABE ~ DCE AA~

Let’s practice some more. Not similar. There is no way of knowing if the corresponding angles are congruent, or if the corresponding sides are proportional. Not enough information is given for us to make a certain determination. Yes, MNL~ QPO The sides are all in the same proportion, having the scale factor of 2. SSS~

Next… Similarity in right triangles Please get: One piece of colored paper Straight edge Scissors Your compass Your pencil Using your compass and straightedge, create a rectangle.

Essential Understanding: Draw a diagonal You have created two right triangles =) In one triangle, draw the altitude from the right angle to the hypotenuse Number the angles as shown Cut out the three triangles. How can you match the angles of the triangles to show that all three triangles are similar? Explain how you know the matching angles are congruent. 9 8 7 6 5 3 4 2 1 Essential Understanding: When you draw the altitude to the hypotenuse of a right triangle, you form three pairs of similar right triangles.

Essential Understanding: 9 8 7 6 5 3 4 2 1 Angles 2, 8 and 9 are all congruent – right angles are congruent Angles 1 and 4 are congruent – alternate interior angles in a parallelogram are congruent Angles 3 and 7 are congruent – alternate interior angles in a parallelogram are congruent Essential Understanding: When you draw the altitude to the hypotenuse of a right triangle, you form three pairs of similar right triangles.

What similarity statement can you write relating the three triangles in the diagram? Z W Y X XYZ ~ YWZ ~ XWY

x b x 15 a = x Geometric Mean What is the geometric mean of 6 and 15? A proportion in which the means are equal. Example What is the geometric mean of 6 and 15? 6 = x x 15 x2 = 90 x = √90 9 ▪ 10 3 ▪ 3 ▪ 2 ▪ 5 x = 3 √10

x 16 x 10 What is the geometric mean of 3 and 16? Your turn: 6 ▪ 8 2 ▪ 3 ▪ 2 ▪ 2 ▪ 2 x = 4 √3 4 = x x 10 x2 = 40 x = √40 4 ▪ 10 2 ▪ 2 ▪ 2 ▪ 5 x = 2 √10

Solve for x and y…. (Pull the triangles apart) Z W Y X y = 3 9 y 3 12 = x x 9 y = 3 √3 x = 6 √3 9 y Z Y X x 12 Z Y W x Y X W y 3 9 y x

Solve for x and y…. (Pull the triangles apart) Z W Y X Solve for x and y…. (Pull the triangles apart) 50 40 X = 20 x Y = 10√5 y Z Y X 50 Z Y W y Y X W x y 10 40 x

7.3 and 7.4 Practice worksheets Your assignment 7.3 and 7.4 Practice worksheets Cross off: 7 & 8 on pg 23 10, 14, 15, 16, 17 on pg 24 and 32 – 38 on pg 34