SIMILARITY, CONGRUENCE, AND PROOFS Review Booklet SIMILARITY, CONGRUENCE, AND PROOFS

5 Ways to Prove Triangles Congruent SSS: All 3 sides are exactly the same SAS: 2 congruent sides and the angle in between ASA: 2 congruent angles are the side in between AAS: 2 congruent angles and a side NOT in between HL: ONLY FOR RIGHT TRIANGLES – Hypotenuse and 1 Leg

Match up corresponding parts. CONGRUENCE STATEMENT Order matters! Match up corresponding parts. Example: ABC  DEF

Triangle Sum Theorem The 3 angles in a triangle add up and equal ______. 180

Exterior Angle Theorem The 2 remote interior angles add up and equal the exterior angle Remote Angle Exterior Angle Remote Angle

Isosceles Triangle 2 congruent sides Opposite of the congruent sides are congruent angles

Rigid Motion – the shape will still be congruent after the move Reflection Translation Rotation

Dilate the figure by 1/2. Use the origin as the center of dilation.

Find the center of dilation

Similar Polygons Corresponding angles are congruent Corresponding sides are proportional Similarity Statement

Solve for x and y. x = 26 cm y = 12 cm L A 5 cm x S 10 cm y 13 cm C B T x = 26 cm y = 12 cm

In similar triangles, angles are congruent and sides are proportional Find the missing angle measures. A L 53 S C B 37 T

Find the perimeter of the smaller triangle. 12 cm 4 cm Perimeter = x Perimeter = 60 cm x = 20 cm

3 ways to Prove Triangles Similar Angle-Angle (AA~) Similarity Postulate Side-Side-Side (SSS~) Similarity Theroem Side-Angle-Side (SAS~) Similarity Thm

Determine whether the triangles are similar Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement. Yes, AA~ 68° 43° 68° 43° V NO Y 7 3 5 Z X W 11 U

SAS~ 12 4 5 15 Prove that RST ~ PSQ 1. Two sides are proportional 2. Included angle is congruent SAS~ R S T P Q 12 4 5 15

A tree cast a shadow 18 feet long A tree cast a shadow 18 feet long. At the same time a person who is 6 feet tall cast a shadow 4 feet long. How tall is the tree?

Review Booklet Right Triangle

The Pythagorean Theorem “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a2 + b2 = c2

Ladder Problem Solution a2 + b2 = c2 72 + b2 = 252 49 + b2 = 625 b2 = 576 b = 24 m How did you do?

Trig Ratios

Trig Ratio What is cos R? What is sin R? What is tan R?

Co-Function Relationships

Co-Function Relationships Cos 64 = Sin ____ 26

Find a Missing Side Solve for x. Round to the nearest tenth. x x = 17.6

 = 31.4 Find a Missing Angle  Solve for . Round to the nearest tenth.   = 31.4

The angle of elevation from a ship to the top of a 35 meter lighthouse on the coast measures 26. How far from the coast is the ship? Round to the nearest tenth. tan 26 = 35/x x = 71.8 m