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Review Booklet SIMILARITY, CONGRUENCE, AND PROOFS 5 Ways to Prove Triangles Congruent 1. SSS: All 3 sides are exactly the same 2. SAS: 2 congruent sides and the angle in between 3. ASA: 2 congruent angles are the side in between 4. AAS: 2 congruent angles and a side NOT in between 5. HL: ONLY FOR RIGHT TRIANGLES – Hypotenuse and 1 Leg CONGRUENCE STATEMENT Order matters! Match up corresponding parts. Example: ABC DEF Triangle Sum Theorem The 3 angles in a triangle add 180 up and equal ______. 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 1. Reflection 2. Translation 3. Rotation Dilate the figure by 1/2. Use the origin as the center of dilation. A 4,4 A ' 2,2 B 2, 6 B ' 1, 3 C 6,0 C ' 3,0 Find the center of dilation Center 2,2 Similar Polygons 1. Corresponding angles are congruent 2. Corresponding sides are proportional 3. Similarity Statement ABC ~ DEF Solve for x and y. ABC ~ SLT L A 10 cm B x 24 cm x = 26 cm y 5 cm S 13 cm C T y = 12 cm In similar triangles, angles are congruent and sides are proportional ABC ~ SLT Find the missing angle measures. A L 53 S B C 37 T mC 37 mL 90 mS 53 Find the perimeter of the smaller triangle. 12 cm Perimeter = 60 cm 4 cm Perimeter = x x = 20 cm 3 ways to Prove Triangles Similar 1)Angle-Angle (AA~) Similarity Postulate 2)Side-Side-Side (SSS~) Similarity Theroem 3)Side-Angle-Side (SAS~) Similarity Thm Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement. 68° 43° 68° 43° V Y 7 W 11 3 U X 5 Z Prove that RST ~ PSQ 1. Two sides are proportional SAS~2. Included angle is congruent S 4 P 12 R 5 16 20 4 5 4 4 1 1 Q 15 T S S reflexive 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? tree's shadow tree's height person's shadow person's height 18 x 4 6 x 27 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.” a 2 + b 2 = c2 Ladder Problem Solution a2 + b2 = c2 72 + b2 = 252 49 + b2 = 625 b2 = 576 b = 24 m How did you do? A=7m Trig Ratios Trig Ratio What is cos R? What is sin R? What is tan R? 21 29 20 29 20 21 Co-Function Relationships sin cos(90 ) cos sin(90 ) 1 tan tan(90 ) Co-Function Relationships 26 Cos 64 = Sin ____ Find a Missing Side Solve for x. Round to the nearest tenth. x = 17.6 x 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