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For each pair of triangles below, determine whether or not the triangles are similar. If they are similar, show your reasoning in a flowchart. If they are not similar, explain how you know.  

  1. Two triangles. First is triangle A, B, C, with interior angles 36 degrees on angle A and 42 degrees on angle C. Second is triangle X, Y, Z, with interior angles 42 degrees on angle Y and 102 degrees on angle X.

    measure of Angle YXZ = 102 degrees,  measure of angle BAC = 36 degrees, and  Angle ACB is congruent to angle ZYX.

    measure of Angle CBA = 102 degrees and measure of Angle XZY = 36 degrees, both by the Triangle Angle Sum Theorem.

    All three angles are the same in both triangles.

    Triangle A, B, C is similar to triangle X, Y, Z by A, A similarity.

  2. Two triangles. First has two sides, each 6 units, and the base, 10 units. The angle opposite the side, 10, has 1 tick mark. Second has the following side lengths: 5 on top side, 3 on bottom side, and 7 on right side.The angle opposite the side, 7, has 1 tick mark

    The triangles are not similar because corresponding sides do not have the same ratio. One triangle is isosceles and the other is not.

  3. Two right triangles. First is Triangle F, E, D with legs F, E, = 3 and E, D = 4. Second is triangle B, U, G with side lengths 6 on side B, U, and 10 on side B, G, the hypotenuse.

    Proof with 10 ovals. 1 & 2 point to 3. 4 & 5 point to 6. 7 and 8 point to 9. 3, 6, and 9 point to 10. 1 and 8 have the justification “Pythagorean theorem”. 2, 4, 5, and 7 have the justification, “given”. Labels in ovals: #1, F D = 5, #2, B G = 10, #3, F D divided by B G = one half, #4, F E = 3, #5, B U = 6, #6, F E divided by B U = one half, #7, E D = 4, #8, U G = 8, #9, E D divided by U G = one half, #10: Triangle F E D is similar to triangle B U G. 10 has the justification “S S S similarity”