Similar Triangles and Proving Similarity · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

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Mathematics · Geometry · ages 14-15
Name ______________________   Date ____________

  1. Triangle A has sides 6, 8 and 10. Triangle B has sides 9, 12 and 15. Are they similar, and by which test?

    • Yes, by SSS
    • The triangles are congruent
    • No, the ratios differ

  2. Two triangles have two pairs of equal corresponding angles. What follows?

    • The triangles are similar by AA
    • The triangles are congruent
    • The triangles have the same area

  3. Both triangles have angles of 50, 60 and 70 degrees. What follows?

    • They must be congruent
    • They must be similar
    • Nothing follows

  4. Two known similar triangles have matching corresponding vertices listed in order. Amir pairs shortest with shortest before writing any ratio. Is Amir right?

    Circle one:   True   False

  5. In two similar triangles, the side between the 50 and 60 degree angles in one triangle matches which side in the other?

    • The side between the 50 and 70 degree angles
    • The side between the 50 and 60 degree angles
    • The side opposite the 50 degree angle

  6. Which pair of triangles must be similar?

    • Sides 3, 4, 5 against sides 4, 5, 6
    • Angles 50, 60, 70 in both triangles
    • Any two right triangles

  7. Two similar triangles have matching sides of 3 cm and 12 cm. A second side of the smaller is 5 cm. How long is its match in cm?

    Answer: ______________

  8. Two similar triangles have matching sides of 15 cm and 6 cm, with the 6 cm side in the smaller triangle. A second side of the larger triangle is 25 cm. How long is its match in the smaller triangle, in cm?

    Answer: ______________

  9. Triangle A has sides 4, 6 and 8 cm. Similar triangle B has shortest side 10 cm. How long is the longest side of B in cm?

    Answer: ______________

  10. Ben writes a ratio from mismatched vertices and gets a wrong side, yet his triangles truly are similar. What went wrong?

    • Wrong pairing, wrong scale factor
    • Similarity was never proved
    • Similar triangles never share a scale factor
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Same shape, new size W1-mt_NPRVlr4lrs-s1

  1. Yes, by SSS · Matching shortest to shortest gives 6 with 9, 8 with 12 and 10 with 15. The ratios 9/6, 12/8 and 15/10 are all 1.5, so the triangles are similar by SSS.
  2. The triangles are similar by AA · All three angle pairs equal means similar by AA. Sizes may differ, so congruence does not follow.
  3. They must be similar · All three angles match, so similarity is certain by AA. The sizes may differ, so congruence does not follow.
  4. True · Similarity is given and the vertices are already matched. Pairing shortest with shortest keeps every ratio correct, so Amir is right.
  5. The side between the 50 and 60 degree angles · Equal angles sit opposite matching sides. The side between the 50 and 60 degree angles is opposite the 70 degree angle in both triangles, so it pairs with the side between the same two angles.
  6. Angles 50, 60, 70 in both triangles · Matching angles guarantee similarity; the other pairs can differ.
  7. 20 · The scale factor from small to large is 12/3 = 4, and 5 times 4 is 20.
  8. 10 · The scale factor from larger to smaller is 6/15 = 0.4. Multiplying 25 by 0.4 gives 10 cm.
  9. 20 · The factor is 2.5, and 8 times 2.5 is 20.
  10. Wrong pairing, wrong scale factor · A ratio written from mismatched vertices can give a wrong scale factor and a wrong side, even when the triangles truly are similar.
Worksheet · LightMySky