Similar figures: corresponding sides, scale factor, and all circles are similar

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   No side lengths are given. Ari plans two triangle sails. In △ABC and △DEF, ∠A = ∠D = 50° and ∠B = ∠E = 60°. What proves they are similar?

   • Not enough: a side length is needed
   • AA: two pairs of equal angles are enough
   • SAS~: two sides in a ratio and the angle between them
   • SSS~: the three pairs of sides are in one ratio
2. 2.

   Bo compares two tent fronts. △ABC has sides 8, 12, 18 and △DEF has sides 12, 18, 27. No angles are given. What proves they are similar?

   • AA: the angles must be equal, because the sides are in a ratio
   • SAS~: two sides in a ratio and the angle between them
   • SSS~: 8/12, 12/18 and 18/27 are all 2/3
   • Not enough: the angles are needed too

3. Read this first

   Use this

   Two triangles. AB = 6, DE = 4, AC = 9, DF = 6, and ∠A = ∠D = 55°. BC and EF are not given. Lengths in cm.

4. 3.

   Esme checks two sail triangles. AB/DE = AC/DF = 3/2 and ∠A = ∠D = 55°. The third sides are unknown. What proves they are similar?

   • Not enough: the third pair of sides is needed
   • SAS~: two sides in one ratio with the equal angle between them
   • SSS~: the three pairs of sides are in one ratio, including BC and EF
   • AA: two pairs of equal angles
5. 4.

   AC and DF are not given. Jun checks two map triangles. AB/DE = BC/EF = 3/2 and ∠A = ∠D = 55°. What can Jun conclude?

   • Not enough: the equal angle is not between the two proportional sides
   • AA: two pairs of equal angles
   • SAS~: two sides are in a ratio and an angle is equal, so the triangles are similar
   • SSS~: the three pairs of sides are in one ratio
6. 5.

   Given: AB/DE = AC/DF = 3/2. Nothing else is known. Kira knows AB/DE = AC/DF = 3/2 for two triangles. Which fact completes a proof of similarity by SAS~?

   • AB = DE, the first sides are equal
   • BC = EF, the third sides are equal
   • ∠A = ∠D, the angle between AB and AC
   • ∠B = ∠E, an angle that is not between those sides
7. 6.

   Nothing else is known. Milo knows ∠C = ∠F = 90° in two triangles. Which fact completes a proof of similarity by AA?

   • BC = EF, one pair of legs is equal
   • AB = DE, the hypotenuses are equal
   • AC/DF = 2, one pair of legs in a ratio
   • ∠A = ∠D, a second pair of equal angles
8. 7.

   Noah marks D on AB and E on AC of △ABC with DE ∥ BC. Which facts prove △ADE ~ △ABC?

   • ∠A is shared, and ∠ADE = ∠AED because DE is parallel
   • ∠A is shared, and ∠ADE = ∠ABC as corresponding angles
   • ∠ADE = ∠DEA as alternate interior angles
   • DE = BC, because parallel segments are equal
9. 8.

   Ren marks D on AB and E on AC. AD = 3, DB = 6, AE = 4 and EC = 8. Is DE parallel to BC?

   • Yes: AD/AB = AE/EC
   • Yes: AD/DB = AE/EC = 1/2, so DE ∥ BC by the converse
   • Not enough: DE and BC are needed
   • No: AD and AE are different lengths, so the lines cannot be parallel
10. 9.

    Tala marks D on AB and E on AC. AD = 3, DB = 6, AE = 5 and EC = 8. Is DE parallel to BC?

    • No: AD and AE are different lengths
    • Yes: both ratios are close to one half, so the lines are close enough to parallel
    • No: AD/DB = 1/2 but AE/EC = 5/8, so DE is not parallel to BC
    • Yes: D and E lie on the sides of the triangle
11. 10.

    Zee draws DE ∥ BC with D on AB and E on AC. AD = 4, AB = 10 and AE = 6. Which equation finds AC?

    • 4/6 = 6/AC, mixing AD, DB and AE
    • 4/10 = AC/6, with the ratio turned over
    • 4/10 = DE/6
    • 4/10 = 6/AC, from AD/AB = AE/AC