Similarity proofs: fill in the missing step or reason

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Given: ∠DCF is a right angle, and CQ ⊥ DF. Prove: DC² = DQ * DF. Ladder: 1. ∠DCF is a right angle, and CQ ⊥ DF -- Given; 2. ∠DQC is a right angle -- Definition of perpendicular lines; 3. ∠DQC ≅ ∠DCF -- Right angles congruence theorem; 4. ∠QDC ≅ ∠CDF -- Reflexive property; 5. triangle DQC ~ triangle DCF -- AA similarity postulate; 6. DQ/DC = DC/DF -- Definition of similar triangles; 7. DC² = DQ * DF -- ?.

2. 1.

   Bo draws a ramp as right triangle CDF and drops a support CQ to the slope DF. Which reason fills the blank?

   • Given
   • Pythagorean theorem
   • Distributive property
   • Cross products property

3. Use this

   Given: ∠DCF is a right angle, and CQ ⊥ DF. Prove: FC² = FQ * FD. Ladder: 1. ∠DCF is a right angle, and CQ ⊥ DF -- Given; 2. ∠FQC is a right angle -- Definition of perpendicular lines; 3. ∠FQC ≅ ∠FCD -- Right angles congruence theorem; 4. ∠QFC ≅ ∠CFD -- Reflexive property; 5. triangle FQC ~ triangle FCD -- AA similarity postulate; 6. FQ/FC = FC/FD -- Definition of similar triangles; 7. FC² = FQ * FD -- ?.

4. 2.

   Esme uses the same ramp CDF and support CQ, now working from corner F. Which reason fills the blank?

   • Division property of equality
   • Pythagorean theorem
   • Given
   • Cross products property

5. Use this

   Given: ∠USW is a right angle, and SY ⊥ UW. Prove: US² = UY * UW. Ladder: 1. ∠USW is a right angle, and SY ⊥ UW -- Given; 2. ∠UYS is a right angle -- Definition of perpendicular lines; 3. ∠UYS ≅ ∠USW -- Right angles congruence theorem; 4. ∠YUS ≅ ∠SUW -- Reflexive property; 5. triangle UYS ~ triangle USW -- ?; 6. UY/US = US/UW -- Definition of similar triangles; 7. US² = UY * UW -- Cross products property.

6. 3.

   Jun draws a tent side as right triangle SUW, with a pole SY to the slope UW. Which reason fills the blank?

   • HL
   • ASA
   • AA similarity postulate
   • SAS similarity theorem

7. Use this

   Given: ∠USW is a right angle, and SY ⊥ UW. Prove: WS² = WY * WU. Ladder: 1. ∠USW is a right angle, and SY ⊥ UW -- Given; 2. ∠WYS is a right angle -- Definition of perpendicular lines; 3. ∠WYS ≅ ∠WSU -- Right angles congruence theorem; 4. ∠YWS ≅ ∠SWU -- Reflexive property; 5. triangle WYS ~ triangle WSU -- ?; 6. WY/WS = WS/WU -- Definition of similar triangles; 7. WS² = WY * WU -- Cross products property.

8. 4.

   Kira uses the same tent side SUW and pole SY, now working from corner W. Which reason fills the blank?

   • HL
   • AA similarity postulate
   • SSS similarity theorem
   • AAS
9. 5.

   Luz proves the Pythagorean theorem with similar triangles on a ramp HQU. Which statement fills the blank?

   • QH² + UH² = QU + QF + UF
   • QH² + UH² = QU²
   • QH + UH = QU
   • QH² + UH² = QU(QF + UF)
10. 6.

    Milo proves the Pythagorean theorem with similar triangles on a roof corner WGY. Which statement fills the blank?

    • GC = YC
    • GW² + YW² = GY²
    • GC + GW = GY
    • GC + YC = GY
11. 7.

    Noah proves the Pythagorean theorem with similar triangles on a road sign VSD. Which statement fills the blank?

    • SV² = DV²
    • SV² + DV² = SH · SD + DH · SD
    • SV² + DV² = SD²
    • SV² + DV² = SH · SD
12. 8.

    Tala proves the Pythagorean theorem with similar triangles on a tent side FYH. Which statement and reason fill the blanks?

    • YF² + HF² = YH² — Pythagorean theorem
    • YF² + HF² = YH² — Addition property of equality
    • YF + HF = YH — Substitution property
    • YF² + HF² = YH² — Substitution property

13. Use this

    Ladder: 1. ∠HGV is a right angle, and GD ⊥ HV -- Given; 2. ∠HDG is a right angle -- Definition of perpendicular lines; 3. ∠HDG ≅ ∠HGV -- Right angles congruence theorem; 4. ∠DHG ≅ ∠GHV -- Reflexive property; 5. triangle HDG ~ triangle HGV -- AA similarity postulate; 6. ? -- ?; 7. HG² = HD * HV -- Cross products property.

14. 9.

    Zee draws a roof corner as right triangle GHV, with a beam GD to the slope HV. Which statement and reason fill the blanks?

    • HD/HG = HV/HG — Definition of similar triangles
    • HD/HG = HG/HV — Definition of similar triangles
    • HD/HG = HG/HV — AA similarity postulate
    • HD/HG = GD/HV — Definition of similar triangles

15. Use this

    Given: ∠CYS is a right angle, and YW ⊥ CS. Prove: SY² = SW * SC. Ladder: 1. ∠CYS is a right angle, and YW ⊥ CS -- Given; 2. ∠SWY is a right angle -- Definition of perpendicular lines; 3. ∠SWY ≅ ∠SYC -- Right angles congruence theorem; 4. ∠WSY ≅ ∠YSC -- Reflexive property; 5. triangle SWY ~ triangle SYC -- AA similarity postulate; 6. ? -- ?; 7. SY² = SW * SC -- Cross products property.

16. 10.

    Ari cuts a right-triangle sandwich YCS, then cuts from Y to the long edge CS at a right angle, meeting it at W. Which statement and reason fill the blanks?

    • SW/SY = SY/SC — Definition of similar triangles
    • SW/SY = YW/SC — Definition of similar triangles
    • SW/SY = SC/SY — Definition of similar triangles
    • SW/SY = SY/SC — AA similarity postulate