Similarity proofs: fill in the missing step or reason
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
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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 -- ?.
Esme uses the same ramp CDF and support CQ, now working from corner F. Which reason fills the blank?
- Given
- Pythagorean theorem
- Cross products property
- Division property of equality
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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.
Kira uses the same tent side SUW and pole SY, now working from corner W. Which reason fills the blank?
- SSS similarity theorem
- AA similarity postulate
- HL
- AAS
Luz proves the Pythagorean theorem with similar triangles on a ramp HQU. Which statement fills the blank?
- QH + UH = QU
- QH² + UH² = QU(QF + UF)
- QH² + UH² = QU + QF + UF
- QH² + UH² = QU²
Milo proves the Pythagorean theorem with similar triangles on a roof corner WGY. Which statement fills the blank?
- GW² + YW² = GY²
- GC = YC
- GC + YC = GY
- GC + GW = GY
Noah proves the Pythagorean theorem with similar triangles on a road sign VSD. Which statement fills the blank?
- SV² + DV² = SH · SD
- SV² + DV² = SD²
- SV² = DV²
- SV² + DV² = SH · SD + DH · SD
Pia proves the Pythagorean theorem with similar triangles on a pyramid face UFQ. Which statement fills the blank?
- FG + FU = FQ
- FG = QG
- FU² + QU² = FQ²
- FG + QG = FQ
Ren proves the Pythagorean theorem with similar triangles on a ramp DWC. Which statement and reason fill the blanks?
- WD² + CD² = WC² — Addition property of equality
- WD + CD = WC — Substitution property
- WD² + CD² = WC² — Pythagorean theorem
- WD² + CD² = WC² — Substitution property
Tala proves the Pythagorean theorem with similar triangles on a tent side FYH. Which statement and reason fill the blanks?
- YF + HF = YH — Substitution property
- YF² + HF² = YH² — Addition property of equality
- YF² + HF² = YH² — Pythagorean theorem
- YF² + HF² = YH² — Substitution property
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.
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 = HG/HV — AA similarity postulate
- HD/HG = HG/HV — Definition of similar triangles
- HD/HG = GD/HV — Definition of similar triangles
- HD/HG = HV/HG — Definition of similar triangles
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.
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 = YW/SC — Definition of similar triangles
- SW/SY = SY/SC — AA similarity postulate
- SW/SY = SY/SC — Definition of similar triangles
- SW/SY = SC/SY — Definition of similar triangles