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Grade 10 · Math · Geometry

10th grade Similarity proofs: fill in the missing step or reason worksheet

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Similarity proofs: fill in the missing step or reason

Grade 10 Math Worksheet - Challenge Me

NameDate

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

    • Pythagorean theorem
    • Distributive property
    • Given
    • 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. Esme uses the same ramp CDF and support CQ, now working from corner F. Which reason fills the blank?

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

  5. 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.

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

    • AAS
    • SSS similarity theorem
    • AA similarity postulate
    • HL

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

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

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

    • GW² + YW² = GY²
    • GC + YC = GY
    • GC = YC
    • GC + GW = GY

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

    • SV² + DV² = SH · SD
    • SV² = DV²
    • SV² + DV² = SD²
    • SV² + DV² = SH · SD + DH · SD

  10. Pia proves the Pythagorean theorem with similar triangles on a pyramid face UFQ. Which statement fills the blank?

    • FG = QG
    • FG + FU = FQ
    • FG + QG = FQ
    • FU² + QU² = FQ²

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

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

  12. 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.

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

  14. 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.

  15. 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 = SC/SY — Definition of similar triangles
    • SW/SY = SY/SC — AA similarity postulate
    • SW/SY = YW/SC — Definition of similar triangles
    • SW/SY = SY/SC — Definition of similar triangles
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10th grade Similarity proofs: fill in the missing step or reason worksheet – printable
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About this worksheet

Similarity proofs: fill in the missing step or reason. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can name the reason for each step of a proof with similar triangles.
  • I can tell a similarity rule from a congruence rule, and the triangle proportionality theorem from its converse.
  • I can follow the similar-triangles proof of the Pythagorean theorem.

Standard: G-SRT.B.4, G-SRT.B.5

Words to know

rung: One line of the proof: a statement with its reason beside it.

Given / Prove: Given: the facts we start with. Prove: the statement the last rung must reach.

converse: The if-then turned around; it is a different statement, true or not.

similar (~): Same shape, possibly a different size: matching angles congruent, sides in one ratio.

AA similarity postulate: Two pairs of congruent angles make two triangles similar.

triangle proportionality theorem: A line parallel to one side cuts the other two sides in the same ratio.

cross products property: If a/b = c/d, then ad = bc.

distributive property: a · b + c · b = b(a + c).

Common mistakes

Your child used a congruence rule (such as ASA) or said "congruent" for triangles that are only similar: the same shape at a different size. Similar triangles use AA, SAS or SSS similarity.

Your child wrote "Given" for the statement the proof is trying to reach. Only the facts on the Given line are given. Cover the Prove line and ask what is known.

Your child named the converse of the triangle proportionality theorem when the theorem itself was needed. The theorem goes from a parallel line to equal ratios; the converse goes back.

What to say

Ask: which fact do we have?

Questions parents ask

What is the most common mistake with similarity proofs: fill in the missing step or reason?

Your child used a congruence rule (such as ASA) or said "congruent" for triangles that are only similar: the same shape at a different size. Similar triangles use AA, SAS or SSS similarity.

Are these 10th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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