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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

NameDate

Circle the best answer.

  1. Read this first

    Use this

    Given: DH ∥ FG. Prove: DQ/GQ = HQ/FQ. Ladder: 1. DH ∥ FG -- Given; 2. ∠HDQ ≅ ∠FGQ -- Alternate interior angles theorem; 3. ∠DQH ≅ ∠GQF -- Vertical angles theorem; 4. triangle DQH ~ triangle GQF -- AA similarity postulate; 5. DQ/GQ = HQ/FQ -- ?.

  2. Bo draws a bike frame where bars DG and FH cross at Q, with DH ∥ FG. Which reason fills the blank?

    • Definition of similar triangles
    • Triangle proportionality theorem
    • Given
    • AA similarity postulate

  3. Use this

    Given: HD ∥ YU. Prove: HF/UF = DF/YF. Ladder: 1. HD ∥ YU -- Given; 2. ∠DHF ≅ ∠YUF -- Alternate interior angles theorem; 3. ∠HFD ≅ ∠UFY -- Vertical angles theorem; 4. triangle HFD ~ triangle UFY -- ?; 5. HF/UF = DF/YF -- Definition of similar triangles.

  4. Jun opens scissors with blades HU and YD crossing at F, with HD ∥ YU. Which reason fills the blank?

    • AA similarity postulate
    • SAS similarity theorem
    • AAS
    • Vertical angles theorem

  5. Luz measures a stripe VQ across a road sign HCS, level with its base CS. Which reason fills the blank?

    • Substitution property
    • Division property of equality
    • Cross products property
    • Triangle proportionality theorem

  6. Milo measures a cord DW across a tent front YFG, level with the floor FG. Which reason fills the blank?

    • Distributive property
    • Cross products property
    • Definition of similar triangles
    • Division property of equality

  7. Use this

    Given: YF ∥ VC. Prove: YH/CH = FH/VH. Ladder: 1. YF ∥ VC -- Given; 2. ? -- Alternate interior angles theorem; 3. ∠YHF ≅ ∠CHV -- Vertical angles theorem; 4. triangle YHF ~ triangle CHV -- AA similarity postulate; 5. YH/CH = FH/VH -- Definition of similar triangles.

  8. Noah draws scissors whose blades YC and VF cross at H, with YF ∥ VC. Which statement fills the blank?

    • ∠FYH ≅ ∠VCH
    • ∠FYH ≅ ∠CVH
    • YH/CH = FH/VH
    • ∠FYH ≅ ∠YFH

  9. Use this

    Sketch, not to scale: segments UH and SQ cross at G (U upper left, H lower right, Q upper right, S lower left). Marked: arrows on UQ and SH. Given: UQ ∥ SH. Prove: UG/HG = QG/SG. Ladder: 1. UQ ∥ SH -- Given; 2. ∠QUG ≅ ∠SHG -- Alternate interior angles theorem; 3. ∠UGQ ≅ ∠HGS -- Vertical angles theorem; 4. triangle UGQ ~ triangle HGS -- AA similarity postulate; 5. ? -- Definition of similar triangles.

  10. Pia draws bike-rack bars UH and SQ that cross at G, with UQ ∥ SH. Which statement fills the blank?

    • UG/HG = QG/SG
    • UG/HG = SG/QG
    • UQ/SH = UG/SG
    • UG ≅ HG

  11. Ren measures a cut HU across a sandwich half WQC, level with the edge QC. Which statement fills the blank?

    • 3/5 = UC/9
    • 3/8 = 9/UC
    • 3/5 = 9/UC
    • UC = 15

  12. Use this

    Given: CY ∥ QW. Prove: CV/WV = YV/QV. Ladder: 1. CY ∥ QW -- Given; 2. ∠YCV ≅ ∠QWV -- Alternate interior angles theorem; 3. ∠CVY ≅ ∠WVQ -- Vertical angles theorem; 4. triangle CVY ~ triangle WVQ -- AA similarity postulate; 5. ? -- Definition of similar triangles.

  13. Tala crosses craft sticks CW and QY at V, with CY ∥ QW. Which statement fills the blank?

    • CV ≅ WV
    • CV/WV = QV/YV
    • CV/WV = YV/QV
    • CY/QW = CV/QV

  14. Zee draws scissors whose blades FS and HG cross at D, with FG ∥ HS. Which statement fills the blank?

    • △FDG ~ △FDH
    • △FDG ~ △SDH
    • △FDG ~ △HDS
    • △FDG ≅ △SDH

  15. Use this

    Marked: arrows on VU and DQ. Given: VU ∥ DQ. Prove: VW/QW = UW/DW.

    Ladder: 1. VU ∥ DQ -- Given; 2. ∠UVW ≅ ∠DQW -- Alternate interior angles theorem; 3. ∠VWU ≅ ∠QWD -- Vertical angles theorem; 4. ? -- AA similarity postulate; 5. VW/QW = UW/DW -- Definition of similar triangles.

  16. Ari ties tent cords VQ and DU that cross at W, with VU ∥ DQ. Which statement fills the blank?

    • △VWU ~ △VWD
    • △VWU ~ △QWD
    • △VWU ~ △DWQ
    • △VWU ≅ △QWD
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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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