Similarity proofs: fill in the missing step or reason

Grade 10 Math Worksheet

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. 1.

   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
   • AA similarity postulate
   • Triangle proportionality theorem
   • Given
3. 2.

   Esme crosses craft sticks SV and UW at C, with SW ∥ UV. Which reason fills the blank?

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

4. 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.

5. 3.

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

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

6. Use this

   Ladder: 1. WC ∥ GD -- Given; 2. ∠CWS ≅ ∠GDS -- Alternate interior angles theorem; 3. ∠WSC ≅ ∠DSG -- Vertical angles theorem; 4. triangle WSC ~ triangle DSG -- ?; 5. WS/DS = CS/GS -- Definition of similar triangles.

7. 4.

   Kira draws two ramps WD and GC that cross at S, with WC ∥ GD. Which reason fills the blank?

   • ASA
   • Vertical angles theorem
   • AA similarity postulate
   • SSS similarity theorem
8. 5.

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

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

9. 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.

10. 6.

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

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

11. 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.

12. 7.

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

    • UQ/SH = UG/SG
    • UG/HG = SG/QG
    • UG ≅ HG
    • UG/HG = QG/SG
13. 8.

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

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

14. 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.

15. 9.

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

    • CV/WV = YV/QV
    • CY/QW = CV/QV
    • CV/WV = QV/YV
    • CV ≅ WV
16. 10.

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

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