Quadrilateral proofs: fill in the missing step or reason

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Use this

   Given: DFGH is a parallelogram. Prove: DF ≅ GH. Ladder: 1. DFGH is a parallelogram -- Given; 2. DF ∥ HG and DH ∥ FG -- Definition of parallelogram; 3. ∠FDG ≅ ∠HGD -- Alternate interior angles theorem; 4. ∠HDG ≅ ∠FGD -- Alternate interior angles theorem; 5. DG ≅ DG -- Reflexive property; 6. triangle DFG ≅ triangle GHD -- ASA; 7. DF ≅ GH -- ?.

2. 1.

   Milo proves the parallelogram opposite sides theorem on a window frame DFGH, using diagonal DG. Which reason fills the blank?

   • CPCTC
   • Given
   • Parallelogram opposite sides theorem
   • ASA

3. Use this

   Given: DFGH is a parallelogram. Prove: GH ≅ DF. Ladder: 1. DFGH is a parallelogram -- Given; 2. DF ∥ HG and DH ∥ FG -- Definition of parallelogram; 3. ∠DFH ≅ ∠GHF -- Alternate interior angles theorem; 4. ∠GFH ≅ ∠DHF -- Alternate interior angles theorem; 5. FH ≅ FH -- Reflexive property; 6. triangle FGH ≅ triangle HDF -- ASA; 7. GH ≅ DF -- ?.

4. 2.

   Noah proves the parallelogram opposite sides theorem on a rug DFGH, using diagonal FH. Which reason fills the blank?

   • ASA
   • Parallelogram opposite sides theorem
   • Given
   • CPCTC

5. Use this

   Given: DFGH is a parallelogram. Prove: ∠DFG ≅ ∠GHD. Ladder: 1. DFGH is a parallelogram -- Given; 2. DF ∥ HG and DH ∥ FG -- Definition of parallelogram; 3. ∠FDG ≅ ∠HGD -- ?; 4. ∠HDG ≅ ∠FGD -- Alternate interior angles theorem; 5. DG ≅ DG -- Reflexive property; 6. triangle DFG ≅ triangle GHD -- ASA; 7. ∠DFG ≅ ∠GHD -- CPCTC.

6. 3.

   Pia proves the parallelogram opposite angles theorem on a tile DFGH, using diagonal DG. Which reason fills the blank?

   • CPCTC
   • Converse of the alternate interior angles theorem
   • Alternate interior angles theorem
   • Corresponding angles postulate

7. Use this

   Given: DFGH is a parallelogram. Prove: ∠FGH ≅ ∠HDF. Ladder: 1. DFGH is a parallelogram -- Given; 2. DF ∥ HG and DH ∥ FG -- Definition of parallelogram; 3. ∠DFH ≅ ∠GHF -- Alternate interior angles theorem; 4. ∠GFH ≅ ∠DHF -- ?; 5. FH ≅ FH -- Reflexive property; 6. triangle FGH ≅ triangle HDF -- ASA; 7. ∠FGH ≅ ∠HDF -- CPCTC.

8. 4.

   Ren proves the parallelogram opposite angles theorem on a table top DFGH, using diagonal FH. Which reason fills the blank?

   • CPCTC
   • Alternate exterior angles theorem
   • Alternate interior angles theorem
   • Converse of the alternate interior angles theorem

9. Use this

   Given: DFGH is a parallelogram. Prove: FG ≅ HD. Ladder: 1. DFGH is a parallelogram -- Given; 2. DF ∥ HG and DH ∥ FG -- Definition of parallelogram; 3. ∠FDG ≅ ∠HGD -- Alternate interior angles theorem; 4. ∠HDG ≅ ∠FGD -- Alternate interior angles theorem; 5. DG ≅ DG -- Reflexive property; 6. triangle DFG ≅ triangle GHD -- ?; 7. FG ≅ HD -- CPCTC.

10. 5.

    Tala proves that FG ≅ HD on a parallelogram door panel DFGH, using diagonal DG. Which reason fills the blank?

    • SAS
    • CPCTC
    • ASA
    • Definition of parallelogram
11. 6.

    Ari proves that the braces of a rectangle window DFGH are equal, using the bottom edge DF. Which statement fills the blank?

    • ∠HDF ≅ ∠GFD
    • DH ≅ FG
    • HF ≅ GD
    • ∠HDF ≅ ∠DFH

12. Use this

    Given: DFGH is a rectangle. Prove: DG ≅ FH. Ladder: 1. DFGH is a rectangle -- Given; 2. DFGH is a parallelogram, and ∠DHG and ∠FGH are right angles -- Definition of rectangle; 3. ? -- Right angles congruence theorem; 4. DH ≅ FG -- Parallelogram opposite sides theorem; 5. HG ≅ HG -- Reflexive property; 6. triangle DHG ≅ triangle FGH -- SAS; 7. DG ≅ FH -- CPCTC.

13. 7.

    Bo proves that the braces of a rectangle door DFGH are equal, using the top edge HG. Which statement fills the blank?

    • DG ≅ FH
    • DH ≅ FG
    • ∠DHG ≅ ∠HGD
    • ∠DHG ≅ ∠FGH
14. 8.

    Esme proves that the diagonals of a rectangle book cover DFGH are equal, using the left edge DH. Which statement fills the blank?

    • △FDH ≅ △GHD
    • △FDH ≅ △DHG
    • △FDH ≅ △FGH
    • FH ≅ GD
15. 9.

    Jun proves that the diagonals of a rectangle box lid DFGH are equal, using the right edge FG. Which statement fills the blank?

    • DG ≅ HF
    • △DFG ≅ △GFH
    • △DFG ≅ △FGH
    • △DFG ≅ △HGF

16. Use this

    Given: DFGH is a parallelogram. Prove: HQ ≅ FQ. Ladder: 1. DFGH is a parallelogram -- Given; 2. DH ∥ FG -- Definition of parallelogram; 3. DH ≅ FG -- Parallelogram opposite sides theorem; 4. ? -- Alternate interior angles theorem; 5. ∠DHQ ≅ ∠GFQ -- Alternate interior angles theorem; 6. triangle DHQ ≅ triangle GFQ -- ASA; 7. HQ ≅ FQ -- CPCTC.

17. 10.

    Luz proves that the folds of a parallelogram paper DFGH cut each other in half at Q. Which statement fills the blank?

    • ∠HDQ ≅ ∠DFQ
    • ∠HDQ ≅ ∠FGQ
    • ∠HDQ ≅ ∠QDF
    • HQ ≅ FQ