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?

   • Parallelogram opposite sides theorem
   • ASA
   • Given
   • CPCTC

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?

   • CPCTC
   • ASA
   • Given
   • Parallelogram opposite sides theorem

5. 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.

6. 3.

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

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

7. 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.

8. 4.

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

   • ASA
   • Definition of parallelogram
   • SAS
   • CPCTC
9. 5.

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

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

10. 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.

11. 6.

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

    • ∠DHG ≅ ∠HGD
    • ∠DHG ≅ ∠FGH
    • DG ≅ FH
    • DH ≅ FG
12. 7.

    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
    • FH ≅ GD
    • △FDH ≅ △FGH
13. 8.

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

    • △DFG ≅ △HGF
    • △DFG ≅ △GFH
    • DG ≅ HF
    • △DFG ≅ △FGH
14. 9.

    Kira proves that the sticks of a parallelogram frame DFGH cut each other in half at Q. Which statement fills the blank?

    • DQ ≅ GQ
    • △DFQ ≅ △HGQ
    • △DFQ ≅ △DHQ
    • △DFQ ≅ △GHQ

15. 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.

16. 10.

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

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