Quadrilateral proofs: fill in the missing step or reason

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Marked: DF and HG with arrows and one tick each. Given: DF ∥ HG and DF ≅ HG. Prove: DFGH is a parallelogram. Ladder: 1. DF ∥ HG -- Given; 2. DF ≅ HG -- Given; 3. ∠DFH ≅ ∠GHF -- Alternate interior angles theorem; 4. FH ≅ FH -- Reflexive property; 5. triangle DFH ≅ triangle GHF -- SAS; 6. ∠DHF ≅ ∠GFH -- CPCTC; 7. DH ∥ FG -- ?; 8. DFGH is a parallelogram -- Definition of parallelogram.

2. 1.

   Noah builds a window frame DFGH with DF and HG parallel and the same length. Which reason fills the blank?

   • Alternate interior angles theorem
   • Definition of parallelogram
   • Converse of the alternate interior angles theorem
   • Converse of the corresponding angles postulate

3. Use this

   Given: DH ∥ FG and DH ≅ FG. Prove: DFGH is a parallelogram. Ladder: 1. DH ∥ FG -- Given; 2. DH ≅ FG -- Given; 3. ∠HDG ≅ ∠FGD -- Alternate interior angles theorem; 4. DG ≅ DG -- Reflexive property; 5. triangle HDG ≅ triangle FGD -- ?; 6. ∠HGD ≅ ∠FDG -- CPCTC; 7. DF ∥ HG -- Converse of the alternate interior angles theorem; 8. DFGH is a parallelogram -- Definition of parallelogram.

4. 2.

   Pia builds a door frame DFGH with DH and FG parallel and the same length. Which reason fills the blank?

   • ASA
   • SAS
   • SSS
   • CPCTC

5. Use this

   Given: DH ∥ FG and DH ≅ FG. Prove: DFGH is a parallelogram. Ladder: 1. DH ∥ FG -- Given; 2. DH ≅ FG -- Given; 3. ∠DHF ≅ ∠GFH -- Alternate interior angles theorem; 4. FH ≅ FH -- Reflexive property; 5. triangle DHF ≅ triangle GFH -- ?; 6. ∠DFH ≅ ∠GHF -- CPCTC; 7. DF ∥ HG -- Converse of the alternate interior angles theorem; 8. DFGH is a parallelogram -- Definition of parallelogram.

6. 3.

   Ren builds a box lid DFGH with DH and FG parallel and the same length. Which reason fills the blank?

   • CPCTC
   • ASA
   • SSS
   • SAS

7. Use this

   Given: DFGH is a rhombus. Prove: DG bisects ∠FGH. Ladder: 1. DFGH is a rhombus -- Given; 2. DFGH is a parallelogram, DF ≅ DH and FG ≅ HG -- Definition of rhombus; 3. DG ≅ DG -- Reflexive property; 4. triangle DFG ≅ triangle DHG -- SSS; 5. ? -- CPCTC; 6. DG bisects ∠FGH -- Definition of angle bisector.

8. 4.

   Zee checks that the line DG on a rhombus tile DFGH splits the corner at G evenly. Which statement fills the blank?

   • ∠FGD ≅ ∠FGH
   • ∠FGD ≅ ∠DFG
   • ∠FGD ≅ ∠HGD
   • DG bisects ∠FGH
9. 5.

   Ari checks that the fold FH of a rhombus envelope flap DFGH splits the corner at F evenly. Which statement fills the blank?

   • FH bisects ∠DFG
   • △FDH ≅ △GFH
   • △FDH ≅ △FGH
   • △FDH ≅ △QFG

10. Use this

    Given: DFGH is a rhombus. Prove: FH bisects ∠DHG. Ladder: 1. DFGH is a rhombus -- Given; 2. DFGH is a parallelogram, FD ≅ FG and DH ≅ GH -- Definition of rhombus; 3. FH ≅ FH -- Reflexive property; 4. triangle FDH ≅ triangle FGH -- SSS; 5. ? -- CPCTC; 6. FH bisects ∠DHG -- Definition of angle bisector.

11. 6.

    Bo checks that the seam FH of a rhombus quilt patch DFGH splits the corner at H evenly. Which statement fills the blank?

    • ∠DHF ≅ ∠HDF
    • ∠DHF ≅ ∠GHF
    • ∠DHF ≅ ∠DHG
    • FH bisects ∠DHG

12. Use this

    Sketch, not to scale: quadrilateral DFGH (D bottom left, F bottom right, G top right, H top left), with diagonals DG and FH crossing at Q. Marked: one tick on each side. Given: DFGH is a rhombus. Prove: FH ⊥ DG. Ladder: 1. DFGH is a rhombus -- Given; 2. DFGH is a parallelogram, and DF ≅ DH -- Definition of rhombus; 3. FQ ≅ QH -- Parallelogram diagonals theorem; 4. DQ ≅ DQ -- Reflexive property; 5. triangle FQD ≅ triangle HQD -- SSS; 6. ∠FQD ≅ ∠HQD -- CPCTC; 7. ? -- ?.

13. 7.

    Esme proves the rhombus diagonals theorem on a rhombus kite DFGH, using the triangles at D. Which statement and reason fill the blanks?

    • FH ⊥ DG — Linear pair perpendicular theorem
    • FH ⊥ DG — Rhombus diagonals theorem
    • FH ≅ DG — Rectangle diagonals theorem
    • FH ∥ DG — Linear pair perpendicular theorem

14. Use this

    Ladder: 1. DFGH is a rhombus -- Given; 2. DFGH is a parallelogram, and GF ≅ GH -- Definition of rhombus; 3. FQ ≅ QH -- Parallelogram diagonals theorem; 4. GQ ≅ GQ -- Reflexive property; 5. triangle FQG ≅ triangle HQG -- SSS; 6. ∠FQG ≅ ∠HQG -- CPCTC; 7. ? -- ?.

15. 8.

    Jun proves the rhombus diagonals theorem on a rhombus tile DFGH, using the triangles at G. Which statement and reason fill the blanks?

    • FH ⊥ DG — Rhombus diagonals theorem
    • FH ⊥ DG — Linear pair perpendicular theorem
    • FH ∥ DG — Linear pair perpendicular theorem
    • FH ≅ DG — Rectangle diagonals theorem
16. 9.

    Kira measures a 72° corner at D on a parallelogram rug DFGH. Which statement and reason fill the blanks?

    • ∠HDF and ∠DFG are complementary — Same-side interior angles theorem
    • ∠HDF ≅ ∠DFG — Parallelogram opposite angles theorem
    • ∠HDF and ∠DFG are supplementary — Same-side interior angles theorem
    • ∠HDF and ∠DFG are supplementary — Alternate interior angles theorem
17. 10.

    Luz measures a 64° corner at F on a parallelogram table top DFGH. Which statement and reason fill the blanks?

    • ∠DFG and ∠FGH are supplementary — Same-side interior angles theorem
    • ∠DFG and ∠FGH are complementary — Same-side interior angles theorem
    • ∠DFG ≅ ∠FGH — Parallelogram opposite angles theorem
    • ∠DFG and ∠FGH are supplementary — Alternate interior angles theorem