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

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

2. 1.

   Milo builds a table top DFGH with DF and HG parallel and the same length. Which reason fills the blank?

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

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?

   • CPCTC
   • ASA
   • SAS
   • SSS

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?

   • SSS
   • SAS
   • CPCTC
   • ASA
7. 4.

   Tala checks that stick DG of a rhombus kite DFGH splits the corner at D evenly. Which statement fills the blank?

   • △DFG ≅ △QFG
   • △DFG ≅ △HDG
   • DG bisects ∠HDF
   • △DFG ≅ △DHG

8. 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.

9. 5.

   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 ≅ ∠HGD
   • DG bisects ∠FGH
   • ∠FGD ≅ ∠DFG
10. 6.

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

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

11. 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.

12. 7.

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

13. 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. ? -- ?.

14. 8.

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

15. 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. ? -- ?.

16. 9.

    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 — Rectangle diagonals theorem
    • FH ⊥ DG — Rhombus diagonals theorem
    • FH ⊥ DG — Linear pair perpendicular theorem
    • FH ∥ DG — Linear pair perpendicular theorem
17. 10.

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