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10th grade Quadrilateral proofs: fill in the missing step or reason worksheet

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Quadrilateral proofs: fill in the missing step or reason

Grade 10 Math Worksheet - Challenge Me

NameDate

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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10th grade Quadrilateral proofs: fill in the missing step or reason worksheet – printable
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About this worksheet

Quadrilateral proofs: fill in the missing step or reason. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can name the reason for each step of a proof about a parallelogram, rectangle or rhombus.
  • I can tell which facts belong to every parallelogram and which only to a rectangle or rhombus.
  • I can use congruent triangles inside a quadrilateral, then CPCTC.

Words to know

rung: One line of the proof: a statement with its reason beside it.

Given / Prove: Given: the facts we start with. Prove: the statement the last rung must reach.

converse: The if-then turned around; it is a different statement, true or not.

parallelogram: A quadrilateral with both pairs of opposite sides parallel.

rectangle / rhombus: A parallelogram with four right angles / with four congruent sides.

opposite / next-door angles: Angles across the shape / angles that share a side.

right angles congruence theorem: All right angles are congruent.

linear pair perpendicular theorem: Two lines making a congruent linear pair are perpendicular.

Common mistakes

Your child wrote "Given" for the statement the proof is trying to reach. Only the facts on the Given line are given. Cover the Prove line and ask what is known.

Your child mixed up two facts: the definition of a parallelogram gives parallel opposite sides, and a theorem gives congruent (equal) opposite sides.

Your child used a fact that only a rectangle has, on a shape known only to be a parallelogram. Draw a leaning parallelogram and check whether the fact still holds.

Questions parents ask

What is the most common mistake with quadrilateral proofs: fill in the missing step or reason?

Your child wrote "Given" for the statement the proof is trying to reach. Only the facts on the Given line are given. Cover the Prove line and ask what is known.

Are these 10th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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