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Parallelograms Special Quadrilaterals Test Answer Key with Scoring – Geometry

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

Geometry Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for Geometry
Looking for a parallelograms special quadrilaterals test answer key? Then this answer key gives the solutions and the scoring guide of the Geometry test on Quadrilateral Properties: Parallelograms to Kites. Each answer shows the property, the work and the scoring split. For proofs, one valid chain is shown and any logically complete chain earns the same credit.

Standards: G-CO.C.11 · Level: Grade 10 Geometry

Question 1 Parallelogram sides (2 points)

Opposite sides are congruent: 7x − 9 = 4x + 12, so 3x = 21 and x = 7. Then PQ = 7(7) − 9 = 40.

x = 7; PQ = 40

Scoring: 1 point for x, 1 point for PQ.

Question 2 Parallelogram angles (3 points)

  1. Consecutive angles are supplementary: (5x + 8) + (7x + 16) = 180, so 12x + 24 = 180 and x = 13.
  2. ∠P = 5(13) + 8 = 73° and ∠Q = 7(13) + 16 = 107°.
  3. Opposite angles are congruent: ∠R = 73° and ∠S = 107°.

x = 13; 73°, 107°, 73°, 107°

Scoring: 1 point per part.

Question 3 Diagonal pieces (2 points)

The diagonals bisect each other: 2x + 5 = 4x − 11, so 16 = 2x and x = 8. Then AE = 21 and AC = 42.

x = 8; AC = 42

Scoring: 1 point for x, 1 point for AC.

Question 4 Test for a parallelogram (3 points)

  1. Yes: diagonals that bisect each other prove a parallelogram.
  2. No: an isosceles trapezoid has one pair of parallel sides and a pair of congruent legs.
  3. Yes: both pairs of opposite angles congruent is a parallelogram test.

(a) yes, (b) no, (c) yes

Scoring: 1 point per part (answer and reason).

Question 5 Rhombus measures (3 points)

ABCDE815
Solved figure: AE = 8, BE = 15, BD = 30.
  1. The diagonals are perpendicular and bisect each other, so AE = 8 and BE = √(17² − 8²) = √225 = 15.
  2. BD = 2 · 15 = 30.
  3. Area = ½ · 16 · 30 = 240.

BE = 15; BD = 30; area 240

Scoring: 1 point per part.

Question 6 Prove diagonals give a parallelogram (3 points)

Statement Reason
1. AE ≅ CE and BE ≅ DE Given
2. ∠AEB ≅ ∠CED Vertical angles are congruent
3. △AEB ≅ △CED SAS
4. AB ≅ CD and ∠EAB ≅ ∠ECD CPCTC
5. AB ∥ CD Converse of the Alternate Interior Angles Theorem

Scoring: 1 point for steps 2 and 3, 1 point for step 4, 1 point for the parallel conclusion with its reason.

Question 7 Trapezoids (2 points)

The midsegment is the average of the bases: (14 + 26) ÷ 2 = 20.

The angles along a leg are supplementary: 180° − 65° = 115°.

Midsegment 20; adjacent angle 115°

Scoring: 1 point per part.

Question 8 Kite angles (2 points)

The angles between the unequal sides are congruent, so ∠D = ∠B = 112°. The four angles add to 360°: ∠C = 360° − 76° − 112° − 112° = 60°.

∠D = 112°; ∠C = 60°

Scoring: 1 point per angle.

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