Relate Adjacent Parallelogram Angles

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A parallelogram has one interior angle 40°. Find an adjacent interior angle. Candidate answers: (A) 139 ° (B) 140 ° (C) 141 ° (D) 280 °.
• Independent check inputs: A parallelogram has one interior angle 43°. Find an adjacent interior angle. Candidate answers: (A) 136 ° (B) 137 ° (C) 138 ° (D) 274 °.

Learn the idea

Geometric properties can be expressed through coordinates as well as diagrams. Parallelogram angle and diagonal theorems, midpoint coordinates, perpendicular slopes and perpendicular base-height relationships provide exact arguments about figures. Opposite parallelogram sides are parallel. A neighboring side acts as a transversal, so adjacent interior angles are supplementary, while opposite interior angles are congruent.

Here is the method on the supplied teaching case. Adjacent angles in a parallelogram are supplementary because opposite sides are parallel. Thus the adjacent angle is 180−40=140°. Opposite angles, rather than adjacent ones, are equal.

Choose adjacent rather than opposite angles for the 180-degree relation. A general quadrilateral need not have this property; the parallelogram condition supplies the required parallelism.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A parallelogram has one interior angle 61°. Find an adjacent interior angle. Candidate answers: (A) 118 ° (B) 119 ° (C) 120 ° (D) 238 °. Show the mathematical reason, not only the selected answer. Expected result: 119 °. Adjacent angles in a parallelogram are supplementary because opposite sides are parallel. Thus the adjacent angle is 180−61=119°. Opposite angles, rather than adjacent ones, are equal.

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