What quadrilateral could possess these two characteristics?....

Question

1. opposite angles have the same measure
2. one pair of opposite angles is acute angles

A. parallelogram

B. rectangle

C. square

D. trapezoid

Answer 1

When two "opposite angles" - angles constituting opposite corners of a quadrilateral - have the same measure, the other pair of angles also share the same measure. This is because the lengths of opposite sides must also be equal - otherwise the  pair of angles would not have the same measure. Taken together, pairs of opposite congruent angles & sides cause the other pairs of angles & sides of a quadrilateral to also be congruent.

 

Computation is useless here. Let's eliminate impossible solutions. Consider the trapezoid, having a pair of parallel sides & a pair of non-parallel sides. Opposite angles of a trapezoid will always have different measures, so it does not meet these criteria. Next, consider the square & rectangle. All their angles are 90 degrees. While opposite angles are indeed the same measure, not even one of the angles is acute - thus causing the square & rectangle to fail the criteria.

 

The only correct answer is parallelogram. Pairs of opposite angles sharing the same measure causes pairs of opposite sides to also share the same measure. This relationship among sides & angles causes parallel pairs of opposite sides. Of the 4 choices, only the parallelogram can have a pair of acute opposite angles.

Answer 2

Use the process of elimination.

 

1.  "One pair of opposite angles is acute angles"

     This requirement eliminates the square and rectangle because all angles in both of these are right angles.

 

That leaves us with: parallelogram and trapezoid

 

2. "Opposite angles have the same measure"

 

This eliminate the trapezoid. Trapezoids have 2 pairs of angles with the same measure but the angles are adjacent to each other instead of opposite.

 

That leaves only parallelogram.