Question

use the given information to complete the proof of the following theorem.
if opposite angles of a quadrilateral are congruent, then it is a parallelogram.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallele
use this definition in your proof.
given: ( mangle a = mangle c )
( mangle b = mangle d )
prove: ( abcd ) is a parallelogram

Explanation:

Step1: Substitute \(m\angle C\) with \(m\angle A\) and \(m\angle D\) with \(m\angle B\) in step 4

Since \(m\angle A = m\angle C\) and \(m\angle B = m\angle D\), we have \(m\angle A+m\angle B + m\angle A+m\angle B=360^{\circ}\)

Step2: Simplify step 4

Factor out 2, we get \(2(m\angle A + m\angle B)=360^{\circ}\)

Step3: Identify supplementary angles in step 7

Since \(m\angle A + m\angle B = 180^{\circ}\), \(\angle A\) and \(\angle B\) are supplementary angles

Step4: Determine parallel sides in step 8

If consecutive - interior angles (\(\angle A\) and \(\angle B\)) are supplementary, then \(AD\parallel BC\) (Consecutive - Interior Angles Theorem)

Step5: Find \(m\angle A + m\angle D\) in step 9

Substitute \(m\angle B\) with \(m\angle D\) in \(m\angle A + m\angle B=180^{\circ}\), we get \(m\angle A + m\angle D = 180^{\circ}\)

Step6: Determine parallel sides in step 11

Since \(\angle A\) and \(\angle D\) are supplementary, \(AB\parallel CD\) (Consecutive - Interior Angles Theorem)

Step7: Conclude in step 12

Since both pairs of opposite sides (\(AD\parallel BC\) and \(AB\parallel CD\)) are parallel, \(ABCD\) is a parallelogram (Definition of a parallelogram)

Answer:

1. \(m\angle A+m\angle B + m\angle A+m\angle B = 360^{\circ}\)
2. \(2(m\angle A + m\angle B)=360^{\circ}\)
3. \(\angle A\) and \(\angle B\)
4. \(BC\); Consecutive - Interior Angles Theorem
5. \(180\)
6. \(CD\); Consecutive - Interior Angles Theorem
7. Definition of a parallelogram