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 parallel
use this definition in your proof.
given: ( mangle a = mangle c )
( mangle b = mangle d )
prove: ( abcd ) is a parallelogram
1 ( mangle a = mangle c ) given
2 ( mangle b = mangle d ) given
3 ( mangle a + mangle b + mangle c + mangle d = 360^{circ} ) quadrilateral angle - sum property
4 ( mangle a + mangle b + manglesquare + manglesquare=square^{circ} ) substitution property
5 ( 2(manglesquare + manglesquare)=square^{circ} ) simplifying
6 ( mangle a + mangle b = 180^{circ} ) multiplication and division properties
7 ( anglesquare ) and ( anglesquare ) are supp. angles definition of supplementary angles
8 ( overline{ad}parallelsquare ) reason?
9 ( mangle a + mangle d=square^{circ} ) substitution property
10 ( angle a ) and ( angle d ) are supp. angles definition of supplementary angles
11 ( overline{ab}parallelsquare ) reason?
12 ( abcd ) is a parallelogram reason?

Explanation:

Step1: Substitute angles in step 4

Since \(m\angle A = m\angle C\) and \(m\angle B = m\angle D\), substitute \(m\angle C\) with \(m\angle A\) and \(m\angle D\) with \(m\angle B\) in \(m\angle A+m\angle B + m\angle C+m\angle D=360^{\circ}\). So \(m\angle A+m\angle B + m\angle A+m\angle B=360^{\circ}\).

Step2: Simplify step 5

Factor out 2 from \(m\angle A+m\angle B + m\angle A+m\angle B\). We get \(2(m\angle A + m\angle B)=360^{\circ}\).

Step3: Find supplementary angles in step 7

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

Step4: Find parallel sides in step 8

If \(\angle A\) and \(\angle B\) are supplementary, then \(\overline{AD}\parallel\overline{BC}\) (Consecutive - interior angles are supplementary, so lines are parallel).

Step5: Substitute in step 9

Since \(m\angle B = m\angle D\) and \(m\angle A + m\angle B=180^{\circ}\), substitute \(m\angle B\) with \(m\angle D\). So \(m\angle A + m\angle D = 180^{\circ}\).

Step6: Find parallel sides in step 11

If \(\angle A\) and \(\angle D\) are supplementary, then \(\overline{AB}\parallel\overline{CD}\) (Consecutive - interior angles are supplementary, so lines are parallel).

Step7: Conclude in step 12

Since \(\overline{AD}\parallel\overline{BC}\) and \(\overline{AB}\parallel\overline{CD}\), by the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), \(ABCD\) is 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. \(\overline{BC}\), Consecutive - interior angles are supplementary, so lines are parallel
5. \(180^{\circ}\)
6. \(\overline{CD}\), Consecutive - interior angles are supplementary, so lines are parallel
7. Definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel)