Question

use the given information to complete the proof of the following theorem.
if a quadrilateral is a parallelogram, then its opposite angles are congruent.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are paralle
use this definition in your proof.
given: ( abcd ) is a parallelogram
prove: ( angle a cong angle c )
1 ( abcd ) is a parallelogram given
2 ( overline{ab} parallel )
definition of a parallelogram
3 ( overline{ad} parallel )
definition of a parallelogram
4 ( angle a ) and ( angle ) are supp. angles reason?
5 ( angle a ) and ( angle ) are supp. angles reason?
6 ( angle b cong angle )
if ( angle s ) are supp. to the same ( angle ), they are ( cong )
7 ( angle c ) and ( angle ) are supp. angles reason?
8 ( angle a cong angle c )
reason?

Explanation:

Step1: Fill in the parallel sides

Since \(ABCD\) is a parallelogram, by the definition of a parallelogram (\(AB\parallel CD\) and \(AD\parallel BC\)). So in statement 2, \(\overline{AB}\parallel\overline{CD}\), and in statement 3, \(\overline{AD}\parallel\overline{BC}\).

Step2: Find supplementary angles for \(\angle A\)

When \(\overline{AB}\parallel\overline{CD}\) and \(\overline{AD}\) is a transversal, \(\angle A\) and \(\angle D\) are same - side interior angles. By the same - side interior angles theorem (if two parallel lines are cut by a transversal, then the same - side interior angles are supplementary), so in statement 4, \(\angle A\) and \(\angle D\) are supp. angles. When \(\overline{AD}\parallel\overline{BC}\) and \(\overline{AB}\) is a transversal, \(\angle A\) and \(\angle B\) are same - side interior angles. By the same - side interior angles theorem, so in statement 5, \(\angle A\) and \(\angle B\) are supp. angles.

Step3: Use the congruence of angles supplementary to the same angle

Since \(\angle A\) and \(\angle B\) are supplementary (\(\angle A+\angle B = 180^{\circ}\)), \(\angle A\) and \(\angle D\) are supplementary (\(\angle A+\angle D=180^{\circ}\)), and if two angles are supplementary to the same angle, they are congruent. So in statement 6, \(\angle B\cong\angle D\).

Step4: Find supplementary angles for \(\angle C\)

When \(\overline{AB}\parallel\overline{CD}\) and \(\overline{BC}\) is a transversal, \(\angle C\) and \(\angle B\) are same - side interior angles. By the same - side interior angles theorem, \(\angle C\) and \(\angle B\) are supp. angles.

Step5: Prove \(\angle A\cong\angle C\)

Since \(\angle A\) and \(\angle B\) are supplementary (\(\angle A+\angle B = 180^{\circ}\)), \(\angle C\) and \(\angle B\) are supplementary (\(\angle C+\angle B=180^{\circ}\)). By the congruence of angles supplementary to the same angle (if \(\angle x+\angle y = 180^{\circ}\) and \(\angle z+\angle y=180^{\circ}\), then \(\angle x\cong\angle z\)), so in statement 8, \(\angle A\cong\angle C\).

Answer:

1. \(\overline{CD}\); 3. \(\overline{BC}\); 4. \(\angle D\), Same - side interior angles theorem; 5. \(\angle B\), Same - side interior angles theorem; 6. \(\angle D\); 7. \(\angle B\), Same - side interior angles theorem; 8. If \(\angle\)s are supp. to the same \(\angle\), they are \(\cong\)