Question

lets take a break
your answer is incorrect.

• statement 7: your answer is incorrect.
• statement 8: your answer is incorrect.
• statement 12: your answer is incorrect.

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 + mangle a + mangle b = 360^{circ} ) substitution property
5 ( 2(mangle a + mangle b)=360^{circ} ) simplifying
6 ( mangle a + mangle b = 180^{circ} ) multiplication and division properties
7 ( angle b ) and ( angle c ) are supp. angles definition of supplementary angles
8 ( overline{ad}paralleloverline{cd} ) if cons. int. ( angle s ) are supp., then lines ( parallel )
9 ( mangle a + mangle d = 180^{circ} ) substitution property
10 ( angle a ) and ( angle d ) are supp. angles definition of supplementary angles
11 ( overline{ab}paralleloverline{cd} ) if cons. int. ( angle s ) are supp., then lines ( parallel )
12 ( abcd ) is a parallelogram if cons. int. ( angle s ) are supp., then lines ( parallel )

Explanation:

Step1: Analyze Statement 7

The correct pair of supplementary angles should be \( \angle A\) and \( \angle B\) (from step 6 \(m\angle A + m\angle B=180^{\circ}\)), not \( \angle B\) and \( \angle C\).

Step2: Analyze Statement 8

Since \(m\angle A + m\angle B = 180^{\circ}\), by the "If consecutive - interior angles are supplementary, then lines are parallel" (consecutive - interior angles for \(AD\parallel BC\) with transversal \(AB\)), the correct parallel lines should be \(AD\parallel BC\), not \(AD\parallel CD\) ( \(AD\) and \(CD\) are adjacent sides, not parallel in general).

Step3: Analyze Statement 12

The reason for \(ABCD\) being a parallelogram is that both pairs of opposite sides are parallel (\(AD\parallel BC\) and \(AB\parallel CD\)). The correct reason is "If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram" (based on the definition of a parallelogram), not "If cons. int. \( \angle\)s are supp., then lines \(||\)"

Answer:

• Statement 7: \( \angle A\) and \( \angle B\) are supp. angles (Reason: Definition of Supplementary Angles)
• Statement 8: \( \overline{AD}\parallel\overline{BC}\) (Reason: If cons. int. \( \angle\)s are supp., then lines \(||\))
• Statement 12: \(ABCD\) is a parallelogram (Reason: If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram)