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 q=mangle s )
( mangle r=mangle t )
prove: ( qrst ) is a parallelogram
1 ( mangle q=mangle s ) given
2 ( mangle r=mangle t ) given
3 ( mangle q + mangle r + mangle s + mangle t = 360^{circ} ) quadrilateral angle - sum property
4 ( mangle q + mangle r + manglesquare + manglesquare=square^{circ} ) substitution property
5 ( 2(manglesquare + manglesquare)=square^{circ} ) simplifying
6 ( mangle q + mangle r = 180^{circ} ) multiplication and division properties
7 ( angle q ) and ( angle r ) are supp. angles reason?
8 ( overline{square}paralleloverline{rs} ) if cons. int. ( angle s ) are supp., then lines ( parallel )
9 ( mangle q + mangle t = 180^{circ} ) reason?
10 ( anglesquare ) and ( anglesquare ) are supp. angles definition of supplementary angles
11 ( overline{square}paralleloverline{st} ) if cons. int. ( angle s ) are supp., then lines ( parallel )
12 ( qrst ) is a parallelogram reason?

Explanation:

Step1: Substitute angles in step 4

Since \(m\angle Q = m\angle S\) and \(m\angle R=m\angle T\) (from steps 1 and 2), in step 4, we substitute \(m\angle S\) with \(m\angle Q\) and \(m\angle T\) with \(m\angle R\). So \(m\angle Q + m\angle R+m\angle Q + m\angle R=360^{\circ}\)

Step2: Simplify step 5

From step 4: \(2(m\angle Q + m\angle R)=360^{\circ}\)

Step7: Reason for \(\angle Q\) and \(\angle R\) being supplementary

If the sum of two angles is \(180^{\circ}\), by the definition of supplementary angles, \(\angle Q\) and \(\angle R\) are supplementary.

Step8: Identify parallel sides

\(\overline{QR}\parallel\overline{RS}\) (because \(\angle Q\) and \(\angle R\) are consecutive - interior angles and they are supplementary, so by the converse of the consecutive - interior angles theorem)

Step9: Reason for \(m\angle Q + m\angle T=180^{\circ}\)

Since \(m\angle R = m\angle T\) (given) and \(m\angle Q + m\angle R=180^{\circ}\) (from step 6), by substitution \(m\angle Q + m\angle T = 180^{\circ}\)

Step10: Identify supplementary angles

\(\angle Q\) and \(\angle T\) are supplementary (by the definition of supplementary angles as \(m\angle Q + m\angle T=180^{\circ}\))

Step11: Identify parallel sides

\(\overline{RQ}\parallel\overline{ST}\) (because \(\angle Q\) and \(\angle T\) are consecutive - interior angles and they are supplementary, so by the converse of the consecutive - interior angles theorem)

Step12: Reason for \(QRST\) being a parallelogram

By the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel, here \(\overline{QR}\parallel\overline{ST}\) and \(\overline{RQ}\parallel\overline{ST}\))

Answer:

1. \(m\angle Q + m\angle R+m\angle Q + m\angle R = 360^{\circ}\)
2. \(2(m\angle Q + m\angle R)=360^{\circ}\)
3. Definition of Supplementary Angles
4. \(\overline{QR}\)
5. Substitution Property (\(m\angle R=m\angle T\) and \(m\angle Q + m\angle R = 180^{\circ}\))
6. \(\angle Q\) and \(\angle T\)
7. \(\overline{RQ}\)
8. Definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel)