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

Since \(m\angle Q = m\angle S\) and \(m\angle R = m\angle T\), substitute \(m\angle S\) with \(m\angle Q\) and \(m\angle T\) with \(m\angle R\) in statement 4. So \(m\angle Q+m\angle R + m\angle Q+m\angle R=360^{\circ}\).

Step2: Simplify the equation

Factor out the common factor 2 in statement 5. We get \(2(m\angle Q + m\angle R)=360^{\circ}\).

Step3: Find the reason for statement 7

Two angles are supplementary if their sum is \(180^{\circ}\). Since \(m\angle Q + m\angle R=180^{\circ}\), by the definition of supplementary angles, \(\angle Q\) and \(\angle R\) are supplementary.

Step4: Find the parallel - side for statement 8

\(\angle Q\) and \(\angle R\) are consecutive interior angles for sides \(RQ\) and \(RS\). So \(RQ\parallel RS\).

Step5: Find the reason for statement 9

From \(m\angle Q + m\angle R=180^{\circ}\) and \(m\angle R = m\angle T\), by substitution property, \(m\angle Q+m\angle T = 180^{\circ}\).

Step6: Find the angles for statement 10

\(\angle Q\) and \(\angle T\) are supplementary.

Step7: Find the parallel - side for statement 11

\(\angle Q\) and \(\angle T\) are consecutive interior angles for sides \(RQ\) and \(ST\). So \(RQ\parallel ST\).

Step8: Find the reason for statement 12

Since both pairs of opposite sides (\(RQ\parallel RS\) and \(RQ\parallel ST\)) are parallel, by the definition of a parallelogram, \(QRST\) is a parallelogram.

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. \(RQ\)
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. \(RQ\)
8. Definition of a parallelogram (both pairs of opposite sides are parallel)