Question

∠1 = ∠2
∠5 = ∠6
prove:
mq = mp

1. ∠3, ∠5 are supplementary and ∠4, ∠6 are supplementary
2. ∠5 = ∠6 and ∠1 = ∠2
3. ∠3 = ∠4
4. mn = mn
5. triangle mnq congruent to triangle mnp
6. mq = mp

Explanation:

Step1: Use the property of supplementary angles

If \(\angle3\) and \(\angle5\) are supplementary (\(\angle3+\angle5 = 180^{\circ}\)), \(\angle4\) and \(\angle6\) are supplementary (\(\angle4+\angle6=180^{\circ}\)), and \(\angle5=\angle6\) (given), then \(\angle3=\angle4\) (supplements of equal angles are equal).

Step2: Use the reflexive property

For any segment \(MN\), \(MN = MN\) (reflexive property of equality).

Step3: Use ASA (Angle - Side - Angle) congruence criterion

We have \(\angle1=\angle2\) (given), \(MN = MN\) (reflexive), \(\angle3=\angle4\) (from step 1). So, \(\triangle MNQ\cong\triangle MNP\) by ASA.

Step4: Use CPCTE (Corresponding Parts of Congruent Triangles are Equal)

Since \(\triangle MNQ\cong\triangle MNP\), then \(MQ = MP\) (CPCTE).

Answer:

1. Given
2. Given
3. Supplements to = angles.
4. Reflexive
5. ASA
6. CPCTE