Question

statements reasons 1. p || q 1. given 2. ∠1 ≅ ∠2 2. 3. m∠1 = m∠2 3. 4. ∠2 and ∠3 are a linear pair. 4. definition of linear pair 5. 5. linear pairs theorem 6. m∠2 + m∠3 = 180° 6. 7. 7. substitution 8. ∠1 and ∠3 are supplementary 8. definition of congruent angles alternate exterior angles theorem definition of corresponding angles ∠2 and ∠3 are complementary. definition of supplementary angles ?m∠1 + m∠3 = 180°? m∠2 + m∠3 = 90° m∠2 + m∠3 = 180 ∠2 and ∠3 are supplementary. alternate interior angles theorem definition of complementary angles

Explanation:

Step1: Identify reason for $\angle1\cong\angle2$

Since $p\parallel q$, by the Alternate Interior Angles Theorem, $\angle1\cong\angle2$. So the reason for statement 2 is Alternate Interior Angles Theorem.

Step2: Identify reason for $m\angle1 = m\angle2$

By the Definition of congruent angles, if two angles are congruent, their measures are equal. So the reason for statement 3 is Definition of congruent angles.

Step3: Fill in statement 5

Since $\angle2$ and $\angle3$ are a linear - pair and by the Linear Pairs Theorem, linear pairs of angles are supplementary. So statement 5 is $\angle2$ and $\angle3$ are supplementary.

Step4: Identify reason for $m\angle2 + m\angle3=180^{\circ}$

By the Definition of supplementary angles, if two angles are supplementary, the sum of their measures is $180^{\circ}$. So the reason for statement 6 is Definition of supplementary angles.

Step5: Fill in statement 7

Since $m\angle1 = m\angle2$ and $m\angle2 + m\angle3=180^{\circ}$, by substitution, we get $m\angle1 + m\angle3=180^{\circ}$. So statement 7 is $m\angle1 + m\angle3 = 180^{\circ}$.

Step6: Identify reason for statement 8

By the Definition of supplementary angles, since $m\angle1 + m\angle3 = 180^{\circ}$, $\angle1$ and $\angle3$ are supplementary. So the reason for statement 8 is Definition of supplementary angles.

Answer:

1. Alternate Interior Angles Theorem
2. Definition of congruent angles
3. $\angle2$ and $\angle3$ are supplementary
4. Definition of supplementary angles
5. $m\angle1 + m\angle3 = 180^{\circ}$
6. Definition of supplementary angles