Question

fill in the reasons for 4 and 5 based on the previous statement given and the diagram below.
statements\treasons

1. ∠4 and ∠6 are supplementary\t1. given
2. ∠6 and ∠2 form a linear pair\t2. definition of a linear pair
3. ∠6 and ∠2 are supplementary\t3. supplement theorem
4. ∠4≅∠2\t4.
5. a||b\t5.

answer choices: alternate exterior angle theorem, corresponding angle converse, ∠4 and ∠2 are supplementary, supplement supplements theorem, alternate interior angles converse, linear pair theorem

Explanation:

Step1: Analyze given angle - supplement relationships

We know that $\angle4$ and $\angle6$ are supplementary, and $\angle6$ and $\angle2$ are supplementary.

Step2: Apply congruent - supplements theorem

If two angles are supplementary to the same angle, then they are congruent. Since $\angle4$ and $\angle2$ are both supplementary to $\angle6$, we have $\angle4\cong\angle2$. So the reason for statement 4 is Congruent Supplements Theorem.

Step3: Analyze parallel - line determination

$\angle4$ and $\angle2$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So the reason for statement 5 is Corresponding Angle Converse.

Answer:

1. Congruent Supplements Theorem
2. Corresponding Angle Converse