Question

for exercises 1 - 5, match the diagram in column a with the name of the theorem in column b to complete the two - column proof.
column a

1. given:

conclusion: ∠1≅∠2

1. given: ∠1 and ∠2 are complementary. ∠2 and ∠3 are complementary.

conclusion: ∠1≅∠3

1. given: ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary.

conclusion: ∠1≅∠3

1. given: ∠1 and ∠2 are right angles.

conclusion: ∠1≅∠2

1. given: ∠1≅∠2, ∠1 and ∠2 are supplementary.

conclusion: ∠1 and ∠2 are right angles.
column b
congruent complements theorem
theorem 1 - 5 (if two angles are congruent and supplementary, then each is a right angle.)
theorem 1 - 4 (all right angles are congruent.)
congruent supplements
vertical angles theorem

Explanation:

Step1: Identify vertical - angle property

In the first case, when two lines intersect, vertical angles are congruent. So for the first problem with the intersection of lines and conclusion $\angle1\cong\angle2$, it is the Vertical Angles Theorem.

Step2: Apply congruent - complements concept

For the second problem, since $\angle1$ and $\angle2$ are complementary, and $\angle2$ and $\angle3$ are complementary, by the Congruent Complements Theorem, $\angle1\cong\angle3$.

Step3: Use congruent - supplements property

In the third problem, as $\angle1$ and $\angle2$ are supplementary, and $\angle2$ and $\angle3$ are supplementary, the Congruent Supplements Theorem gives $\angle1\cong\angle3$.

Step4: Recall right - angle congruence

For the fourth problem, given that $\angle1$ and $\angle2$ are right angles, by Theorem 1 - 4 (All right angles are congruent), $\angle1\cong\angle2$.

Step5: Apply Theorem 1 - 5

In the fifth problem, given $\angle1\cong\angle2$ and $\angle1$ and $\angle2$ are supplementary, by Theorem 1 - 5 (If two angles are congruent and supplementary, then each is a right angle), $\angle1$ and $\angle2$ are right angles.

Answer:

1. Vertical Angles Theorem
2. Congruent Complements Theorem
3. Congruent Supplements Theorem
4. Theorem 1 - 4 (All right angles are congruent)
5. Theorem 1 - 5 (If two angles are congruent and supplementary, then each is a right angle)