Question

example 3
proof write a two - column proof.
6 given: ∠2 ≅ ∠4
prove: ∠1 ≅ ∠3

example 4
proof write a two - column proof.

1. given: ∠5 ≅ ∠7

prove: ∠5 ≅ ∠8

1. given: ∠1 ≅ ∠3

prove: ∠2 ≅ ∠4

Explanation:

Step1: Use the definition of complementary angles

If two angles are complementary to the same angle, then they are congruent.
We know that \(\angle1\) and \(\angle2\) are complementary (\(\angle1+\angle2 = 90^{\circ}\)), and \(\angle3\) and \(\angle4\) are complementary (\(\angle3+\angle4=90^{\circ}\)).

Step2: Substitute using the given \(\angle2\cong\angle4\)

Since \(\angle2\cong\angle4\), we can substitute \(\angle2\) for \(\angle4\) in the equation \(\angle3+\angle4 = 90^{\circ}\). So \(\angle3+\angle2=90^{\circ}\).

Step3: Apply the congruent - complements theorem

We have \(\angle1+\angle2=90^{\circ}\) and \(\angle3 + \angle2=90^{\circ}\). By the congruent - complements theorem (if two angles are complementary to the same angle, then the two angles are congruent), we can conclude that \(\angle1\cong\angle3\).

Answer:

Statements	Reasons
2. \(\angle2\cong\angle4\)	Given
3. \(\angle3+\angle2=90^{\circ}\)	Substitution Property (substitute \(\angle2\) for \(\angle4\) in \(\angle3+\angle4 = 90^{\circ}\))
4. \(\angle1\cong\angle3\)	Congruent - Complements Theorem (if \(\angle1+\angle2=\angle3+\angle2 = 90^{\circ}\), then \(\angle1\cong\angle3\))