Question

given: $angle 4 cong angle 3$ and $c parallel d$
prove: $angle 4 cong angle 2$

statements	reasons
2. $angle 3 cong angle 1$	2. select an answer
3. $angle 1 cong angle 2$	3. select an answer
4. $angle 4 cong angle 2$	4. select an answer

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Explanation:

Step1: Justify ∠3 ≅ ∠1

Since \(c \parallel d\), ∠3 and ∠1 are alternate interior angles, which are congruent for parallel lines cut by a transversal.

Step2: Justify ∠1 ≅ ∠2

∠1 and ∠2 are vertical angles, and vertical angles are always congruent.

Step3: Justify ∠4 ≅ ∠2

We know ∠4 ≅ ∠3 (given), ∠3 ≅ ∠1, and ∠1 ≅ ∠2. By the transitive property of congruence, if \(a \cong b\) and \(b \cong c\), then \(a \cong c\).

Answer:

STATEMENTS	REASONS
2. ∠3 ≅ ∠1	2. Alternate Interior Angles Theorem
3. ∠1 ≅ ∠2	3. Vertical Angles Congruence Theorem
4. ∠4 ≅ ∠2	4. Transitive Property of Congruence