Question

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

1. $\angle 4 \cong \angle 3$ and $c \parallel d$
2. $\angle 3 \cong \angle 1$
3. $\angle 1 \cong \angle 2$
4. $\angle 4 \cong \angle 2$

reasons
1 given

1. select an answer
2. select an answer
3. select an answer

parallel lines cut by a transversal - aia are congruent
vertical angles are congruent.
parallel lines cut by a transversal - corresponding angles are congruent
angle addition postulate
subtraction property of equality
substitution
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question 5

Explanation:

Step1: Match ∠3≅∠1 to rule

∠3 and ∠1 are corresponding angles formed by parallel lines \(c \parallel d\) cut by a transversal. So the reason is Parallel lines cut by a transversal - Corresponding angles are congruent.

Step2: Match ∠1≅∠2 to rule

∠1 and ∠2 are opposite (vertical) angles formed by intersecting lines. So the reason is Vertical angles are congruent.

Step3: Match ∠4≅∠2 to rule

We know \(∠4≅∠3\) (given) and \(∠3≅∠1≅∠2\), so we substitute the congruent angles. The reason is substitution.

Answer:

1. Parallel lines cut by a transversal - Corresponding angles are congruent
2. Vertical angles are congruent
3. substitution