Question

Provide reasons for the proof.

Given: ∠2 = ∠4 and
∠2 and ∠3 are supplementary

Prove: ∠1 = ∠3

Answer 1

The fact that ∠2 equals ∠4 and ∠2 and ∠3 are supplementary, leads us by substituting equal angles and using properties of supplementary angles to the conclusion that ∠1 therefore equals ∠3.

Step-by-step explanation:

To provide reasons for the proof of '∠1 = ∠3', we need to leverage the given information - '∠2 = ∠4' and '∠2 and ∠3 are supplementary'. Verify whether the following assumptions are true:

1. ∠1 and ∠2 are vertically opposite angles by the intersecting lines, hence, ∠1 = ∠2 because vertically opposite angles are equal. This notes our assumption as '∠1 = ∠2'.
2. Because it's given that '∠2 = ∠4', ∠1 can equally be written as ∠4 - taking into account transitive property of equality (if ∠1 = ∠2 and ∠2 = ∠4, then ∠1 = ∠4).
3. The same principle applies to '∠2 and ∠3 are supplementary', meaning ∠2 + ∠3 = 180°. Given that ∠2 = ∠4, we can substitute ∠4 in the equation to get ∠4 + ∠3 = 180°. Thus, ∠3 = 180° - ∠4. Since, ∠4 equals to ∠1 it implies also that ∠3 = ∠1.

In conclusion, using the supplied information, we can deduct that ∠1 = ∠3.

Learn more about Angle Equality