Question

Kelly and Daniel wrote the following proofs to prove that vertical angles are congruent. Who is correct?

Line segment NT intersects line segment MR forming four angles. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles. Kelly's Proof Statement Justification ∠2 = ∠4 Vertical angles are congruent. ∠1 = ∠3 Vertical angles are congruent. Vertical Angles are congruent. Vertical Angle Theorem Daniel's Proof Statement Justification ∠1 + ∠2 = 180° Definition of Supplementary Angles ∠1 + ∠4 = 180° Definition of Supplementary Angles ∠1 + ∠2 = ∠1 + ∠4 Transitive Property of Equality ∠2 = ∠4 Subtraction Property of Equality
Both Kelly and Daniel are correct.
Neither Kelly or Daniel is correct.
Kelly is correct, but Daniel is not.
Daniel is correct, but Kelly is not.

Answer 1

The answer is Daniel is correct, but Kelly is not.

Explanation:

Answer 2

(D) Daniel is correct, but Kelly is not.

Explanation:

It is given that Line segment NT intersects line segment MR forming four angles. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles.

Kelly's Proof:∠2 = ∠4 (Vertical angles are congruent)

∠1 = ∠3 (Vertical angles are congruent)

Using Vertical Angle Theorem the vertical angles are equal.

since, we cannot use the direct information given in the question in order to prove the same.

Thus, the justification given by Kelly is incorrect.

Now, Daniel's proof:

∠1 + ∠2 = 180°(Definition of Supplementary Angles)

and ∠1 + ∠4 = 180° (Definition of Supplementary Angles)

⇒∠1 + ∠2 = ∠1 + ∠4(Transitive Property of Equality)

Hence, ∠2 = ∠4(Subtraction Property of Equality)

Thus, the justification of Daniel is correct.

Hence, option (D) is correct.