Correcting a Two-Column Proof | Prioritize Congruence Between Angles QRY and PQR in Math Content Optimization

Below is a two-column proof incorrectly proving that the three angles of ΔPQR sum to 180°:StatementsReasonsDraw line ZY parallel to Constructionm∠ ZRP + m∠ PRQ + m∠ QRY = m∠ ZRYAngle Addition Postulate∠ ZRP ≅ ∠ RPQAlternate Interior Angles Theorem∠ QRY ≅ ∠ PQRAlternate Interior Angles Theoremm∠ RPQ + m∠ PRQ + m∠ PQR = m∠ ZRYSubstitutionm∠ ZRY = 180°Definition of a Straight Anglem∠ RPQ + m∠ PRQ + m∠ PQR = 180°SubstitutionWhich statement will accurately correct the two-column proof?Line ZY should be drawn parallel to segment PQ.The measure of angle ZRY equals 180° by definition of supplementary angles.Angles QRY and PQR should be proven congruent before the construction of line ZY.The three angles of ΔPQR equal 180° according to the Transitive Property of Equality.

The correct statement to correct the two-column proof is:

Angles QRY and PQR should be proven congruent before the construction of line ZY

The correct statement to correct the two-column proof is:

Angles QRY and PQR should be proven congruent before the construction of line ZY.

Explanation:
In the given two-column proof, the incorrect step is when the alternate interior angles theorem is used to claim that ∠ZRP is congruent to ∠RPQ and ∠QRY is congruent to ∠PQR. Before applying the alternate interior angles theorem, we need to establish that the two lines, ZR and YP, are parallel to each other. However, this information is not provided in the given proof. Hence, the proof is not valid.

To correct the proof, we need to establish the congruence between angles QRY and PQR before considering the parallel lines construction. Once we have proven that QRY and PQR are congruent, we can then proceed with the parallel lines construction and apply the angle addition postulate to conclude that the sum of the angles of triangle PQR is 180°.

More Answers: