Question

below is a two - column proof incorrectly proving that the three angles of (\triangle pqr) add up to (180^{circ}):

statements	reasons
(mangle zrp+mangle prq + mangle qry=mangle zry)	angle addition postulate
(angle zrpcongangle rpq)	alternate interior angles theorem
(angle qrycongangle pqr)	alternate interior angles theorem
(mangle rpq+mangle prq + mangle pqr=mangle zry)	substitution
(mangle zry = 180^{circ})	definition of supplementary angles
(mangle rpq+mangle prq + mangle pqr=180^{circ})	substitution

which statement will accurately correct the two - column proof?

the measure of angle (zry) equals (180^{circ}) by definition of a straight angle

angles (qry) and (pqr) should be proven congruent before the construction of line (zy)

the three angles of (\triangle pqr) equal (180^{circ}) according to the transitive property of equality.

line (zy) should be drawn parallel to segment (qr).

Explanation:

Step1: Analyze the error in the reason for \(m\angle ZRY = 180^{\circ}\)

The original proof used "Definition of Supplementary Angles" for \(m\angle ZRY=180^{\circ}\). But \(\angle ZRY\) is a straight - angle. A straight - angle is an angle whose measure is \(180^{\circ}\) by the definition of a straight - angle (a straight line forms an angle of \(180^{\circ}\)), not by the definition of supplementary angles (which is about two angles whose sum is \(180^{\circ}\)).

Step2: Check other options

• For the option "Angles \(QRY\) and \(PQR\) should be proven congruent before the construction of line \(ZY\)": The congruence of \(\angle QRY\) and \(\angle PQR\) (by Alternate Interior Angles Theorem) depends on the construction of \(ZY\parallel PQ\). So this option is wrong.
• For the option "The three angles of \(\triangle PQR\) equal \(180^{\circ}\) according to the Transitive Property of Equality": The proof uses substitution (substituting congruent angles) and the measure of a straight - angle, not the transitive property (\(a = b\) and \(b = c\) implies \(a = c\)). So this option is wrong.
• For the option "Line \(ZY\) should be drawn parallel to segment \(QR\)": If \(ZY\parallel QR\), we cannot use the Alternate Interior Angles Theorem to relate \(\angle ZRP\) and \(\angle RPQ\) (since we want to relate angles of \(\triangle PQR\)). The construction of \(ZY\parallel PQ\) is correct for the angle - addition and substitution steps related to the angles of \(\triangle PQR\).

Answer:

The measure of angle \(ZRY\) equals \(180^{\circ}\) by definition of a straight angle.