Question

Below is a two-column proof incorrectly proving that the three angles of ΔPQR add up to 180°:

Statements Reasons
Draw line ZY parallel to segment PQ Construction
m∠ZRP + m∠PRQ + m∠QRY = m∠ZRY Angle Addition Postulate
∠ZRP ≅ ∠RPQ Alternate Interior Angles Theorem
∠QRY ≅ ∠PQR Alternate Interior Angles Theorem
m∠RPQ + m∠PRQ + m∠PQR = m∠ZRY Substitution
m∠ZRY = 180° Definition of Supplementary Angles
m∠RPQ + m∠PRQ + m∠PQR = 180° Substitution


Which statement will accurately correct the two-column proof?
The measure of angle ZRY equals 180° by definition of a straight angle.
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.
Line ZY should be drawn parallel to segment QR.

Answer 1

The proof can be corrected by stating that the measure of angle ZRY equals 180° by definition of a straight angle after drawing the line ZY parallel to PQ, showing that angles ZRP and RPQ are corresponding angles, and angles QRY and PQR are alternate interior angles.

Step-by-step explanation:

The statement that will accurately correct the two-column proof is that the measure of angle ZRY equals 180° by definition of a straight angle. This is because whenever you draw a line, such as ZY, the angles that lie on that straight line will sum to 180°, hence why they are called supplementary.

In a two-column proof to show that the three angles of ΔPQR add up to 180°, after constructing line ZY parallel to PQ, you can show that angles ZRP and RPQ are corresponding angles, and angles QRY and PQR are also corresponding angles due to the parallel lines and a transversal.

Then you can use alternate interior angles being congruent (as angles ZRP and RPQ, and angles QRY and PQR are alternate interior angles with respect to the parallel lines and transversal RY) and substitute those values into the angle sum ZRY. Finally, since ZRY is a straight angle, it measures 180°, therefore by substitution, the three angles of ΔPQR add up to 180°.