Question

question: 3 heather must prove this theorem: if a quadrilateral is a parallelogram, then the opposite sides are congruent. she draws parallelogram abcd with diagonal ac to start her proof. below is her proof of the theorem. statement reason 1. abcd is a parallelogram. given 2. (overline{ab}paralleloverline{cd}) and (overline{bc}paralleloverline{ad}) definition of parallelogram 3. (angle1congangle4) alternate interior angles theorem 4. (angle2congangle3) alternate interior angles theorem 5. (overline{ac}congoverline{ac}) reflexive property of congruence 6. (\triangle abccong\triangle cda) 7. (overline{bc}congoverline{da}) and (overline{ab}congoverline{cd}) corresponding parts of congruent triangles are congruent. which of the following is the missing reason for statement six in the proof? a. asa b. sss

Explanation:

We have two - pairs of congruent angles ($\angle1\cong\angle4$ and $\angle2\cong\angle3$) and a common side ($\overline{AC}\cong\overline{AC}$). The Angle - Side - Angle (ASA) congruence postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Here, for $\triangle ABC$ and $\triangle CDA$, $\angle1,\overline{AC},\angle2$ of $\triangle ABC$ are congruent to $\angle4,\overline{AC},\angle3$ of $\triangle CDA$ respectively.

Answer:

A. ASA