Question

According to the given information, quadrilateral RECT is a rectangle. By the definition of a rectangle, all four angles measure 90°. Segment ER is parallel to segment CT and segment EC is parallel to segment RT by the Converse of the Same-Side Interior Angles Theorem. Quadrilateral RECT is then a parallelogram by definition of a parallelogram. Now, construct diagonals ET and CR. Because RECT is a parallelogram, opposite sides are congruent. Therefore, one can say that segment ER is congruent to segment CT. Segment TR is congruent to itself by the Reflexive Property of Equality. The _______________ says triangle ERT is congruent to triangle CTR. And because corresponding parts of congruent triangles are congruent (CPCTC), diagonals ET and CR are congruent.

Which of the following completes the proof?


A Angle-Side-Angle (ASA) Theorem

B Hypotenuse-Leg (HL) Theorem

C Side-Angle-Side (SAS) Theorem

D Side-Side-Side (SSS) Theorem

Answer 1

The Side-Angle-Side (SAS) Theorem proves that triangle ERT is congruent to triangle CTR, leading to the conclusion that diagonals ET and CR are congruent by CPCTC.

Step-by-step explanation:

The correct theorem that completes the proof showing triangle ERT is congruent to triangle CTR is the Side-Angle-Side (SAS) Theorem. In our case, segment ER is congruent to segment CT by parallelogram properties (opposite sides of a parallelogram are congruent), and segment TR is congruent to itself by the Reflexive Property of Equality. Additionally, because RECT is a rectangle and therefore all angles are 90°, both triangles have a right angle, satisfying the angle requirement for SAS. Consequently, by the SAS Theorem, the triangles are congruent. Using CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we can deduce that diagonals ET and CR are congruent as well.

Answer 2

A. Angle-Side-Angle (ASA) Theorem