Question

Please answer quick! Thank You! Math

Spencer wrote the following paragraph proof showing that rectangles are parallelograms with congruent diagonals.




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 ________________. 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 Side-Angle-Side (SAS) Theorem 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. Alternate Interior Angles Theorem

B. Converse of the Alternate Interior Angles Theorem

C. Converse of the Same-Side Interior Angles Theorem

D. Converse of the Same-Side Interior Angles Theorem

Answer 1

I believe it is B.) Alternate Interior.

Answer 2

C and D.

Explanation:

We are given that Spencer wrote a paragraph for proving that rectangles are parallelograms with congruent diagonals

We have to find the missing step in Spencer's proof

Proof:

According to the given information quadrilateral RECT is a rectangle. By definition of a rectangle , all four angles measure
`90^(\circ)`

.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.

Converse of the same- side angles theorem: When the sum of same side interior angles of a transversal line is
`180^(\circ)` then the lines are parallel.

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 Side-Angle-Side(SAS) Theorem says triangle ERT is congruent to triangle CTR.And because corresponding parts of congruent triangles are congruent (CPCTC),diagonals ET and Cr are congruent.

Answer :Option C and D

Converse of the same-Side Interior angles Theorem