Question

Complete the two-column proof by providing a reason for each of the five statements. Given: Rectangle R U T S Prove: ∠ U S R = ∠ S U T
Statement Reason
1. R U T S is a rectangle
2. R U = S T ; U T = R S
3. ∠ S T U and ∠ S R U are right angles
4. △ U R S ≅ △ S T U
5. ∠ U S R = ∠ S U T

Answer 1

Answer:

The two column proof can be presented as follows;

Statement                 ${}$                              Reason

1. RUST is a rectangle              ${}$              Given

2. RU = ST ; UT = RS             ${}$                  Definition of a rectangle

3. ∠STU and ∠SRU are right angles  ${}$    Definition of a rectangle

4. ΔURS ≅ ΔSTU                  ${}$                   By SAS rule of congruency

5. ∠USR = ∠SUT                 ${}$                     By CPCTC            

Step-by-step explanation:

Given that the side RU, the angle ∠SRU and the side RS of triangle ΔURS are congruent to the side ST the angle ∠STU and the side UT of triangle ΔSTU, then ΔURS is congruent to ΔSTU, by the Side-Angle-Side (SAS) rule of congruency

Therefore, we have that ∠USR ≅ ∠SUT and therefore, it can be shown that ∠USR = ∠SUT using the Congruent Parts of Congruent Triangle are Congruent (CPCTC) postulate.

Answer 2

Answer:

Proof is stated below.

Step-by-step explanation:

Given: R U T S is a rectangle.

To Prove: < U S R = < S U T

Therefore,

Given: R U T S is a rectangle.

R U = S T ; U T = R S - Definition of a parallelogram / rectangle (sides are equal).

<S T U & < S R U are right angles - Definition of a rectangle (a rectangle has four right angles).

Triangle U R S is congruent to triangle S T U - SAS (Side-Angle-Side)

< U S R = < S U T - CPCTE

Hence, proved.