Question

complete the proof to show that $\triangle pqr \cong \triangle tbr$.

given: $r$ is the midpoint of $\overline{qb}$, $\angle qpr \cong \angle btr$, $\angle q$ and $\angle b$ are right angles.

prove: $\triangle pqr \cong \triangle tbr$

statements	reasons
2. statement box with some text	2. given
3. statement box with some text	3. reason box with some text
4. statement box with some text	4. all right angles are congruent.
5. statement box with some text	5. the midpoint separates the segment into two congruent segments.
6. statement box with some text	6. reason box with some text

Explanation:

Step1: Analyze Statement 1 Reason

The reason for "R is the midpoint of \(\overline{QB}\)" is "Given" (since it's part of the problem's given information).

Step2: Determine Statement 2

From the given, " \(\angle QPR \cong \angle BTR\)" should be the statement for reason "Given". So statement 2: \(\angle QPR \cong \angle BTR\)

Step3: Determine Statement 3 Reason

Statement 3's reason should be related to the right angles. The given says "\(\angle Q\) and \(\angle B\) are right angles", so statement 3: \(\angle Q\) and \(\angle B\) are right angles, reason: "Given"

Step4: Determine Statement 4

From reason "All right angles are congruent", statement 4: \(\angle Q \cong \angle B\)

Step5: Determine Statement 5

From reason "The midpoint separates the segment into two congruent segments", statement 5: \(\overline{QR} \cong \overline{BR}\) (since R is midpoint of QB)

Step6: Determine Statement 6 and Reason

Now, we have two angles and a side: \(\angle QPR \cong \angle BTR\), \(\angle Q \cong \angle B\), \(\overline{QR} \cong \overline{BR}\). So by AAS (Angle - Angle - Side) congruence criterion, \(\triangle PQR \cong \triangle TBR\). So statement 6: \(\triangle PQR \cong \triangle TBR\), reason: "AAS Congruence Criterion"

Answer:

1. Reason for Statement 1: Given
2. Statement 2: \(\boldsymbol{\angle QPR \cong \angle BTR}\)
3. Statement 3: \(\boldsymbol{\angle Q}\) and \(\boldsymbol{\angle B}\) are right angles; Reason: Given
4. Statement 4: \(\boldsymbol{\angle Q \cong \angle B}\)
5. Statement 5: \(\boldsymbol{\overline{QR} \cong \overline{BR}}\)
6. Statement 6: \(\boldsymbol{\triangle PQR \cong \triangle TBR}\); Reason: AAS Congruence Criterion