Question

two pieces of toast will make a perfect match if the triangles shown are congruent. fill in the blanks with the options from the drag and drop to complete the two column proof. given: \\( \overline{pr} \cong \overline{tv} \\), \\( \overline{pq} \cong \overline{tu} \\), \\( \angle q \\) and \\( \angle u \\) are right angles prove: \\( \triangle pqr \cong \triangle tuv \\) \

$$\begin{tabular}{|c|c|} \\hline statements & reasons \\\\ \\hline 1. \\( \\square \\) & 1. given \\\\ \\hline 2. \\( \\triangle pqr \\) and \\( \\triangle tuv \\) are right triangles & 2. definition of a right triangle \\\\ \\hline 3. \\( \\overline{pr} \\cong \\overline{tv} \\) & 3. \\( \\square \\) \\\\ \\hline 4. \\( \\square \\) & 4. given \\\\ \\hline 5. \\( \\square \\) & 5. \\( \\square \\) \\\\ \\hline 6. \\( \\square \\) & 6. \\( \\square \\) \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Analyze Given Information

We know \( \overline{PR} \cong \overline{TV} \), \( \overline{PQ} \cong \overline{TU} \), and \( \angle Q \) and \( \angle U \) are right angles. So we start by listing the given statements.
Statement 1: \( \overline{PQ} \cong \overline{TU} \), \( \angle Q \) and \( \angle U \) are right angles, \( \overline{PR} \cong \overline{TV} \) (Reason: Given)

Step2: Identify Right Triangles

Since \( \angle Q \) and \( \angle U \) are right angles, by the definition of a right triangle, \( \triangle PQR \) and \( \triangle TUV \) are right triangles. So Statement 2: \( \triangle PQR \) and \( \triangle TUV \) are right triangles (Reason: Definition of a Right Triangle)

Step3: Apply Hypotenuse - Leg (HL) Congruence

In right triangles \( \triangle PQR \) and \( \triangle TUV \), we have \( \overline{PQ} \cong \overline{TU} \) (leg) and \( \overline{PR} \cong \overline{TV} \) (hypotenuse). By the Hypotenuse - Leg (HL) Congruence Theorem, \( \triangle PQR \cong \triangle TUV \). So Statement 3: \( \triangle PQR \cong \triangle TUV \) (Reason: HL Congruence Theorem)

Step4: Check Given for Statement 4

We are given \( \overline{PR} \cong \overline{TV} \), so Statement 4: \( \overline{PR} \cong \overline{TV} \) (Reason: Given)

Answer:

1. Statement: \( \overline{PQ} \cong \overline{TU} \), \( \angle Q \) and \( \angle U \) are right angles, \( \overline{PR} \cong \overline{TV} \); Reason: Given
2. Statement: \( \triangle PQR \) and \( \triangle TUV \) are right triangles; Reason: Definition of a Right Triangle
3. Statement: \( \triangle PQR \cong \triangle TUV \); Reason: HL Congruence Theorem
4. Statement: \( \overline{PR} \cong \overline{TV} \); Reason: Given