Question

problem 7:

problem 8:

problem 9:

Explanation:

Problem 7

Step1: Determine the reason for $\angle PQT\cong\angle RQS$

Vertical angles are congruent. So, the reason for $\angle PQT\cong\angle RQS$ is "Vertical Angles Theorem".

Step2: Determine the congruence criterion for $\triangle PQT\cong\triangle SQR$

We have $\overline{PQ}\cong\overline{QS}$ (given), $\angle PQT\cong\angle RQS$ (vertical angles), and $\overline{QT}\cong\overline{QR}$ (given). By the Side - Angle - Side (SAS) congruence criterion, $\triangle PQT\cong\triangle SQR$.

Problem 8

Step1: Determine the reason for $\angle V\cong\angle X$

We have $\overline{UV}\cong\overline{UX}$ (given), $\angle VWU\cong\angle XWU$ (given), and $\overline{UW}\cong\overline{UW}$ (reflexive property). By the Side - Angle - Side (SAS) congruence criterion, $\triangle UWV\cong\triangle UWX$. And if two triangles are congruent, then their corresponding angles are congruent. So, $\angle V\cong\angle X$ because of "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Problem 9

Step1: Determine the reason for $\angle Y\cong\angle C$

Let's assume it is given.

Step2: Fill in the second statement

Since we are dealing with congruent triangles and there are marked equal sides in the figure, the second statement could be $\overline{YA}\cong\overline{CA}$ (given, based on the side - markings in the figure).

Step3: Fill in the third statement

The third statement (since we know vertical angles are congruent) is $\angle YAZ\cong\angle CAB$ (Vertical Angles Theorem).

Step4: Determine the congruence criterion for $\triangle YZA\cong\triangle CBA$

We have $\angle Y\cong\angle C$ (assumed given), $\overline{YA}\cong\overline{CA}$ (given), and $\angle YAZ\cong\angle CAB$ (vertical angles). By the Angle - Side - Angle (ASA) congruence criterion, $\triangle YZA\cong\triangle CBA$.

Answer:

• Problem 7:
• 3. Vertical Angles Theorem
• 4. SAS (Side - Angle - Side)
• Problem 8:
• 4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
• 5. SAS (Side - Angle - Side)
• Problem 9:
• 1. Given (assumed)
• 2. $\overline{YA}\cong\overline{CA}$
• 3. $\angle YAZ\cong\angle CAB$
• 4. ASA (Angle - Side - Angle)