Question

use the given information to prove that $\triangle rst \cong \triangle vut$.
given: $\overline{st} \cong \overline{ut}$
$\angle srt \cong \angle uvt$
prove: $\triangle rst \cong \triangle vut$
statement reason
1 $\overline{st} \cong \overline{ut}$ reason?
2 $\angle srt \cong \angle \square$ given
3 $\angle rts \cong \angle vtu$ reason?
4 $\triangle rst \cong \triangle vut$ reason?

Explanation:

Step1: Identify Given Information

We know $\overline{ST} \cong \overline{UT}$ (given), $\angle SRT \cong \angle UVT$ (given), and we need to find the third pair of congruent parts for triangle congruence.

Step2: Identify Vertical Angles

$\angle RTS$ and $\angle VTU$ are vertical angles. Vertical angles are congruent, so $\angle RTS \cong \angle VTU$ (Vertical Angles Theorem).

Step3: Apply AAS Congruence

We have:

• $\angle SRT \cong \angle UVT$ (given)
• $\angle RTS \cong \angle VTU$ (vertical angles)
• $\overline{ST} \cong \overline{UT}$ (given)

By the Angle - Angle - Side (AAS) Congruence Theorem, $\triangle RST \cong \triangle VUT$.

Now, filling in the blanks:

• For Statement 2: $\angle SRT \cong \angle UVT$ (given), so the angle is $\angle UVT$.
• For Reason 1: $\overline{ST} \cong \overline{UT}$ is given, so Reason 1 is "Given".
• For Reason 3: $\angle RTS \cong \angle VTU$ because they are vertical angles, so Reason 3 is "Vertical Angles are Congruent".
• For Reason 4: $\triangle RST \cong \triangle VUT$ by AAS, so Reason 4 is "AAS (Angle - Angle - Side) Congruence Theorem".

Answer:

• Statement 2: $\angle SRT \cong \angle \boldsymbol{UVT}$
• Reason 1: Given
• Reason 3: Vertical Angles are Congruent
• Reason 4: AAS (Angle - Angle - Side) Congruence Theorem
• $\triangle RST \cong \triangle VUT$ by AAS