Question

use the given information to prove that $\triangle rst \cong \triangle tur$.
given: $\overline{rs} \cong \overline{tu}$, $\angle srt \cong \angle utr$
prove: $\triangle rst \cong \triangle tur$

statement	reason
2 $\angle srt \cong \angle utr$	given
3 $\overline{rt} \cong \overline{rt}$	reason?
4 $\triangle rst \cong \triangle tur$	reason?

Explanation:

Step1: Identify Given Information

We know \( \overline{RS} \cong \overline{TU} \) (given), \( \angle SRT \cong \angle UTR \) (given), and \( \overline{RT} \cong \overline{RT} \) (reflexive property of congruence, since it's the same side for both triangles).

Step2: Determine Congruence Criterion

Using the Side - Angle - Side (SAS) congruence criterion, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. For \( \triangle RST \) and \( \triangle TUR \):

• \( \overline{RS} \cong \overline{TU} \) (given)
• \( \angle SRT \cong \angle UTR \) (given)
• \( \overline{RT} \cong \overline{RT} \) (reflexive property)

So, by SAS, \( \triangle RST \cong \triangle TUR \).

Now, filling in the blanks:

For Statement 2:

The angle should be \( \angle UTR \) because we are given \( \angle SRT \cong \angle UTR \).

For Reason 1:

The reason for \( \overline{RS} \cong \overline{TU} \) is "Given".

For Reason 3:

The reason for \( \overline{RT} \cong \overline{RT} \) is "Reflexive Property of Congruence" (a segment is congruent to itself).

For Reason 4:

The reason for \( \triangle RST \cong \triangle TUR \) is "SAS (Side - Angle - Side) Congruence Criterion".

Answer:

• Statement 2: \( \angle UTR \)
• Reason 1: Given
• Reason 3: Reflexive Property of Congruence
• Reason 4: SAS (Side - Angle - Side) Congruence Criterion