Question

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sv bisects ∠tsu and su ≅ st. complete the proof that tv ≅ uv.
1sv bisects ∠tsu
2su ≅ st
3∠tsv ≅ ∠usv
4sv ≅ sv
5δstv ≅ δsuv
6tv ≅ uv

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of angle - bisector

Since \(\overleftrightarrow{SV}\) bisects \(\angle TSU\), by the definition of an angle - bisector, \(\angle TSV\cong\angle USV\).

Step3: Reflexive property of congruence

For any segment \(AB\), \(AB\cong AB\). So, \(\overline{SV}\cong\overline{SV}\) (reflexive property).

Step4: SAS (Side - Angle - Side) congruence criterion

We have \(\overline{SU}\cong\overline{ST}\) (given), \(\angle TSV\cong\angle USV\) (from the angle - bisector), and \(\overline{SV}\cong\overline{SV}\) (reflexive property). So, \(\triangle STV\cong\triangle SUV\) by the SAS (Side - Angle - Side) congruence criterion.

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

If \(\triangle STV\cong\triangle SUV\), then their corresponding parts are congruent. So, \(\overline{TV}\cong\overline{UV}\) by CPCTC.

Answer:

1. Given
2. Given
3. Definition of angle - bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)