question 4 of 10
the isosceles triangle theorem says \if two sides of a triangle are congruent,
then the angles opposite those sides are congruent.\
if you are using this figure to prove the isosceles triangle theorem, which of
the following would be the best strategy?
a. draw $ \overline { t v } $ so that $ v $ is the midpoint of $ \overline { s u } $, then prove $ \delta s t v \cong \delta $
utv using sas.
b. draw $ \overline { t v } $ bisecting $ \angle t $, with $ v $ on $ \overline { s u } $, then prove $ \delta s t v \cong \delta u t v $
using aas.
c. draw $ \overline { t v } $ so that $ v $ is the midpoint of $ \overline { s u } $, then prove $ \delta s t v \cong \delta $
utv using sss.
d. draw $ \overline { t v } $ bisecting $ \angle t $, with $ v $ on $ \overline { s u } $, then prove $ \delta s t v \cong \delta u t v $
using asa.

Question

Accepted Answer

# Explanation:
## Step1: Analyze the given triangle
We have an isosceles triangle $ \triangle STU$ with $ST = UT$.
## Step2: Consider the angle - bisector approach
If we draw $ \overline{TV}$ bisecting $ \angle T$ (so $ \angle STV=\angle UTV$), and $TV = TV$ (common side). Also, $ST = UT$ (given).
By the ASA (Angle - Side - Angle) congruence criterion:
In $ \triangle STV$ and $ \triangle UTV$, we have $ \angle STV=\angle UTV$ (by construction of the angle bisector), $ST = UT$ (given), and $ \angle TSV=\angle TUV$ (if we can show the triangles are congruent). But more precisely, for the congruence of $ \triangle STV$ and $ \triangle UTV$:
We know $ST = UT$ (given), $ \angle STV=\angle UTV$ (angle - bisector), and $TV$ is common. So $ \triangle STV\cong\triangle UTV$ by ASA.
## Step3: Analyze other options
- Option A: If $V$ is the mid - point of $SU$ ($SV = UV$), $ST = UT$, $TV=TV$. But we don't know the included angles ($ \angle STV$ and $ \angle UTV$) for SAS.
- Option B: AAS requires two angles and a non - included side. We don't have the non - included side information in an obvious way for this proof of the isosceles triangle theorem.
- Option C: If $V$ is the mid - point of $SU$ ($SV = UV$), $ST = UT$, $TV = TV$. But we need to establish the congruence based on the given property of the isosceles triangle (two sides equal). The angle - bisector approach is more in line with the theorem's logic.

# Answer:
D. Draw $ \overline{TV}$ bisecting $ \angle T$, with $V$ on $ \overline{SU}$, then prove $ \triangle STV\cong\triangle UTV$ using ASA.

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the given triangle

Step2: Consider the angle - bisector approach

Step3: Analyze other options

Answer: