Question

complete the statement explain why it is true

choose the correct answer below.

a. \\( \overline { v t } \cong \overline { t x } \\) because if two angles are congruent, then the sides opposite those angles are congruent

b. \\( \overline { v t } \cong \overline { v x } \\) because if two angles are congruent. then the sides connecting those angles are congruent

c. \\( \overline { v t } \cong \overline { t x } \\) because if two angles are congruent, then the sides connecting those angles are congruent

d. \\( \overline { v t } \cong \overline { v x } \\) because if two angles are congruent, then the sides opposite those angles are congruent

Explanation:

In a triangle, the side - angle relationship states that if two angles are congruent, the sides opposite those angles are congruent. In the given triangle, we need to check the congruence of sides based on angle congruence.

• For option A: $\overline{VT}\cong\overline{TX}$ is correct. The side $\overline{VT}$ is opposite an angle at $X$ and $\overline{TX}$ is opposite an angle at $T$. If those two angles (the ones they are opposite to) are congruent, then by the isosceles triangle theorem (if two angles of a triangle are congruent, then the sides opposite those angles are congruent), $\overline{VT}\cong\overline{TX}$.
• Option B: The statement about sides connecting the angles is incorrect. The theorem is about sides opposite the congruent angles.
• Option C: The notation $\overline{VT}\cong\overline{TX}$ is correct in terms of side - angle relationship (opposite sides), but the description of sides connecting the angles is wrong.
• Option D: The notation $\overline{VT}\cong\overline{VX}$ is incorrect. $\overline{VT}$ and $\overline{VX}$ are not in the correct side - angle (opposite) relationship as per the given angle congruence in the triangle.

Answer:

A. $\overline{VT}\cong\overline{TX}$ because if two angles are congruent, then the sides opposite those angles are congruent.