Question

given \\(\widehat{stq} \cong \widehat{rqt}\\), which statement is true?

select one:
a. \\(\overline{sv} \cong \overline{vq}\\)
b. \\(\overline{sr} \cong \overline{tq}\\)
c. \\(\overline{sq}\\) and \\(\overline{rt}\\) bisect each other
d. \\(\overline{sq} \cong \overline{rt}\\)

Explanation:

Analyze the given arc congruence

$$ \widehat{STQ} \cong \widehat{RQT} $$

$$ m\widehat{STQ} = m\widehat{ST} + m\widehat{TQ} $$

$$ m\widehat{RQT} = m\widehat{RQ} + m\widehat{QT} $$

$$ m\widehat{ST} + m\widehat{TQ} = m\widehat{RQ} + m\widehat{QT} \implies m\widehat{ST} = m\widehat{RQ} \implies \widehat{ST} \cong \widehat{RQ} $$

Relate arc measures to inscribed angles

$$ m\angle S = \frac{1}{2} m\widehat{RQT} $$

$$ m\angle R = \frac{1}{2} m\widehat{STQ} $$

$$ \widehat{STQ} \cong \widehat{RQT} \implies \angle S \cong \angle R $$

$$ \angle SVT \cong \angle RVQ \quad (\text{vertical angles}) $$

Determine segment relationships

$$ \triangle SVR \sim \triangle TVQ \implies \frac{SV}{TV} = \frac{VR}{VQ} = \frac{SR}{TQ} $$

$$ \widehat{ST} \cong \widehat{RQ} \implies \angle SQR \cong \angle QST \implies \triangle SVQ \text{ is isosceles with } SV = VQ $$

$$ SV = VQ \implies \overline{SV} \cong \overline{VQ} $$

Answer:

• a. \(\overline{SV} \cong \overline{VQ}\) (Correct answer)
• b. \(\overline{SR} \cong \overline{TQ}\)
• c. \(\overline{SQ}\) and \(\overline{RT}\) bisect each other
• d. \(\overline{SQ} \cong \overline{RT}\)