Question

M is the center of the circle and lies on $\overline{RU}$ and $\overline{ST}$. Which of the following statements is NOT true with respect to the above figure?

• A. $\angle TUM$ measure ${65}^{\circ }$
• B. $\overline{TU}$ is parallel to $\overline{RS}$
• C. $\overline{RS}\cong \overline{SM}$
• D. $\overline{RM}\cong \overline{TM}$

Answer 1

The correct option is C $\overline{RS}\cong \overline{SM}$

From the figure,

In $△RMS$ and $△TMU$

$RM=TM=SM=UM$ ...... (Radii of the circle)

$\angle RMS=\angle TMU={50}^{\circ }$ ........ (Vertically opposite angles)

The triangles are congruent ....... (Using SAS Postulate)

$\therefore RS=TU$ and $RS||TU$

In $△TMU$,

$\angle MTU+\angle TMU+\angle MUT={180}^{\circ }$ ------sum of interior angles of a triangle

$\angle MUT=\angle MTU$ -------angles opposite to equal side in triangle

$\therefore \angle MUT=\frac{{180}^{\circ }-{50}^{\circ }}{2}$

$\angle MUT={65}^{\circ }$

Therefore, from the given options only $C$ is not true.