Answer:
m = 36
Step-by-step explanation:
Points P, Q, and R lie on the circumference of the circle with center O.
The line segments connecting the center O to points P, Q and R represent the radii of the circle and are therefore of equal length:
[tex]\sf \overline{OP}=\overline{OQ}=\overline{OR}[/tex]
This means that triangles OPQ and OQR are isosceles triangles, since two of their sides are equal in length.
In an isosceles triangle, the angles opposite the congruent sides are equal in measure. Therefore:
- As OP = OQ in triangle OPQ, then m∠OPQ = m∠PQO = m°.
- As OQ = OR in triangle OQR, then m∠OQR = m∠QRO = 2m°.
Line segment OQ is the transversal of parallel lines PQ and OR.
According to the Alternate Interior Angle Theorem, m∠ROQ = m∠PQO.
Since m∠PQO = m°, then m∠ROQ = m°.
Now, we have determined the measures of all three angles of triangle OQR:
- m∠ROQ = m°
- m∠OQR = 2m°
- m∠QRO = 2m°
Since the interior angles of a triangle sum to 180°, then:
[tex]\sf m\angle ROQ + m\angle OQR + m\angle QRO = 180^{\circ}[/tex]
Substitute the expression for each angle into the sum equation and solve for m:
[tex]\sf m^{\circ} + 2m^{\circ} + 2m^{\circ} = 180^{\circ}\\\\\\5m^{\circ} = 180^{\circ}\\\\\\\5m=180\\\\\\\dfrac{5m}{5}=\dfrac{180}{5}\\\\\\m=36[/tex]
Therefore, the value of m is:
[tex]\LARGE\boxed{\boxed{\sf m=36}}[/tex]
