Question

in \\( \triangle opq, \overline{oq} \\) is extended through point \\( q \\) to point \\( r, m \angle opq=(3 x+5)^{\circ}, m \angle qop=(2 x+5)^{\circ} \\), and \\( m \angle pqr=(8 x-11)^{\circ} \\). find \\( m \angle qop \\).

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
In \(\triangle OPQ\), \(\angle PQR\) is an exterior angle. So, \(m\angle PQR=m\angle OPQ + m\angle QOP\).
Substitute the given angle expressions: \((8x - 11)=(3x + 5)+(2x + 5)\).

Step2: Solve the equation for \(x\)

Simplify the right - hand side of the equation: \((3x + 5)+(2x + 5)=3x+2x + 5 + 5=5x + 10\).
The equation becomes \(8x-11 = 5x+10\).
Subtract \(5x\) from both sides: \(8x-5x-11=5x - 5x+10\), which gives \(3x-11 = 10\).
Add \(11\) to both sides: \(3x-11 + 11=10 + 11\), so \(3x=21\).
Divide both sides by \(3\): \(x=\frac{21}{3}=7\).

Step3: Find \(m\angle QOP\)

Substitute \(x = 7\) into the expression for \(m\angle QOP\).
Since \(m\angle QOP=(2x + 5)^{\circ}\), then \(m\angle QOP=(2\times7 + 5)^{\circ}\).
Calculate \(2\times7+5=14 + 5=19\).

Answer:

\(19\)