Question

in \\( \triangle p q r, overline{p r} \\) is extended through point \\( r \\) to point \\( s, m angle p q r=(3 x+8)^{circ} \\), \\( m angle q r s=(9 x-17)^{circ} \\), and \\( m angle r p q=(3 x+2)^{circ} \\). find \\( m angle q r s \\).

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
In \(\triangle PQR\), \(\angle QRS\) is an exterior angle. So, \(m\angle QRS=m\angle PQR + m\angle RPQ\).
Substitute the given angle expressions: \((9x - 17)=(3x + 8)+(3x + 2)\).

Step2: Simplify the right - hand side of the equation

Simplify \((3x + 8)+(3x + 2)\):

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The equation becomes \(9x - 17=6x + 10\).

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(9x-6x - 17=6x-6x + 10\), which gives \(3x-17 = 10\).
Add \(17\) to both sides: \(3x-17 + 17=10 + 17\), so \(3x=27\).
Divide both sides by \(3\): \(x=\frac{27}{3}=9\).

Step4: Find \(m\angle QRS\)

Substitute \(x = 9\) into the expression for \(m\angle QRS=(9x - 17)^{\circ}\).
\(m\angle QRS=(9\times9 - 17)^{\circ}=(81 - 17)^{\circ}=64^{\circ}\).

Answer:

\(64^{\circ}\)