Question

1. in △ pqr, if ( mangle p ) is 14 less than five times ( x ), ( mangle q ) is five less than ( x ), and ( mangle r ) is nine less than twice ( x ), find ( x ) and the measure of each angle.

( x=)
( mangle p=)
( mangle q=)
( mangle r=)

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle P + m\angle Q + m\angle R = 180\).

Step2: Express each angle in terms of \(x\)

• \(m\angle P = 5x - 14\)
• \(m\angle Q = x - 5\)
• \(m\angle R = 2x - 9\)

Step3: Substitute into angle sum formula

\((5x - 14) + (x - 5) + (2x - 9) = 180\)
Simplify left side: \(5x - 14 + x - 5 + 2x - 9 = 8x - 28\)
So, \(8x - 28 = 180\)

Step4: Solve for \(x\)

Add 28 to both sides: \(8x = 180 + 28 = 208\)
Divide by 8: \(x = \frac{208}{8} = 26\)

Step5: Find each angle

• \(m\angle P = 5(26) - 14 = 130 - 14 = 116\)
• \(m\angle Q = 26 - 5 = 21\)
• \(m\angle R = 2(26) - 9 = 52 - 9 = 43\)

Answer:

\(x = 26\)
\(m\angle P = 116^\circ\)
\(m\angle Q = 21^\circ\)
\(m\angle R = 43^\circ\)