Question

6 find the value of x.
9 find the m∠mni.

Explanation:

Step1: Use the exterior - angle theorem for triangle \(HIJ\)

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.
For \(\triangle HIJ\), we have \((12x - 4)^{\circ}=(6x + 4)^{\circ}+(5x + 2)^{\circ}\)

Step2: Simplify the equation

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Step3: Solve for \(x\)

Subtract \(11x\) from both sides of the equation:
\(12x-11x-4=11x - 11x+6\)
\(x-4 = 6\)
Add \(4\) to both sides: \(x=6 + 4=10\)

Step1: Use the exterior - angle theorem for triangle \(LMN\)

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.
For \(\triangle LMN\), we have \((19x + 2)^{\circ}=(6x)^{\circ}+(15x-12)^{\circ}\)

Step2: Simplify the equation

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Step3: Solve for \(x\)

Subtract \(19x\) from both sides: \(19x-19x + 2=21x-19x-12\)
\(2 = 2x-12\)
Add \(12\) to both sides: \(2+12=2x-12 + 12\)
\(14 = 2x\)
Divide both sides by \(2\): \(x = 7\)

Step4: Find \(m\angle MNI\)

First, find \(m\angle LMN=(15x-12)^{\circ}\)
Substitute \(x = 7\) into \(15x-12\): \(15\times7-12=105 - 12=93^{\circ}\)

Answer:

\(x = 10\)

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