Question

solve for x in the figure below.
(figure with triangle fgh and exterior angle at h, and another triangle with angles x + 66, x + 86, 50°)
left figure: triangle fgh, with f, g, h. at h, exterior angle is 9x + 12. angle at f is 7x + 6, angle at g is 22°. right figure: triangle with angles x + 66, x + 86, 50°.

Explanation:

First Figure (Left)

Step1: Recall Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(9x + 12=(7x + 6)+22\)

Step2: Simplify the equation

First, expand the right - hand side: \(9x + 12=7x+6 + 22\)
Then, combine like terms on the right - hand side: \(9x + 12=7x + 28\)
Subtract \(7x\) from both sides: \(9x-7x + 12=7x-7x + 28\), which gives \(2x+12 = 28\)
Subtract 12 from both sides: \(2x+12 - 12=28 - 12\), so \(2x=16\)
Divide both sides by 2: \(x=\frac{16}{2}=8\)

Second Figure (Right)

Step1: Recall Triangle Angle Sum Theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \((x + 66)+(x + 86)+50=180\)

Step2: Simplify the equation

First, combine like terms: \(x+x+66 + 86+50=180\)
\(2x+(66 + 86+50)=180\)
\(2x+(152 + 50)=180\)
\(2x + 202=180\)
Subtract 202 from both sides: \(2x+202-202=180 - 202\)
\(2x=- 22\)
Divide both sides by 2: \(x=\frac{-22}{2}=- 11\)

Answer:

For the left - hand figure, \(x = 8\); for the right - hand figure, \(x=-11\)