Question

1. there is a triangle uvw, with a line extending from u to c. the angle between uc and uv is 106 degrees. the length of uv is 11x + 4, and the length of uw is 15x - 2.

Explanation:

Step1: Apply 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 \(UVW\), \(\angle CUV\) is an exterior angle. So, \(106^{\circ}=(11x + 4)+(15x-2)\).

Step2: Simplify the equation

First, combine like terms on the right - hand side of the equation: \((11x + 4)+(15x-2)=11x+15x + 4 - 2=26x+2\). So the equation becomes \(26x+2 = 106\).

Step3: Solve for \(x\)

Subtract 2 from both sides of the equation: \(26x+2-2=106 - 2\), which gives \(26x=104\). Then divide both sides by 26: \(x=\frac{104}{26}\).

Answer:

\(x = 4\)