Question

1. find ( mangle h ).

triangle diagram with vertices g, h, f; angle at g is ( 89^circ ), side gh is ( 5x - 7 ), angle at f (exterior angle) is ( 14x + 1 ), and point d on the extension of f.

Explanation:

Step1: Identify 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 \(FGH\), \(\angle DFG\) (with measure \(14x + 1\)) is an exterior angle, and the two non - adjacent interior angles are \(\angle G=89^{\circ}\) and \(\angle H = 5x-7\). So, we can set up the equation \(14x + 1=89+(5x - 7)\).

Step2: Simplify the Equation

First, simplify the right - hand side of the equation: \(89+(5x - 7)=5x+(89 - 7)=5x + 82\). So our equation becomes \(14x+1 = 5x + 82\).

Step3: Solve for \(x\)

Subtract \(5x\) from both sides: \(14x-5x + 1=5x-5x + 82\), which simplifies to \(9x+1 = 82\). Then subtract \(1\) from both sides: \(9x+1 - 1=82 - 1\), so \(9x=81\). Divide both sides by \(9\): \(x=\frac{81}{9}=9\).

Step4: Find \(m\angle H\)

We know that \(m\angle H = 5x-7\). Substitute \(x = 9\) into the expression: \(5(9)-7=45 - 7=38\).

Answer:

\(38^{\circ}\)