Question

1. find the length of $overline{gh}$. 14. $angle a$ and $angle b$ are supplementary. $mangle a=(x + 10)^{circ}$ and $mangle b=(2x + 8)^{circ}$, find $mangle a$.

Explanation:

Step1: Determine coordinates of points

Assume from the grid, \(H(-3, 3)\) and \(G(0, - 2)\).

Step2: Apply distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-3,y_1 = 3,x_2=0,y_2=-2\). Then \(d=\sqrt{(0+3)^2+(-2 - 3)^2}=\sqrt{3^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\).

Step3: Solve supplementary - angle equation

Since \(\angle A\) and \(\angle B\) are supplementary, \(m\angle A+m\angle B = 180^{\circ}\). Substitute \(m\angle A=(x + 10)^{\circ}\) and \(m\angle B=(2x + 8)^{\circ}\) into the equation: \((x + 10)+(2x+8)=180\). Combine like - terms: \(3x+18 = 180\). Subtract 18 from both sides: \(3x=180 - 18=162\). Divide both sides by 3: \(x = 54\).

Step4: Find \(m\angle A\)

Substitute \(x = 54\) into \(m\angle A=x + 10\). Then \(m\angle A=54+10=64^{\circ}\).

Answer:

The length of \(\overline{GH}\) is \(\sqrt{34}\), \(m\angle A = 64^{\circ}\)