Question

1. determine the missing angle measure(s) in each problem at right using the geometric relationships shown in the diagram at right. be sure to write down the conjecture that justifies each calculation. each part is a separate problem.

a. if (d = 110^{circ}) and (k=5x - 20^{circ}), write an equation and solve for (x).
b. if (b = 4x - 11^{circ}) and (n=x + 26^{circ}), write an equation and solve for (x). then determine the measure of (n).

Explanation:

Step1: Identify angle - relationship for part a

Assume $d$ and $k$ are corresponding angles (or vertical - like angles depending on parallel lines), so $d = k$.
$110^{\circ}=5x - 20^{\circ}$

Step2: Solve the equation for $x$ in part a

Add $20^{\circ}$ to both sides.
$110^{\circ}+20^{\circ}=5x-20^{\circ}+20^{\circ}$
$130^{\circ}=5x$
Divide both sides by 5.
$x=\frac{130^{\circ}}{5}=26^{\circ}$

Step3: Identify angle - relationship for part b

Assume $b$ and $n$ are corresponding angles (or vertical - like angles depending on parallel lines), so $b = n$.
$4x-11^{\circ}=x + 26^{\circ}$

Step4: Solve the equation for $x$ in part b

Subtract $x$ from both sides.
$4x-x-11^{\circ}=x - x+26^{\circ}$
$3x-11^{\circ}=26^{\circ}$
Add $11^{\circ}$ to both sides.
$3x-11^{\circ}+11^{\circ}=26^{\circ}+11^{\circ}$
$3x=37^{\circ}$
Divide both sides by 3.
$x=\frac{37^{\circ}}{3}$

Step5: Find the measure of $n$ in part b

Substitute $x = \frac{37^{\circ}}{3}$ into the expression for $n$.
$n=x + 26^{\circ}=\frac{37^{\circ}}{3}+26^{\circ}=\frac{37^{\circ}+78^{\circ}}{3}=\frac{115^{\circ}}{3}\approx38.33^{\circ}$

Answer:

a. $x = 26^{\circ}$
b. $x=\frac{37^{\circ}}{3}$, $n=\frac{115^{\circ}}{3}$