Question

1. given that the m∠doc = 3/4(m∠boa), find the measures of the marked angles showing and justifying all steps.

hint: write an equation that relates y to x. what must x + y+120 equal? don’t forget to use the given information in order to substitute for one of the variables.

Explanation:

Step1: Recall angle - sum property

Since the angles around a point sum to 360°, we have the equation $x + y+120^{\circ}=360^{\circ}$. This can be rewritten as $x + y=360^{\circ}- 120^{\circ}=240^{\circ}$, so $y = 240^{\circ}-x$.

Step2: Use the given relationship

We are given that $m\angle DOC=\frac{3}{4}(m\angle BOA)$, so $y=\frac{3}{4}x$.

Step3: Substitute and solve

Substitute $y = \frac{3}{4}x$ into $y = 240^{\circ}-x$. We get $\frac{3}{4}x=240^{\circ}-x$. Add $x$ to both sides: $\frac{3}{4}x+x=240^{\circ}$. Combine like - terms: $\frac{3x + 4x}{4}=240^{\circ}$, or $\frac{7x}{4}=240^{\circ}$. Multiply both sides by $\frac{4}{7}$: $x=\frac{4\times240^{\circ}}{7}=\frac{960^{\circ}}{7}\approx137.14^{\circ}$.

Step4: Find the value of y

Substitute $x=\frac{960^{\circ}}{7}$ into $y=\frac{3}{4}x$. Then $y=\frac{3}{4}\times\frac{960^{\circ}}{7}=\frac{720^{\circ}}{7}\approx102.86^{\circ}$.

Answer:

$x=\frac{960^{\circ}}{7}\approx137.14^{\circ}$, $y=\frac{720^{\circ}}{7}\approx102.86^{\circ}$