Question

if (mangle aoc = 85^{circ}), (mangle boc=2x + 10), and (mangle aob = 4x-15), find the degree - measure of (angle boc) and (angle aob). the diagram is not to scale.

a. (mangle boc = 30^{circ}), (mangle aob = 55^{circ})
b. (mangle boc = 40^{circ}), (mangle aob = 45^{circ})
c. (mangle boc = 45^{circ}), (mangle aob = 40^{circ})
d. (mangle boc = 55^{circ}), (mangle aob = 30^{circ})

Explanation:

Step1: Use angle - addition postulate

Since $\angle AOC=\angle AOB+\angle BOC$, we have $85=(4x - 15)+(2x + 10)$.

Step2: Simplify the right - hand side

Combine like terms: $(4x - 15)+(2x + 10)=4x+2x-15 + 10=6x-5$. So, $85 = 6x-5$.

Step3: Solve for $x$

Add 5 to both sides: $85 + 5=6x-5 + 5$, which gives $90 = 6x$. Then divide both sides by 6: $x=\frac{90}{6}=15$.

Step4: Find the measure of $\angle BOC$

Substitute $x = 15$ into the expression for $\angle BOC$: $m\angle BOC=2x + 10=2\times15+10=30 + 10=40^{\circ}$.

Step5: Find the measure of $\angle AOB$

Substitute $x = 15$ into the expression for $\angle AOB$: $m\angle AOB=4x-15=4\times15-15=60 - 15=45^{\circ}$.

Answer:

B. $m\angle BOC = 40^{\circ},m\angle AOB = 45^{\circ}$