Question

question 14 (1 point)
what is the value of x? identify the missing justifications

diagram: point o with rays oa, ob, oc (oa left - upward, ob upward, oc horizontal right); ∠aob = (2x)°, ∠boc = 6(x−3)°; note: drawing not to scale
m∠aoc = 150

m∠aob + m∠boc = m∠aoc a. _____
2x + 6(x − 3) = 150 b. _____
2x + 6x − 18 = 150 c. _____
8x − 18 = 150 d. _____
8x = 168 e. _____
x = 21 f. _____

Explanation:

Step1: Identify the angle addition postulate

The equation \( m\angle AOB + m\angle BOC = m\angle AOC \) is based on the Angle Addition Postulate, which states that if a point \( B \) lies in the interior of \( \angle AOC \), then \( m\angle AOB + m\angle BOC = m\angle AOC \).

Step2: Substitute the angle measures

We know \( m\angle AOB = (2x)^\circ \) and \( m\angle BOC = 6(x - 3)^\circ \), and \( m\angle AOC = 150^\circ \). Substituting these values into the angle addition postulate equation gives \( 2x + 6(x - 3)=150 \). This is the Substitution Property of Equality.

Step3: Distribute the 6

Using the Distributive Property (which states that \( a(b + c)=ab + ac \), here \( a = 6 \), \( b=x \), \( c=-3 \)), we expand \( 6(x - 3) \) to get \( 6x-18 \). So the equation becomes \( 2x + 6x-18 = 150 \).

Step4: Combine like terms

Combine the \( x \)-terms \( 2x \) and \( 6x \). Since \( 2x+6x = 8x \), the equation simplifies to \( 8x - 18 = 150 \). This is combining like terms.

Step5: Add 18 to both sides

To isolate the term with \( x \), we use the Addition Property of Equality (if \( a = b \), then \( a + c = b + c \)). Adding 18 to both sides of \( 8x - 18 = 150 \) gives \( 8x=150 + 18=168 \).

Step6: Divide both sides by 8

Using the Division Property of Equality (if \( a = b \) and \( c
eq0 \), then \( \frac{a}{c}=\frac{b}{c} \)), we divide both sides of \( 8x = 168 \) by 8. So \( x=\frac{168}{8}=21 \).

Answer:

The value of \( x \) is \( \boldsymbol{21} \).

Justifications:
a. Angle Addition Postulate
b. Substitution Property of Equality
c. Distributive Property
d. Combine Like Terms
e. Addition Property of Equality
f. Division Property of Equality