Question

question 14 (1 point)
what is the value of x? identify the missing justifications.
diagram: angles at o, ∠aob = (2x)°, ∠boc = 6(x−3)°, ∠aoc = 150° (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. _____

question 15 (1 point)
a conditional can have a _____ of true or false.
○ a truth value
○ b hypothesis
○ c counterexample
○ d conclusion

Explanation:

Question 14

Step1: Identify Angle Addition Postulate

The equation \( m\angle AOB + m\angle BOC = m\angle AOC \) is by the Angle Addition Postulate (a: Angle Addition Postulate).

Step2: Substitute Angle Measures

Substitute \( m\angle AOB = 2x^\circ \) and \( m\angle BOC = 6(x - 3)^\circ \), \( m\angle AOC = 150^\circ \) (b: Substitution Property).

Step3: Distribute 6

Use the Distributive Property: \( 6(x - 3)=6x - 18 \), so \( 2x + 6x - 18 = 150 \) (c: Distributive Property).

Step4: Combine Like Terms

Combine \( 2x \) and \( 6x \) to get \( 8x \), so \( 8x - 18 = 150 \) (d: Combine Like Terms).

Step5: Add 18 to Both Sides

Add 18 to both sides: \( 8x - 18 + 18 = 150 + 18 \) (e: Addition Property of Equality), resulting in \( 8x = 168 \).

Step6: Divide by 8

Divide both sides by 8: \( \frac{8x}{8}=\frac{168}{8} \) (f: Division Property of Equality), giving \( x = 21 \).

A conditional statement (like "If p, then q") has a truth value (true or false) based on whether the conclusion follows from the hypothesis. A hypothesis is the "if" part, a counterexample disproves a statement, and a conclusion is the "then" part. So the correct term is "truth value".

Answer:

\( x = 21 \) (Justifications: a: Angle Addition Postulate; b: Substitution Property; c: Distributive Property; d: Combine Like Terms; e: Addition Property of Equality; f: Division Property of Equality)

Question 15