Question

question 10 (1 point)
what is the value of x? identify the missing justifications.
m∠pqr = x − 5, m∠sqr = x − 7, and m∠pqs = 100.

image of angles at q: p, r, s with q as vertex
drawing not to scale

m∠pqr + m∠sqr = m∠pqs a. _______
x − 5 + x − 7 = 100 b. substitution property
2x − 12 = 100 c. simplify
2x = 112 d. _______
x = 56 e. division property of equality

options:
○ a angle addition postulate; subtraction property of equality
○ b protractor postulate; addition property of equality
○ c angle addition postulate; addition property of equality
○ d protractor postulate; subtraction property of equality

Explanation:

Step1: Analyze part a

The equation \( m\angle PQR + m\angle SQR = m\angle PQS \) is derived from the Angle Addition Postulate, which states that if a point lies in the interior of an angle, the sum of the two smaller angles formed is equal to the measure of the larger angle. So part a's justification is the Angle Addition Postulate.

Step2: Analyze part d

To get from \( 2x - 12 = 100 \) to \( 2x = 112 \), we add 12 to both sides of the equation. This is the Addition Property of Equality, which states that if we add the same number to both sides of an equation, the equation remains true.

Now, looking at the options:

• Option a: Has Angle Addition Postulate for a and Addition Property of Equality for d? Wait, no, option a says subtraction for d. Wait, let's re - check the options.

Wait, the options are about a and d? Wait, no, the options are labeled a - d but the question is about the missing justifications (a and d). Wait, the options are:

• Option c: Angle Addition Postulate (for a) and Addition Property of Equality (for d). Let's verify:

For part a: \( m\angle PQR + m\angle SQR = m\angle PQS \) is Angle Addition Postulate.
For part d: From \( 2x - 12 = 100 \) to \( 2x=112 \), we add 12 to both sides, which is Addition Property of Equality. So option c is correct.

Answer:

c. Angle Addition Postulate; Addition Property of Equality