Question

statements

1. ( mangle rst = mangle rsu + mangle ust );

( mangle rsu = 3x - 5 ); ( mangle ust = x + 4 );
( mangle rst = 6x - 7 )

1. ( 6x - 7 = (3x - 5) + (x + 4) )
2. blank
3. ( 2x - 7 = -1 )
4. ( 2x = 6 )
5. blank

reasons

1. given
2. blank
3. simplify
4. subtraction property of equality
5. blank
6. division property of equality

options:
a. ( x = 3 )
b. distributive property of equality
c. substitution property of equality
d. addition property of equality (partially visible)

Explanation:

Step 1: Analyze Step 2 (Reason)

We know the measures of \(\angle RSU\), \(\angle UST\), and \(\angle RST\) from the given information. To get the equation \(6x - 7=(3x - 5)+(x + 4)\), we substitute the given angle measures into the angle - addition postulate (\(m\angle RST=m\angle RSU + m\angle UST\)). So the reason for step 2 is the Substitution Property of Equality (option c).

Step 2: Simplify Step 2's Equation for Step 3

Simplify the right - hand side of the equation \(6x - 7=(3x - 5)+(x + 4)\). Combine like terms: \(3x+x=4x\) and \(- 5 + 4=-1\). So the equation becomes \(6x-7 = 4x-1\).

Step 3: Analyze Step 5 (Reason)

We go from \(2x-7=-1\) to \(2x = 6\). To do this, we add 7 to both sides of the equation. This is the Addition Property of Equality (option d).

Step 4: Analyze Step 6 (Statement)

We have the equation \(2x = 6\). Using the Division Property of Equality, we divide both sides of the equation by 2. \(\frac{2x}{2}=\frac{6}{2}\), so \(x = 3\) (option a).

Final Answers:

• Step 2 Reason: c. Substitution Property of Equality
• Step 3 Statement: \(6x-7 = 4x-1\)
• Step 5 Reason: d. Addition Property of Equality
• Step 6 Statement: a. \(x = 3\)

Answer:

Step 1: Analyze Step 2 (Reason)

We know the measures of \(\angle RSU\), \(\angle UST\), and \(\angle RST\) from the given information. To get the equation \(6x - 7=(3x - 5)+(x + 4)\), we substitute the given angle measures into the angle - addition postulate (\(m\angle RST=m\angle RSU + m\angle UST\)). So the reason for step 2 is the Substitution Property of Equality (option c).

Step 2: Simplify Step 2's Equation for Step 3

Simplify the right - hand side of the equation \(6x - 7=(3x - 5)+(x + 4)\). Combine like terms: \(3x+x=4x\) and \(- 5 + 4=-1\). So the equation becomes \(6x-7 = 4x-1\).

Step 3: Analyze Step 5 (Reason)

We go from \(2x-7=-1\) to \(2x = 6\). To do this, we add 7 to both sides of the equation. This is the Addition Property of Equality (option d).

Step 4: Analyze Step 6 (Statement)

We have the equation \(2x = 6\). Using the Division Property of Equality, we divide both sides of the equation by 2. \(\frac{2x}{2}=\frac{6}{2}\), so \(x = 3\) (option a).

Final Answers:

• Step 2 Reason: c. Substitution Property of Equality
• Step 3 Statement: \(6x-7 = 4x-1\)
• Step 5 Reason: d. Addition Property of Equality
• Step 6 Statement: a. \(x = 3\)