Question

1. fill in the blanks to complete the proof. (segment addition) given: ( gd = 3x - 4 ), ( dp = 13 ), ( gp = 27 ). prove: ( x = 6 ). table with columns “statements” (rows: ( gd = 3x - 4 ), ( dp = 13 ), ( gp = 27 ), ( gd + dp = gp ), ( 3x + 9 = 27 ), blank, ( x = 6 )) and “reasons” (rows: given, given, given, blank, blank, subtraction property, division property). diagram: segment ( gp ) with point ( d ) between ( g ) (left) and ( p ) (right).

Explanation:

Step1: Identify the segment addition postulate

The segment addition postulate states that if \( D \) is between \( G \) and \( P \), then \( GD + DP = GP \). So the reason for \( GD + DP = GP \) is the Segment Addition Postulate.

Step2: Substitute the known values

We know \( GD = 3x - 4 \), \( DP = 13 \), and \( GP = 27 \). Substituting these into \( GD + DP = GP \), we get \( (3x - 4)+13 = 27 \). Simplify the left - hand side: \( 3x+(- 4 + 13)=27\), so \( 3x + 9 = 27 \). The reason for this step is Substitution Property (substituting the given values of \( GD \), \( DP \), and \( GP \) into the segment addition equation).

Step3: Use the subtraction property of equality

To isolate the term with \( x \), we subtract 9 from both sides of the equation \( 3x + 9 = 27 \). By the subtraction property of equality (if \( a=b \), then \( a - c=b - c \)), we have \( 3x+9 - 9=27 - 9 \), which simplifies to \( 3x=18 \).

Step4: Use the division property of equality

To solve for \( x \), we divide both sides of the equation \( 3x = 18 \) by 3. By the division property of equality (if \( a=b \), then \( \frac{a}{c}=\frac{b}{c} \) for \( c
eq0 \)), we get \( \frac{3x}{3}=\frac{18}{3} \), so \( x = 6 \).

Filling in the table:

Statements	Reasons
\( GD + DP = GP \)	Segment Addition Postulate
\( (3x - 4)+13 = 27 \)	Substitution Property
\( 3x + 9 = 27 \)	Simplification (from previous step)
\( 3x+9 - 9=27 - 9 \) (i.e., \( 3x = 18 \))	Subtraction Property of Equality
\( \frac{3x}{3}=\frac{18}{3} \) (i.e., \( x = 6 \))	Division Property of Equality

Answer:

The completed proof table has the following reasons (and corresponding statements as above): For \( GD + DP = GP \), the reason is "Segment Addition Postulate"; for \( (3x - 4)+13 = 27 \), the reason is "Substitution Property"; for \( 3x = 18 \), the reason is "Subtraction Property of Equality"; and the final step to get \( x = 6 \) uses the "Division Property of Equality". The key steps fill in the blanks with the appropriate properties and the derived equations.