Question

objective 4: i can write proofs involving segments.

1. fill in the blanks to complete the proof. use items from the bank below – not all items will be used. (3 points)

given, ( ab = 4x - 9 ) and ( bc = 15 ) and ( b ) is the midpoint of ( ac )
prove ( x = 6 )
diagram: segment ( ac ) with ( b ) as midpoint
statements:

• ( ab = 4x - 9 ) and ( bc = 15 ) and ( b ) is the midpoint of ( ac )
• ( ab = bc )
• blank
• ( 4x = 24 )
• ( x = 6 )

reasons:

• given
• blank (substitution property of equality)
• blank
• division property of equality

bank (from the box): definition of midpoint, subtraction property of equality, addition property of equality, symmetric property of equality, ( 4x - 9 = 15 )

1. using your answer from part a, find ( de ) (1 point)

Explanation:

To solve the proof, we analyze each step:

Step 1: Justify \( AB = BC \)

Since \( B \) is the midpoint of \( AC \), by the Definition of Midpoint, a midpoint divides a segment into two equal parts. Thus, \( AB = BC \).

Step 2: Justify \( 4x - 9 = 15 \)

We know \( AB = 4x - 9 \) (given) and \( BC = 15 \) (given). From Step 1, \( AB = BC \). By the Substitution Property of Equality, we substitute \( AB \) with \( 4x - 9 \) and \( BC \) with \( 15 \), so \( 4x - 9 = 15 \).

Step 3: Justify \( 4x = 24 \)

Starting with \( 4x - 9 = 15 \), we use the Addition Property of Equality (add 9 to both sides):
\( 4x - 9 + 9 = 15 + 9 \)
Simplifying, \( 4x = 24 \).

Step 4: Justify \( x = 6 \)

Starting with \( 4x = 24 \), we use the Division Property of Equality (divide both sides by 4):
\( \frac{4x}{4} = \frac{24}{4} \)
Simplifying, \( x = 6 \).

Filling the Table:

STATEMENTS	REASONS
\( AB = BC \)	Definition of Midpoint
\( 4x - 9 = 15 \)	Substitution Property of Equality
\( 4x = 24 \)	Addition Property of Equality
\( x = 6 \)	Division Property of Equality

Answer:

To solve the proof, we analyze each step:

Step 1: Justify \( AB = BC \)

Since \( B \) is the midpoint of \( AC \), by the Definition of Midpoint, a midpoint divides a segment into two equal parts. Thus, \( AB = BC \).

Step 2: Justify \( 4x - 9 = 15 \)

We know \( AB = 4x - 9 \) (given) and \( BC = 15 \) (given). From Step 1, \( AB = BC \). By the Substitution Property of Equality, we substitute \( AB \) with \( 4x - 9 \) and \( BC \) with \( 15 \), so \( 4x - 9 = 15 \).

Step 3: Justify \( 4x = 24 \)

Starting with \( 4x - 9 = 15 \), we use the Addition Property of Equality (add 9 to both sides):
\( 4x - 9 + 9 = 15 + 9 \)
Simplifying, \( 4x = 24 \).

Step 4: Justify \( x = 6 \)

Starting with \( 4x = 24 \), we use the Division Property of Equality (divide both sides by 4):
\( \frac{4x}{4} = \frac{24}{4} \)
Simplifying, \( x = 6 \).

Filling the Table:

STATEMENTS	REASONS
\( AB = BC \)	Definition of Midpoint
\( 4x - 9 = 15 \)	Substitution Property of Equality
\( 4x = 24 \)	Addition Property of Equality
\( x = 6 \)	Division Property of Equality