4.3 practice: two - column geometric proofs
prove the conjecture: if a, b, and d are collinear and ad = 40 and bd = 29, then ab = 11.
given: a, b, and d are collinear and ad = 40 and bd = 29
prove: ab = 11

prove the conjecture: if b is the midpoint of $ \overline { a c } $, and ab = 3x + 4 and bc = 5x - 6, then x = 5.
given: b is the midpoint of $ \overline { a c } $, ab = 3x + 4, and bc = 5x - 6.
prove: x = 5

Question

4.3 practice: two - column geometric proofs
prove the conjecture: if a, b, and d are collinear and ad = 40 and bd = 29, then ab = 11.
given: a, b, and d are collinear and ad = 40 and bd = 29
prove: ab = 11

prove the conjecture: if b is the midpoint of $ \overline { a c } $, and ab = 3x + 4 and bc = 5x - 6, then x = 5.
given: b is the midpoint of $ \overline { a c } $, ab = 3x + 4, and bc = 5x - 6.
prove: x = 5

Accepted Answer

# Explanation:
## Step1: Fill in the first proof
- **Statement**: $A,B,$ and $D$ are collinear
  - **Reason**: Given
- **Statement**: $AB + BD=AD$
  - **Reason**: Segment Addition Postulate (If three points $A$, $B$, and $D$ are collinear with $B$ between $A$ and $D$, then the length of $AD$ is the sum of the lengths of $AB$ and $BD$)
- **Statement**: $AD = 40$, $BD=29$
  - **Reason**: Given
- **Statement**: $AB + 29=40$
  - **Reason**: Substitution (Substitute $AD = 40$ and $BD = 29$ into $AB + BD=AD$)
- **Statement**: $AB=11$
  - **Reason**: Subtraction Property of Equality ($AB=40 - 29$)

## Step2: Fill in the second proof
- **Statement**: $B$ is the midpoint of $\overline{AC}$
  - **Reason**: Given
- **Statement**: $AB = BC$
  - **Reason**: Definition of Midpoint (A midpoint of a segment divides the segment into two equal - length parts)
- **Statement**: $AB = 3x + 4$, $BC=5x - 6$
  - **Reason**: Given
- **Statement**: $3x + 4=5x - 6$
  - **Reason**: Substitution (Since $AB = BC$, substitute $AB = 3x + 4$ and $BC=5x - 6$)
- **Statement**: $4+6=5x - 3x$
  - **Reason**: Subtraction Property of Equality (Subtract $3x$ from both sides and add $6$ to both sides)
- **Statement**: $10 = 2x$
  - **Reason**: Simplification ($4 + 6=10$ and $5x-3x = 2x$)
- **Statement**: $x = 5$
  - **Reason**: Division Property of Equality ($x=\frac{10}{2}$)

# Answer:
For the first proof:
| Statements | Reasons |
|---|---|
| $A,B,$ and $D$ are collinear | Given |
| $AB + BD=AD$ | Segment Addition Postulate |
| $AD = 40$, $BD = 29$ | Given |
| $AB + 29=40$ | Substitution |
| $AB=11$ | Subtraction Property of Equality |

For the second proof:
| Statements | Reasons |
|---|---|
| $B$ is the midpoint of $\overline{AC}$ | Given |
| $AB = BC$ | Definition of Midpoint |
| $AB = 3x + 4$, $BC=5x - 6$ | Given |
| $3x + 4=5x - 6$ | Substitution |
| $4 + 6=5x-3x$ | Subtraction Property of Equality |
| $10 = 2x$ | Simplification |
| $x = 5$ | Division Property of Equality |

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QUESTION IMAGE

Question

Explanation:

Step1: Fill in the first proof

Step2: Fill in the second proof

Answer:

Statements	Reasons
\(AB + BD=AD\)	Segment Addition Postulate
\(AD = 40\), \(BD = 29\)	Given
\(AB + 29=40\)	Substitution
\(AB=11\)	Subtraction Property of Equality

Statements	Reasons
\(AB = BC\)	Definition of Midpoint
\(AB = 3x + 4\), \(BC=5x - 6\)	Given
\(3x + 4=5x - 6\)	Substitution
\(4 + 6=5x-3x\)	Subtraction Property of Equality
\(10 = 2x\)	Simplification
\(x = 5\)	Division Property of Equality