10 given: y is the midpoint of $\overline{xz}$ prove: $xy = \frac{1}{2}xz$ \begin{tabular}{|c|c|} \hline statements & reasons \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular} 11 given: $\overline{ac} \cong \overline{df}$, $\overline{bc} \cong \overline{de}$ prove: $\overline{ab} \cong \overline{ef}$ \begin{tabular}{|c|c|} \hline statements & reasons \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular} 12 given: $\overline{ab} \cong \overline{cd}$; prove: $\overline{ac} \cong \overline{bd}$ \begin{tabular}{|c|c|} \hline statements & reasons \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular}

Question

Accepted Answer

### Problem 10 (Proving $ XY = \frac{1}{2}XZ $ given $ Y $ is the midpoint of $ \overline{XZ} $)
# Explanation:
## Step 1: Recall the midpoint definition
A midpoint of a segment divides it into two congruent (equal - length) segments. So, if $ Y $ is the midpoint of $ \overline{XZ} $, then $ XY = YZ $.
## Step 2: Use the segment addition postulate
The segment addition postulate states that for a segment $ \overline{XZ} $ with a point $ Y $ on it, $ XZ=XY + YZ $.
## Step 3: Substitute $ YZ $ with $ XY $
Since $ XY = YZ $ (from the midpoint definition), we can substitute $ YZ $ in the equation $ XZ = XY+YZ $ with $ XY $. So $ XZ=XY + XY=2XY $.
## Step 4: Solve for $ XY $
Divide both sides of the equation $ XZ = 2XY $ by 2. We get $ XY=\frac{1}{2}XZ $.

### Filling the two - column proof:
| Statements | Reasons |
| --- | --- |
| 1. $ Y $ is the midpoint of $ \overline{XZ} $ | Given |
| 2. $ XY = YZ $ | Definition of midpoint |
| 3. $ XZ=XY + YZ $ | Segment Addition Postulate |
| 4. $ XZ=XY + XY $ (or $ XZ = 2XY $) | Substitution (from step 2) |
| 5. $ XY=\frac{1}{2}XZ $ | Division Property of Equality |

### Problem 11 (Proving $ \overline{AB}\cong\overline{EF} $ given $ \overline{AC}\cong\overline{DF} $ and $ \overline{BC}\cong\overline{DE} $)
# Explanation:
## Step 1: Recall the definition of congruent segments
If two segments are congruent, their lengths are equal. So, if $ \overline{AC}\cong\overline{DF} $, then $ AC = DF $, and if $ \overline{BC}\cong\overline{DE} $, then $ BC = DE $.
## Step 2: Use the segment addition postulate for $ \overline{AC} $ and $ \overline{DF} $
For $ \overline{AC} $, $ AC=AB + BC $ (Segment Addition Postulate). For $ \overline{DF} $, $ DF = DE+EF $ (Segment Addition Postulate).
## Step 3: Substitute $ AC $ and $ DF $ and $ BC $ and $ DE $
Since $ AC = DF $ and $ BC = DE $, we substitute into the equations from step 2: $ AB + BC=DE + EF $.
## Step 4: Substitute $ DE $ with $ BC $
Since $ BC = DE $, we can substitute $ DE $ in the equation $ AB + BC=DE + EF $ with $ BC $. So $ AB + BC=BC + EF $.
## Step 5: Subtract $ BC $ from both sides
Subtract $ BC $ from both sides of the equation $ AB + BC=BC + EF $. By the Subtraction Property of Equality, we get $ AB = EF $.
## Step 6: Recall the definition of congruent segments
If $ AB = EF $, then $ \overline{AB}\cong\overline{EF} $ (Definition of congruent segments).

### Filling the two - column proof:
| Statements | Reasons |
| --- | --- |
| 1. $ \overline{AC}\cong\overline{DF} $, $ \overline{BC}\cong\overline{DE} $ | Given |
| 2. $ AC = DF $, $ BC = DE $ | Definition of congruent segments |
| 3. $ AC=AB + BC $, $ DF = DE + EF $ | Segment Addition Postulate |
| 4. $ AB + BC=DE + EF $ | Substitution (from step 2) |
| 5. $ AB + BC=BC + EF $ | Substitution (from step 2, $ DE = BC $) |
| 6. $ AB = EF $ | Subtraction Property of Equality |
| 7. $ \overline{AB}\cong\overline{EF} $ | Definition of congruent segments |

### Problem 12 (Proving $ \overline{AC}\cong\overline{BD} $ given $ \overline{AB}\cong\overline{CD} $)
# Explanation:
## Step 1: Recall the definition of congruent segments
If $ \overline{AB}\cong\overline{CD} $, then $ AB = CD $ (Definition of congruent segments).
## Step 2: Use the segment addition postulate
For the segment $ \overline{AC} $, $ AC=AB + BC $. For the segment $ \overline{BD} $, $ BD = BC+CD $.
## Step 3: Substitute $ CD $ with $ AB $
Since $ AB = CD $ (from step 1), we substitute $ CD $ in the equation $ BD = BC + CD $ with $ AB $. So $ BD=BC + AB $.
## Step 4: Compare $ AC $ and $ BD $
From step 2, $ AC = AB + BC $, and from step 3, $ BD=AB + BC $. So $ AC = BD $.
## Step 5: Recall the definition of congruent segments
If $ AC = BD $, then $ \overline{AC}\cong\overline{BD} $ (Definition of congruent segments).

### Filling the two - column proof:
| Statements | Reasons |
| --- | --- |
| 1. $ \overline{AB}\cong\overline{CD} $ | Given |
| 2. $ AB = CD $ | Definition of congruent segments |
| 3. $ AC=AB + BC $, $ BD = BC + CD $ | Segment Addition Postulate |
| 4. $ BD=BC + AB $ | Substitution (from step 2) |
| 5. $ AC = BD $ | Substitution (from steps 3 and 4) |
| 6. $ \overline{AC}\cong\overline{BD} $ | Definition of congruent segments |

# Answers:
### Problem 10
The proof is completed as shown in the two - column table above, and we have successfully proven that $ XY=\frac{1}{2}XZ $.

### Problem 11
The proof is completed as shown in the two - column table above, and we have successfully proven that $ \overline{AB}\cong\overline{EF} $.

### Problem 12
The proof is completed as shown in the two - column table above, and we have successfully proven that $ \overline{AC}\cong\overline{BD} $.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 10 (Proving \( XY = \frac{1}{2}XZ \) given \( Y \) is the midpoint of \( \overline{XZ} \))

Step 1: Recall the midpoint definition

Step 2: Use the segment addition postulate

Step 3: Substitute \( YZ \) with \( XY \)

Step 4: Solve for \( XY \)

Filling the two - column proof:

Problem 11 (Proving \( \overline{AB}\cong\overline{EF} \) given \( \overline{AC}\cong\overline{DF} \) and \( \overline{BC}\cong\overline{DE} \))

Step 1: Recall the definition of congruent segments

Step 2: Use the segment addition postulate for \( \overline{AC} \) and \( \overline{DF} \)

Step 3: Substitute \( AC \) and \( DF \) and \( BC \) and \( DE \)

Step 4: Substitute \( DE \) with \( BC \)

Step 5: Subtract \( BC \) from both sides

Step 6: Recall the definition of congruent segments

Filling the two - column proof:

Problem 12 (Proving \( \overline{AC}\cong\overline{BD} \) given \( \overline{AB}\cong\overline{CD} \))

Step 1: Recall the definition of congruent segments

Step 2: Use the segment addition postulate

Step 3: Substitute \( CD \) with \( AB \)

Step 4: Compare \( AC \) and \( BD \)

Step 5: Recall the definition of congruent segments

Filling the two - column proof:

Answer:

Problem 10

Problem 11

Problem 12

Statements	Reasons
2. \( XY = YZ \)	Definition of midpoint
3. \( XZ=XY + YZ \)	Segment Addition Postulate
4. \( XZ=XY + XY \) (or \( XZ = 2XY \))	Substitution (from step 2)
5. \( XY=\frac{1}{2}XZ \)	Division Property of Equality

Statements	Reasons
2. \( AC = DF \), \( BC = DE \)	Definition of congruent segments
3. \( AC=AB + BC \), \( DF = DE + EF \)	Segment Addition Postulate
4. \( AB + BC=DE + EF \)	Substitution (from step 2)
5. \( AB + BC=BC + EF \)	Substitution (from step 2, \( DE = BC \))
6. \( AB = EF \)	Subtraction Property of Equality
7. \( \overline{AB}\cong\overline{EF} \)	Definition of congruent segments

Statements	Reasons
2. \( AB = CD \)	Definition of congruent segments
3. \( AC=AB + BC \), \( BD = BC + CD \)	Segment Addition Postulate
4. \( BD=BC + AB \)	Substitution (from step 2)
5. \( AC = BD \)	Substitution (from steps 3 and 4)
6. \( \overline{AC}\cong\overline{BD} \)	Definition of congruent segments