fill in the missing information in each proof.
4. given: $\overline{gh} \cong \overline{kl}$, $\angle g \cong \angle k$, and $\overline{gi} \cong \overline{kj}$
prove: $\overline{hi} \cong \overline{lj}$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1. \overline{gh} \cong \overline{kl} & 1. \text{given} \\\hline 2. & 2. \text{given} \\\hline 3. \overline{gi} \cong \overline{kj} & 3. \\\hline 4. & 4. \text{sas} \\\hline 5. \overline{hi} \cong \overline{lj} & 5. \\\hline \end{array}

5. given: \(\angle mnp \cong \angle opn$, and $\overline{mn} \cong \overline{op}$
prove: $\overline{mp} \cong \overline{no}$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1. & 1. \text{given} \\\hline 2. \overline{mn} \cong \overline{op} & 2. \\\hline 3. \overline{np} \cong \overline{np} & 3. \\\hline 4. \triangle mnp \cong \triangle opn & 4. \\\hline 5. & 5. \text{cpctc} \\\hline \end{array}

6. given: \(\overline{st} \cong \overline{vu}$
prove: $\angle svt \cong \angle utv$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1. & 1. \text{given} \\\hline 2. & 2. \text{reflexive property} \\\hline 3. & 3. \text{hl} \\\hline 4. \angle svt \cong \angle utv & 4. \\\hline \end{array}

7. given: \(\overline{ac} \cong \overline{ce}$, $\overline{dc} \cong \overline{bc}$
prove: $\angle b \cong \angle d$
\(\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1. & 1. \\\hline 2. & 2. \text{given} \\\hline 3. \angle acb \cong \angle dce & 3. \\\hline 4. \triangle abc \cong \triangle def & 4. \\\hline 5. \angle b \cong \angle d & 5. \\\hline \end{array}

Question

fill in the missing information in each proof.
4. given: $\overline{gh} \cong \overline{kl}$, $\angle g \cong \angle k$, and $\overline{gi} \cong \overline{kj}$
prove: $\overline{hi} \cong \overline{lj}$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1. \overline{gh} \cong \overline{kl} & 1. \text{given} \\\hline 2.  & 2. \text{given} \\\hline 3. \overline{gi} \cong \overline{kj} & 3.  \\\hline 4.  & 4. \text{sas} \\\hline 5. \overline{hi} \cong \overline{lj} & 5.  \\\hline \end{array}

5. given: \(\angle mnp \cong \angle opn$, and $\overline{mn} \cong \overline{op}$
prove: $\overline{mp} \cong \overline{no}$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1.  & 1. \text{given} \\\hline 2. \overline{mn} \cong \overline{op} & 2.  \\\hline 3. \overline{np} \cong \overline{np} & 3.  \\\hline 4. \triangle mnp \cong \triangle opn & 4.  \\\hline 5.  & 5. \text{cpctc} \\\hline \end{array}

6. given: \(\overline{st} \cong \overline{vu}$
prove: $\angle svt \cong \angle utv$
$\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1.  & 1. \text{given} \\\hline 2.  & 2. \text{reflexive property} \\\hline 3.  & 3. \text{hl} \\\hline 4. \angle svt \cong \angle utv & 4.  \\\hline \end{array}

7. given: \(\overline{ac} \cong \overline{ce}$, $\overline{dc} \cong \overline{bc}$
prove: $\angle b \cong \angle d$
\(\begin{array}{|l|l|}\hline \text{statements} & \text{reasons} \\\hline 1.  & 1.  \\\hline 2.  & 2. \text{given} \\\hline 3. \angle acb \cong \angle dce & 3.  \\\hline 4. \triangle abc \cong \triangle def & 4.  \\\hline 5. \angle b \cong \angle d & 5.  \\\hline \end{array}

Accepted Answer

# Explanation:
## Step1: Analyze Problem 4
We are given $\overline{GH} \cong \overline{KL}$, $\angle G \cong \angle K$, and $\overline{GI} \cong \overline{KJ}$. We need to prove $\overline{HI} \cong \overline{LJ}$.
### Step 1.1: Identify Given Angles
Statement 2: $\angle G \cong \angle K$ (Reason: Given)
### Step 1.2: Identify Given Sides (GI and KJ)
Statement 3: $\overline{GI} \cong \overline{KJ}$ (Reason: Given)
### Step 1.3: Apply SAS Congruence
We have two sides and the included angle. So, $\triangle GHI \cong \triangle KLJ$ by SAS (Side - Angle - Side) congruence criterion.
Statement 4: $\triangle GHI \cong \triangle KLJ$ (Reason: SAS)
### Step 1.4: Apply CPCTC
Since $\triangle GHI \cong \triangle KLJ$, corresponding parts are congruent. So, $\overline{HI} \cong \overline{LJ}$ (Reason: CPCTC - Corresponding Parts of Congruent Triangles are Congruent)

## Step2: Analyze Problem 5
We are given $\angle MNP \cong \angle OPN$ and $\overline{MN} \cong \overline{OP}$. We need to prove $\overline{MP} \cong \overline{NO}$.
### Step 2.1: Identify Given Angle
Statement 1: $\angle MNP \cong \angle OPN$ (Reason: Given)
### Step 2.2: Identify Given Side (MN and OP)
Statement 2: $\overline{MN} \cong \overline{OP}$ (Reason: Given)
### Step 2.3: Identify Common Side (NP)
Statement 3: $\overline{NP} \cong \overline{NP}$ (Reason: Reflexive Property of Congruence)
### Step 2.4: Apply SAS Congruence
We have two sides and the included angle. So, $\triangle MNP \cong \triangle OPN$ by SAS (Side - Angle - Side) congruence criterion.
Statement 4: $\triangle MNP \cong \triangle OPN$ (Reason: SAS)
### Step 2.5: Apply CPCTC
Since $\triangle MNP \cong \triangle OPN$, corresponding parts are congruent. So, $\overline{MP} \cong \overline{NO}$ (Reason: CPCTC)

## Step3: Analyze Problem 6
We are given $\overline{ST} \cong \overline{VU}$ and right angles at $S$ and $U$ ($\angle S = \angle U=90^{\circ}$). We need to prove $\angle SVT \cong \angle UTV$.
### Step 3.1: Identify Given Side (ST and VU)
Statement 1: $\overline{ST} \cong \overline{VU}$ (Reason: Given)
### Step 3.2: Identify Common Side (TV)
Statement 2: $\overline{TV} \cong \overline{TV}$ (Reason: Reflexive Property)
### Step 3.3: Apply HL Congruence
Since $\angle S=\angle U = 90^{\circ}$, $\overline{ST} \cong \overline{VU}$ and $\overline{TV} \cong \overline{TV}$, $\triangle STV \cong \triangle UVT$ by HL (Hypotenuse - Leg) congruence criterion.
Statement 3: $\triangle STV \cong \triangle UVT$ (Reason: HL)
### Step 3.4: Apply CPCTC
Since $\triangle STV \cong \triangle UVT$, corresponding angles are congruent. So, $\angle SVT \cong \angle UTV$ (Reason: CPCTC)

## Step4: Analyze Problem 7
We are given $\overline{AC} \cong \overline{CE}$ and $\overline{DC} \cong \overline{BC}$. We need to prove $\angle B \cong \angle D$.
### Step 4.1: Identify Given Sides (AC and CE, DC and BC)
Statement 1: $\overline{AC} \cong \overline{CE}$ (Reason: Given)
Statement 2: $\overline{DC} \cong \overline{BC}$ (Reason: Given)
### Step 4.2: Identify Vertical Angles
Statement 3: $\angle ACB \cong \angle DCE$ (Reason: Vertical Angles are Congruent)
### Step 4.3: Apply SAS Congruence
We have two sides and the included angle. So, $\triangle ABC \cong \triangle EDC$ by SAS (Side - Angle - Side) congruence criterion. Wait, in the problem it is written as $\triangle ABC \cong \triangle DEF$, but it should be $\triangle ABC \cong \triangle EDC$. Assuming it is a typo and $\triangle ABC \cong \triangle EDC$
Statement 4: $\triangle ABC \cong \triangle EDC$ (Reason: SAS)
### Step 4.4: Apply CPCTC
Since $\triangle ABC \cong \triangle EDC$, corresponding angles are congruent. So, $\angle B \cong \angle D$ (Reason: CPCTC)

# Answer:
### Problem 4:
| Statements | Reasons |
| --- | --- |
| 1. $\overline{GH} \cong \overline{KL}$ | 1. Given |
| 2. $\angle G \cong \angle K$ | 2. Given |
| 3. $\overline{GI} \cong \overline{KJ}$ | 3. Given |
| 4. $\triangle GHI \cong \triangle KLJ$ | 4. SAS |
| 5. $\overline{HI} \cong \overline{LJ}$ | 5. CPCTC |

### Problem 5:
| Statements | Reasons |
| --- | --- |
| 1. $\angle MNP \cong \angle OPN$ | 1. Given |
| 2. $\overline{MN} \cong \overline{OP}$ | 2. Given |
| 3. $\overline{NP} \cong \overline{NP}$ | 3. Reflexive Property |
| 4. $\triangle MNP \cong \triangle OPN$ | 4. SAS |
| 5. $\overline{MP} \cong \overline{NO}$ | 5. CPCTC |

### Problem 6:
| Statements | Reasons |
| --- | --- |
| 1. $\overline{ST} \cong \overline{VU}$ | 1. Given |
| 2. $\overline{TV} \cong \overline{TV}$ | 2. Reflexive Property |
| 3. $\triangle STV \cong \triangle UVT$ | 3. HL |
| 4. $\angle SVT \cong \angle UTV$ | 4. CPCTC |

### Problem 7:
| Statements | Reasons |
| --- | --- |
| 1. $\overline{AC} \cong \overline{CE}$ | 1. Given |
| 2. $\overline{DC} \cong \overline{BC}$ | 2. Given |
| 3. $\angle ACB \cong \angle DCE$ | 3. Vertical Angles are Congruent |
| 4. $\triangle ABC \cong \triangle EDC$ (Assuming typo, should be $\triangle ABC \cong \triangle EDC$ instead of $\triangle ABC \cong \triangle DEF$) | 4. SAS |
| 5. $\angle B \cong \angle D$ | 5. CPCTC |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze Problem 4

Step 1.1: Identify Given Angles

Step 1.2: Identify Given Sides (GI and KJ)

Step 1.3: Apply SAS Congruence

Step 1.4: Apply CPCTC

Step2: Analyze Problem 5

Step 2.1: Identify Given Angle

Step 2.2: Identify Given Side (MN and OP)

Step 2.3: Identify Common Side (NP)

Step 2.4: Apply SAS Congruence

Step 2.5: Apply CPCTC

Step3: Analyze Problem 6

Step 3.1: Identify Given Side (ST and VU)

Step 3.2: Identify Common Side (TV)

Step 3.3: Apply HL Congruence

Step 3.4: Apply CPCTC

Step4: Analyze Problem 7

Step 4.1: Identify Given Sides (AC and CE, DC and BC)

Step 4.2: Identify Vertical Angles

Step 4.3: Apply SAS Congruence

Answer:

Problem 4:

Problem 5:

Problem 6:

Problem 7:

Statements	Reasons
2. \(\angle G \cong \angle K\)	2. Given
3. \(\overline{GI} \cong \overline{KJ}\)	3. Given
4. \(\triangle GHI \cong \triangle KLJ\)	4. SAS
5. \(\overline{HI} \cong \overline{LJ}\)	5. CPCTC

Statements	Reasons
2. \(\overline{MN} \cong \overline{OP}\)	2. Given
3. \(\overline{NP} \cong \overline{NP}\)	3. Reflexive Property
4. \(\triangle MNP \cong \triangle OPN\)	4. SAS
5. \(\overline{MP} \cong \overline{NO}\)	5. CPCTC

Statements	Reasons
2. \(\overline{TV} \cong \overline{TV}\)	2. Reflexive Property
3. \(\triangle STV \cong \triangle UVT\)	3. HL
4. \(\angle SVT \cong \angle UTV\)	4. CPCTC

Statements	Reasons
2. \(\overline{DC} \cong \overline{BC}\)	2. Given
3. \(\angle ACB \cong \angle DCE\)	3. Vertical Angles are Congruent
4. \(\triangle ABC \cong \triangle EDC\) (Assuming typo, should be \(\triangle ABC \cong \triangle EDC\) instead of \(\triangle ABC \cong \triangle DEF\))	4. SAS
5. \(\angle B \cong \angle D\)	5. CPCTC