36. complete the proof.
given: (overline{ac}congoverline{ar}) and (angle1congangle2)
prove: (angle3congangle4)

Question

Accepted Answer

# Explanation:
## Step1: State the given information
Given: $\overline{AC}\cong\overline{AR}$, $\angle1\cong\angle2$
## Step2: Use the vertical - angle theorem
$\angle LAC$ and $\angle SAR$ are vertical angles. So, $\angle LAC\cong\angle SAR$ (Vertical Angles Theorem)
## Step3: Prove $\triangle LAC\cong\triangle SAR$
In $\triangle LAC$ and $\triangle SAR$:
- $\overline{AC}\cong\overline{AR}$ (Given)
- $\angle1\cong\angle2$ (Given)
- $\angle LAC\cong\angle SAR$ (Vertical Angles Theorem)

By the Angle - Side - Angle (ASA) congruence criterion, $\triangle LAC\cong\triangle SAR$
## Step4: Conclude $\angle3\cong\angle4$
Since $\triangle LAC\cong\triangle SAR$, then $\angle3\cong\angle4$ (Corresponding parts of congruent triangles are congruent - CPCTC)

# Answer:
| Statements | Reasons |
|--|--|
| 1. $\overline{AC}\cong\overline{AR}$, $\angle1\cong\angle2$ | Given |
| 2. $\angle LAC\cong\angle SAR$ | Vertical Angles Theorem |
| 3. $\triangle LAC\cong\triangle SAR$ | ASA ($\angle1,\overline{AC},\angle LAC$ and $\angle2,\overline{AR},\angle SAR$) |
| 4. $\angle3\cong\angle4$ | CPCTC |

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QUESTION IMAGE

Question

Explanation:

Step1: State the given information

Step2: Use the vertical - angle theorem

Step3: Prove \(\triangle LAC\cong\triangle SAR\)

Step4: Conclude \(\angle3\cong\angle4\)

Answer:

Statements	Reasons
2. \(\angle LAC\cong\angle SAR\)	Vertical Angles Theorem
3. \(\triangle LAC\cong\triangle SAR\)	ASA (\(\angle1,\overline{AC},\angle LAC\) and \(\angle2,\overline{AR},\angle SAR\))
4. \(\angle3\cong\angle4\)	CPCTC