https://file.aiquickdraw.com/mathbot/file/1726091835574b8a8099l.jpg

Let's solve the **Congruent Supplements Theorem** proof step by step, using the diagram and flowchart you've provided. Here's a breakdown of the problem and the reasoning behind each step:

### Given:
- \(\angle RAJ \cong \angle LOD\)
- We are to prove that \(\angle JAW \cong \angle DOG\).

### Congruent Supplements Theorem:
**Statement**: If two angles are supplements of the same angle (or of congruent angles), then the angles are congruent.

### Diagram Analysis:
- \(\angle LOD\) and \(\angle DOG\) are supplementary (they add up to 180° because they form a linear pair).
- \(\angle RAJ\) and \(\angle JAW\) are also supplementary (since \(\angle RAJ + \angle JAW = 180^\circ\)).

We will now fill in the flowchart and provide reasoning for each part.

---

### Flowchart Solution:

1. **Step 1: Identify Supplementary Angles**
   - **Statement**: \(\angle LOD\) and \(\angle DOG\) are supplementary.
   - **Reason**: This is given from the diagram since these two angles form a linear pair.
   
2. **Step 2: \(\angle RAJ\) and \(\angle JAW\) are supplementary**
   - **Statement**: \(\angle RAJ\) and \(\angle JAW\) are supplementary.
   - **Reason**: This follows from the diagram as well since these two angles also form a linear pair.

3. **Step 3: Definition of Supplementary Angles**
   - **Statement**: Definition of supplementary angles: \( m\angle RAJ + m\angle JAW = 180^\circ \) and \( m\angle LOD + m\angle DOG = 180^\circ \).
   - **Reason**: By definition, supplementary angles add up to 180°.

4. **Step 4: Congruence of \(\angle RAJ\) and \(\angle LOD\)**
   - **Statement**: \(\angle RAJ \cong \(\angle LOD\).
   - **Reason**: This is given as part of the initial condition.

5. **Step 5: Measures of Congruent Angles are Equal**
   - **Statement**: \( m\angle RAJ = m\angle LOD \).
   - **Reason**: By definition of congruent angles, their measures are equal.

6. **Step 6: Substitution**
   - **Statement**: Substituting in the equations: \( m\angle RAJ + m\angle JAW = m\angle LOD + m\angle DOG \).
   - **Reason**: Using the fact that \( m\angle RAJ = m\angle LOD \), we substitute this into the equation for supplementary angles.

7. **Step 7: Conclusion**
   - **Statement**: \( \angle JAW \cong \angle DOG \).
   - **Reason**: By the Congruent Supplements Theorem, since both angles are supplements of congruent angles, they must also be congruent.

---

This completes the proof of the **Congruent Supplements Theorem** using the flowchart.

---

### Key Concepts Recap:
1. **Supplementary Angles**: Angles that add up to 180°.
2. **Congruent Angles**: Angles that have the same measure.
3. **Congruent Supplements Theorem**: If two angles are supplements of congruent angles, then they are congruent.

---

Do you need further details or have any questions on this? Here are some follow-up questions:

1. Why are \(\angle LOD\) and \(\angle DOG\) supplementary?
2. How does the definition of supplementary angles apply in this proof?
3. What does it mean for two angles to be congruent?
4. What role does the Congruent Supplements Theorem play in proving the final result?
5. Can you explain why we substitute \( m\angle RAJ = m\angle LOD \) in the equation?

### Tip:
Remember, substitution is a key technique in many geometry proofs. Always look for opportunities to replace one expression with an equivalent one to simplify the proof.

Learn how to prove the Congruent Supplements Theorem with a step-by-step solution. This geometry problem covers concepts like supplementary angles, congruent angles, and substitution techniques. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation