Answer:
Step-by-step explanation:
To prove that \( \angle 1 \) is equal to \( \angle 3 \), given that \( \angle 1 \) and \( \angle 2 \) are supplementary, and \( \angle 2 \) and \( \angle 3 \) are supplementary, we can use the fact that supplementary angles are pairs of angles whose measures add up to \( 180^\circ \).
Here's the step-by-step reasoning:
1. **Given:**
- \( \angle 1 \) and \( \angle 2 \) are supplementary, meaning \( \angle 1 + \angle 2 = 180^\circ \).
- \( \angle 2 \) and \( \angle 3 \) are supplementary, meaning \( \angle 2 + \angle 3 = 180^\circ \).
2. **To prove:**
- \( \angle 1 = \angle 3 \).
3. **Proof:**
- From the given information, we have:
\[ \angle 1 + \angle 2 = 180^\circ \quad \text{(1)} \]
\[ \angle 2 + \angle 3 = 180^\circ \quad \text{(2)} \]
- Now, subtract equation (1) from equation (2):
\[ (\angle 2 + \angle 3) - (\angle 1 + \angle 2) = 180^\circ - 180^\circ \]
\[ \angle 2 + \angle 3 - \angle 1 - \angle 2 = 0 \]
\[ \angle 3 - \angle 1 = 0 \]
- Simplifying, we find:
\[ \angle 3 = \angle 1 \]
4. **Conclusion:**
- Therefore, \( \angle 1 \) is equal to \( \angle 3 \), as required.
This completes the proof that \( \angle 1 \) equals \( \angle 3 \) based on the given conditions of supplementary angles \( \angle 1 \) with \( \angle 2 \) and \( \angle 2 \) with \( \angle 3 \).
Hope this helps a little^^'