We are given a diagram with two triangles, $ \triangle ABE $ and $ \triangle DCE $, and we are to
prove that:
$$
\triangle ABE \cong \triangle DCE
$$
using a
two-column proof.
---
Step 1: Analyze the Diagram
From the image:
- Segment $ AB $ has a single tick mark.
- Segment $ DC $ has a single tick mark → So $ AB \cong DC $
- Segment $ BE $ has a double tick mark.
- Segment $ CE $ has a double tick mark → So $ BE \cong CE $
- Angles $ \angle 1 $ and $ \angle 2 $ are marked as equal (they are vertical angles at point $ E $) → $ \angle AEB \cong \angle DEC $
Note: The two triangles share point $ E $, and lines $ AD $ and $ BC $ cross at $ E $. So $ \angle 1 $ and $ \angle 2 $ are
vertical angles, which are always congruent.
---
Step 2: Use SAS Congruence Postulate
We have:
- $ AB \cong DC $ (side)
- $ \angle AEB \cong \angle DEC $ (included angle)
- $ BE \cong CE $ (side)
This is
SAS (Side-Angle-Side) congruence.
---
Two-Column Proof
|
STATEMENTS |
REASONS |
|------------------------------------|----------------------------------------------|
| 1. $ AB \cong DC $ | 1. Given (marked with one tick) |
| 2. $ BE \cong CE $ | 2. Given (marked with two ticks) |
| 3. $ \angle AEB \cong \angle DEC $ | 3. Vertical angles are congruent |
| 4. $ \triangle ABE \cong \triangle DCE $ | 4. SAS Congruence Postulate (Steps 1, 2, 3) |
---
Final Answer:
$$
\boxed{
\begin{array}{|l|l|}
\hline
\text{STATEMENTS} & \text{REASONS} \\
\hline
1.\ AB \cong DC & \text{Given (marked)} \\
\hline
2.\ BE \cong CE & \text{Given (marked)} \\
\hline
3.\ \angle AEB \cong \angle DEC & \text{Vertical angles are congruent} \\
\hline
4.\ \triangle ABE \cong \triangle DCE & \text{SAS Congruence Postulate} \\
\hline
\end{array}
}
$$
✔ This completes the proof that $ \triangle ABE \cong \triangle DCE $ by
SAS.
Parent Tip: Review the logic above to help your child master the concept of congruent triangles proofs worksheet answers.
