Congruent Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

It looks like you've uploaded an image containing three geometry proofs involving triangles and midpoints. Since I can't view images directly, I’ll help you solve these problems based on the text you’ve provided. Let's go through each proof step by step.

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Problem 1:

> Given: $ \angle B \cong \angle C $, $ \angle AED \cong \angle CEB $
> Prove: $ AB \cong AC $

#### Proof Table:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle B \cong \angle C $ | 1. Given |
| 2. $ \angle AED \cong \angle CEB $ | 2. Given |
| 3. $ \angle A \cong \angle A $ | 3. Reflexive Property (or common angle) |
| 4. $ \triangle AED \cong \triangle CEB $ | 4. ASA (Angle-Side-Angle) Congruence |
| 5. $ AE \cong CE $ | 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 6. $ \angle AED \cong \angle CEB $ (again?) | 6. Given or from congruency? |
| 7. $ AB \cong AC $ | 7. Isosceles Triangle Theorem or from triangle congruence |

Wait — this seems to have some issues in logic. Let’s reevaluate carefully.

Actually, let's clarify:

We are given:
- $ \angle B \cong \angle C $
- $ \angle AED \cong \angle CEB $

But $ \angle AED $ and $ \angle CEB $ are vertical angles if point $ E $ is the intersection of diagonals or lines $ AD $ and $ BE $. But without a diagram, we assume that $ D $ and $ E $ lie on sides of triangle $ ABC $.

Let’s suppose $ E $ is a point on $ BC $, and $ D $ is on $ AB $, and $ AE $ and $ CD $ intersect at $ E $? That might not be clear.

Alternatively, perhaps $ \angle AED $ and $ \angle CEB $ are vertical angles formed by intersecting lines $ AD $ and $ BE $, but again, without a diagram, it's ambiguous.

But wait — maybe there's a typo. Perhaps it's meant to be:

> Given: $ \angle B \cong \angle C $, $ \angle AED \cong \angle CEB $, and $ AE $ and $ CE $ are parts of the triangle?

But actually, if $ \angle B \cong \angle C $, then triangle $ ABC $ is isosceles with $ AB = AC $, by the Isosceles Triangle Theorem.

So perhaps the proof is simpler than expected.

Let’s rewrite it logically.

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✔ Corrected Proof for Problem 1:

> Given: $ \angle B \cong \angle C $
> Prove: $ AB \cong AC $

| Statements | Reasons |
|----------|--------|
| 1. $ \angle B \cong \angle C $ | 1. Given |
| 2. $ AB \cong AC $ | 2. Converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent. |

✔ So the conclusion follows directly.

Note: The mention of $ \angle AED \cong \angle CEB $ may be extraneous or part of a different diagram. If those angles are involved, then perhaps $ E $ is a point such that lines from $ A $ and $ C $ intersect, but unless more context is given, we can’t use them.

So unless $ \angle AED \cong \angle CEB $ is needed to prove $ \angle B \cong \angle C $, it might be irrelevant here.

Assuming the goal is just to prove $ AB \cong AC $ from $ \angle B \cong \angle C $, then the proof is complete via the Isosceles Triangle Theorem.

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Problem 2:

> Given: $ \triangle ABC $, $ AT $ is the midpoint of $ \overline{BC} $, $ \angle ATB \cong \angle ATC $
> Prove: $ AB \cong AC $

This is a classic median and angle bisector situation.

Let’s analyze.

Given:
- $ T $ is the midpoint of $ \overline{BC} $ → $ BT \cong TC $
- $ \angle ATB \cong \angle ATC $
- $ AT $ is common side

So we can try to prove $ \triangle ATB \cong \triangle ATC $

| Statements | Reasons |
|----------|--------|
| 1. $ BT \cong TC $ | 1. Definition of midpoint |
| 2. $ \angle ATB \cong \angle ATC $ | 2. Given |
| 3. $ AT \cong AT $ | 3. Reflexive Property |
| 4. $ \triangle ATB \cong \triangle ATC $ | 4. SAS (Side-Angle-Side) Congruence |
| 5. $ AB \cong AC $ | 5. CPCTC |

✔ Therefore, $ AB \cong AC $, so triangle $ ABC $ is isosceles.

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Problem 3:

> Given: $ \triangle ABC \cong \triangle EDC $, $ C $ is the midpoint of $ \overline{BD} $
> Prove: $ AB \cong DE $

Let’s break this down.

Given:
- $ \triangle ABC \cong \triangle EDC $
- $ C $ is the midpoint of $ \overline{BD} $

From triangle congruence:
- All corresponding parts are congruent
- So $ AB \cong ED $, $ BC \cong DC $, $ AC \cong EC $, etc.

But we are to prove $ AB \cong DE $

Wait — $ AB $ and $ DE $ are corresponding sides of the congruent triangles.

Since $ \triangle ABC \cong \triangle EDC $, then:
- $ AB \leftrightarrow ED $
- So $ AB \cong ED $

Thus, the proof is immediate.

But let’s write it formally.

| Statements | Reasons |
|----------|--------|
| 1. $ \triangle ABC \cong \triangle EDC $ | 1. Given |
| 2. $ AB \cong ED $ | 2. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 3. $ AB \cong DE $ | 3. $ ED \cong DE $, same segment; so $ AB \cong DE $ |

Note: $ DE $ is the same as $ ED $, so yes.

Also, the fact that $ C $ is the midpoint of $ BD $ likely supports the congruence (i.e., $ BC \cong CD $), which is used in proving the triangles congruent, but since it's already given that the triangles are congruent, we don’t need to derive it.

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✔ Final Answers:

#### Problem 1:
> Given: $ \angle B \cong \angle C $
> Prove: $ AB \cong AC $

| Statements | Reasons |
|----------|--------|
| 1. $ \angle B \cong \angle C $ | 1. Given |
| 2. $ AB \cong AC $ | 2. Converse of Isosceles Triangle Theorem |

*(Note: The mention of $ \angle AED \cong \angle CEB $ may be extra or belong to another diagram.)*

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#### Problem 2:
> Given: $ AT $ is the midpoint of $ \overline{BC} $, $ \angle ATB \cong \angle ATC $
> Prove: $ AB \cong AC $

| Statements | Reasons |
|----------|--------|
| 1. $ BT \cong TC $ | 1. Definition of midpoint |
| 2. $ \angle ATB \cong \angle ATC $ | 2. Given |
| 3. $ AT \cong AT $ | 3. Reflexive Property |
| 4. $ \triangle ATB \cong \triangle ATC $ | 4. SAS |
| 5. $ AB \cong AC $ | 5. CPCTC |

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#### Problem 3:
> Given: $ \triangle ABC \cong \triangle EDC $, $ C $ is the midpoint of $ \overline{BD} $
> Prove: $ AB \cong DE $

| Statements | Reasons |
|----------|--------|
| 1. $ \triangle ABC \cong \triangle EDC $ | 1. Given |
| 2. $ AB \cong ED $ | 2. CPCTC |
| 3. $ AB \cong DE $ | 3. Same segment; $ ED \cong DE $ |

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Summary:

All three proofs rely on basic triangle congruence theorems (SAS, ASA, CPCTC) and properties of isosceles triangles.

Let me know if you'd like diagrams or further clarification!