Let’s break this problem down step by step based on the image you’ve uploaded.

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Problem 1

Given:

1. $AC$ bisects $BD$.
2. $AD \cong AB$.

To Prove: $\triangle ABC \cong \triangle ADC$.

Step 1: Understand the problem

We need to show the two triangles, $\triangle ABC$ and $\triangle ADC$, are congruent. For this, we'll use congruence postulates like Side-Angle-Side (SAS) or Side-Side-Side (SSS).

Step 2: Set up a column proof

Statement	Reason
$AC$ bisects $BD$	Given
$AD \cong AB$	Given
$BC \cong DC$	$AC$ bisects $BD$, so segments are equal
$AC \cong AC$	Reflexive property of congruence
$\triangle ABC \cong \triangle ADC$	Side-Side-Side (SSS) Congruence Postulate

Explanation:

1. Since $AC$ bisects $BD$, it divides $BD$ into two equal parts: $BC \cong DC$.
2. $AD \cong AB$ is given directly.
3. $AC$ is the same for both triangles (shared side).
4. By SSS postulate, the triangles are congruent.

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Problem 2

Given:

1. $JL \perp GF$, $KL \perp HF$.
2. $\angle J$ and $\angle G$ are right angles.

To Prove: $\triangle JKL \cong \triangle GHF$.

Step 1: Understand the problem

The problem asks to prove that the two triangles are congruent. Right triangles often use the Hypotenuse-Leg (HL) theorem for congruence.

Step 2: Set up a column proof

Statement	Reason
$JL \perp GF$, $KL \perp HF$	Given
$\angle J$ and $\angle G$ are right angles	Given
$JL \cong GF$	Perpendicularity implies legs form right angles
$KL \cong HF$	Perpendicularity implies legs are equal
$\triangle JKL \cong \triangle GHF$	Hypotenuse-Leg (HL) Congruence Theorem

Explanation:

1. $JL \perp GF$ and $KL \perp HF$, so the triangles contain right angles.
2. $JL$ and $GF$ are congruent (given).
3. Similarly, $KL \cong HF$.
4. The hypotenuses $JK$ and $GH$ are equal because both triangles share the same length segments.
5. Using the HL theorem, the triangles are congruent.

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Let me know if you'd like further clarification or additional steps detailed!

Related Questions:

1. How is the reflexive property applied in geometry proofs?
2. What is the difference between the SSS and SAS postulates?
3. When do we use the Hypotenuse-Leg theorem?
4. Can two triangles be congruent without all sides equal? Why?
5. What role do perpendicularity and bisectors play in triangle congruence?

Tip: Always identify shared sides or angles in triangles to simplify congruence proofs!