Geometry Proof Worksheet (3) featuring four angle relationship problems with diagrams and statements to prove.

Problem Analysis:

We are tasked with solving four geometric proof problems. Each problem involves proving a specific relationship between angles based on given information. Let's solve each problem step by step.

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Problem 1:

Given: \( \angle 3 \cong \angle 4 \)
Prove: \( \angle 1 \cong \angle 2 \)

#### Solution:
1. Identify the given information and the diagram:
- The diagram shows a triangle \( \triangle ABC \) with a transversal line intersecting \( AB \) at points \( A \) and \( B \).
- \( \angle 3 \) and \( \angle 4 \) are alternate interior angles formed by the transversal and the sides of the triangle.
- We are given that \( \angle 3 \cong \angle 4 \).

2. Use the properties of alternate interior angles:
- When a transversal intersects two parallel lines, alternate interior angles are congruent.
- Since \( \angle 3 \cong \angle 4 \), it implies that \( AC \parallel BC \).

3. Analyze the triangle:
- In \( \triangle ABC \), since \( AC \parallel BC \), the base angles \( \angle 1 \) and \( \angle 2 \) are congruent because they are base angles of an isosceles triangle (formed by the parallel sides).

4. Conclusion:
- Therefore, \( \angle 1 \cong \angle 2 \).

Proof:
- Given: \( \angle 3 \cong \angle 4 \)
- By the property of alternate interior angles, \( AC \parallel BC \).
- In \( \triangle ABC \), \( \angle 1 \) and \( \angle 2 \) are base angles of an isosceles triangle.
- Hence, \( \angle 1 \cong \angle 2 \).

Final Answer for Problem 1:
\[
\boxed{\angle 1 \cong \angle 2}
\]

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Problem 2:

Given: \( \angle 1 \cong \angle 4 \)
Prove: \( \angle 2 \cong \angle 3 \)

#### Solution:
1. Identify the given information and the diagram:
- The diagram shows a quadrilateral with a transversal intersecting two lines.
- \( \angle 1 \) and \( \angle 4 \) are corresponding angles.
- We are given that \( \angle 1 \cong \angle 4 \).

2. Use the properties of corresponding angles:
- When a transversal intersects two parallel lines, corresponding angles are congruent.
- Since \( \angle 1 \cong \angle 4 \), it implies that the two lines are parallel.

3. Analyze the angles:
- Since the lines are parallel, the alternate interior angles \( \angle 2 \) and \( \angle 3 \) are also congruent.

4. Conclusion:
- Therefore, \( \angle 2 \cong \angle 3 \).

Proof:
- Given: \( \angle 1 \cong \angle 4 \)
- By the property of corresponding angles, the lines are parallel.
- In the parallel setup, \( \angle 2 \) and \( \angle 3 \) are alternate interior angles.
- Hence, \( \angle 2 \cong \angle 3 \).

Final Answer for Problem 2:
\[
\boxed{\angle 2 \cong \angle 3}
\]

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Problem 3:

Given: \( \angle 1 \cong \angle 3 \)
Prove: \( \angle 2 \) is supplementary to \( \angle 3 \)

#### Solution:
1. Identify the given information and the diagram:
- The diagram shows two intersecting lines forming vertical angles and linear pairs.
- \( \angle 1 \) and \( \angle 3 \) are vertical angles.
- We are given that \( \angle 1 \cong \angle 3 \).

2. Use the properties of vertical angles:
- Vertical angles are always congruent.
- Since \( \angle 1 \cong \angle 3 \), this is consistent with the property of vertical angles.

3. Analyze the supplementary angles:
- \( \angle 2 \) and \( \angle 1 \) form a linear pair (they are adjacent angles on a straight line).
- Angles in a linear pair are supplementary, meaning their measures add up to \( 180^\circ \).

4. Conclusion:
- Since \( \angle 1 \cong \angle 3 \), and \( \angle 2 \) is supplementary to \( \angle 1 \), it follows that \( \angle 2 \) is also supplementary to \( \angle 3 \).

Proof:
- Given: \( \angle 1 \cong \angle 3 \)
- By the property of vertical angles, \( \angle 1 \cong \angle 3 \).
- \( \angle 2 \) and \( \angle 1 \) are supplementary (linear pair).
- Hence, \( \angle 2 \) is supplementary to \( \angle 3 \).

Final Answer for Problem 3:
\[
\boxed{\angle 2 \text{ is supplementary to } \angle 3}
\]

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Problem 4:

Given: \( \angle 4 \cong \angle 6 \)
Prove: \( \angle 5 \cong \angle 6 \)

#### Solution:
1. Identify the given information and the diagram:
- The diagram shows two intersecting lines forming vertical angles and corresponding angles.
- \( \angle 4 \) and \( \angle 6 \) are corresponding angles.
- We are given that \( \angle 4 \cong \angle 6 \).

2. Use the properties of corresponding angles:
- When a transversal intersects two parallel lines, corresponding angles are congruent.
- Since \( \angle 4 \cong \angle 6 \), it implies that the two lines are parallel.

3. Analyze the angles:
- Since the lines are parallel, the vertical angles \( \angle 5 \) and \( \angle 6 \) are congruent.

4. Conclusion:
- Therefore, \( \angle 5 \cong \angle 6 \).

Proof:
- Given: \( \angle 4 \cong \angle 6 \)
- By the property of corresponding angles, the lines are parallel.
- \( \angle 5 \) and \( \angle 6 \) are vertical angles.
- Hence, \( \angle 5 \cong \angle 6 \).

Final Answer for Problem 4:
\[
\boxed{\angle 5 \cong \angle 6}
\]

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Final Answers:

1. \( \boxed{\angle 1 \cong \angle 2} \)
2. \( \boxed{\angle 2 \cong \angle 3} \)
3. \( \boxed{\angle 2 \text{ is supplementary to } \angle 3} \)
4. \( \boxed{\angle 5 \cong \angle 6} \)