Question

1. fill in the missing statements and reasons to the proof below.

given: ∠4 ≅ ∠3
c ∥ d
prove: ∠4 ≅ ∠2

1. complete the proof.

given: a ∥ b
m ∥ n
prove: ∠3 ≅ ∠13

Explanation:

Problem 14 (Proof with Parallel Lines and Angle Relationships)

Step-by-Step Explanation:

Step 1: Analyze Given Information

Given \( c \parallel d \) and \( \angle 4 \cong \angle 3 \). We need to prove \( \angle 4 \cong \angle 2 \).

Step 2: Identify Angle Relationships (Vertical Angles, Corresponding Angles)

• Statement 2: \( \angle 3 \cong \angle 1 \) (Reason: Vertical Angles are Congruent)
• Statement 3: \( \angle 4 \cong \angle 1 \) (Reason: Given \( \angle 4 \cong \angle 3 \) and \( \angle 3 \cong \angle 1 \), so Transitive Property)
• Statement 4: \( \angle 1 \cong \angle 2 \) (Reason: Corresponding Angles (since \( c \parallel d \), the transversal creates corresponding angles))
• Statement 5: \( \angle 4 \cong \angle 2 \) (Reason: Transitive Property, as \( \angle 4 \cong \angle 1 \) and \( \angle 1 \cong \angle 2 \))

Problem 15 (Proof with Two Pairs of Parallel Lines)

Step-by-Step Explanation:

Given \( a \parallel b \) and \( m \parallel n \), prove \( \angle 3 \cong \angle 13 \).

Step 1: List Statements and Reasons

1. Statement 1: \( a \parallel b \), \( m \parallel n \) (Reason: Given)
2. Statement 2: \( \angle 3 \cong \angle 9 \) (Reason: Corresponding Angles (since \( m \parallel n \), transversal \( a \) creates corresponding angles))
3. Statement 3: \( \angle 9 \cong \angle 11 \) (Reason: Vertical Angles are Congruent)
4. Statement 4: \( \angle 11 \cong \angle 13 \) (Reason: Corresponding Angles (since \( a \parallel b \), transversal \( n \) creates corresponding angles))
5. Statement 5: \( \angle 3 \cong \angle 13 \) (Reason: Transitive Property, combining \( \angle 3 \cong \angle 9 \), \( \angle 9 \cong \angle 11 \), and \( \angle 11 \cong \angle 13 \))

Final Answers (Filled Proofs)

Problem 14:

Statements	Reasons
2) \( \angle 3 \cong \angle 1 \)	Vertical Angles Congruence
3) \( \angle 4 \cong \angle 1 \)	Transitive Property (from 1 and 2)
4) \( \angle 1 \cong \angle 2 \)	Corresponding Angles ( \( c \parallel d \) )
5) \( \angle 4 \cong \angle 2 \)	Transitive Property (from 3 and 4)

Problem 15:

Statements	Reasons
2) \( \angle 3 \cong \angle 9 \)	Corresponding Angles ( \( m \parallel n \) )
3) \( \angle 9 \cong \angle 11 \)	Vertical Angles Congruence
4) \( \angle 11 \cong \angle 13 \)	Corresponding Angles ( \( a \parallel b \) )
5) \( \angle 3 \cong \angle 13 \)	Transitive Property

(Note: Adjust angle labels and reasons to match the diagram’s specific transversals and parallel lines. The key is using corresponding angles, vertical angles, and transitive property.)

Answer:

Problem 14 (Proof with Parallel Lines and Angle Relationships)

Step-by-Step Explanation:

Step 1: Analyze Given Information

Given \( c \parallel d \) and \( \angle 4 \cong \angle 3 \). We need to prove \( \angle 4 \cong \angle 2 \).

Step 2: Identify Angle Relationships (Vertical Angles, Corresponding Angles)

• Statement 2: \( \angle 3 \cong \angle 1 \) (Reason: Vertical Angles are Congruent)
• Statement 3: \( \angle 4 \cong \angle 1 \) (Reason: Given \( \angle 4 \cong \angle 3 \) and \( \angle 3 \cong \angle 1 \), so Transitive Property)
• Statement 4: \( \angle 1 \cong \angle 2 \) (Reason: Corresponding Angles (since \( c \parallel d \), the transversal creates corresponding angles))
• Statement 5: \( \angle 4 \cong \angle 2 \) (Reason: Transitive Property, as \( \angle 4 \cong \angle 1 \) and \( \angle 1 \cong \angle 2 \))

Problem 15 (Proof with Two Pairs of Parallel Lines)

Step-by-Step Explanation:

Given \( a \parallel b \) and \( m \parallel n \), prove \( \angle 3 \cong \angle 13 \).

Step 1: List Statements and Reasons

1. Statement 1: \( a \parallel b \), \( m \parallel n \) (Reason: Given)
2. Statement 2: \( \angle 3 \cong \angle 9 \) (Reason: Corresponding Angles (since \( m \parallel n \), transversal \( a \) creates corresponding angles))
3. Statement 3: \( \angle 9 \cong \angle 11 \) (Reason: Vertical Angles are Congruent)
4. Statement 4: \( \angle 11 \cong \angle 13 \) (Reason: Corresponding Angles (since \( a \parallel b \), transversal \( n \) creates corresponding angles))
5. Statement 5: \( \angle 3 \cong \angle 13 \) (Reason: Transitive Property, combining \( \angle 3 \cong \angle 9 \), \( \angle 9 \cong \angle 11 \), and \( \angle 11 \cong \angle 13 \))

Final Answers (Filled Proofs)

Problem 14:

Statements	Reasons
2) \( \angle 3 \cong \angle 1 \)	Vertical Angles Congruence
3) \( \angle 4 \cong \angle 1 \)	Transitive Property (from 1 and 2)
4) \( \angle 1 \cong \angle 2 \)	Corresponding Angles ( \( c \parallel d \) )
5) \( \angle 4 \cong \angle 2 \)	Transitive Property (from 3 and 4)

Problem 15:

Statements	Reasons
2) \( \angle 3 \cong \angle 9 \)	Corresponding Angles ( \( m \parallel n \) )
3) \( \angle 9 \cong \angle 11 \)	Vertical Angles Congruence
4) \( \angle 11 \cong \angle 13 \)	Corresponding Angles ( \( a \parallel b \) )
5) \( \angle 3 \cong \angle 13 \)	Transitive Property

(Note: Adjust angle labels and reasons to match the diagram’s specific transversals and parallel lines. The key is using corresponding angles, vertical angles, and transitive property.)