Question

lines and quadrilaterals review
circle each statement that can be used to prove m || n and p || q. there may be more than one answer. explain your reasoning.
(image of intersecting lines with angles labeled 1 - 16)
statements (some circled):
∠2 ≅ ∠5 and ∠13 ≅ ∠15
m∠1 + m∠4 = 180° and ∠8 ≅ ∠16
m∠6 + m∠7 = 180° and ∠3 ≅ ∠11
m∠9 + m∠12 = 180° and ∠1 ≅ ∠3
∠9 ≅ ∠16 and m∠11 + m∠7 = 180°
m∠14 + m∠15 = 180° and ∠1 ≅ ∠16

Explanation:

Step1: Analyze Parallel Line Theorems

To prove \( m \parallel n \) and \( p \parallel q \), we use angle - related theorems (corresponding angles, alternate interior angles, same - side interior angles). For example, if \( \angle 2\cong\angle5 \) and \( \angle13\cong\angle15 \):

• For \( \angle 2\cong\angle5 \), \( \angle2 \) and \( \angle5 \) are corresponding angles. By the Corresponding Angles Postulate, if corresponding angles are congruent, then the lines cut by a transversal are parallel. So this can be used to prove \( m\parallel n \) (assuming the transversal is the line that creates these angles).
• For \( \angle13\cong\angle15 \), \( \angle13 \) and \( \angle15 \) are alternate interior angles. By the Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines (cut by a transversal) are parallel. This can be used to prove \( p\parallel q \).

Step2: Analyze \( m\angle1 + m\angle4=180^{\circ} \) and \( \angle8\cong\angle16 \)

• \( m\angle1 + m\angle4 = 180^{\circ} \): \( \angle1 \) and \( \angle4 \) are same - side interior angles. By the Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, the lines are parallel. This can be used to prove \( m\parallel n \).
• \( \angle8\cong\angle16 \): \( \angle8 \) and \( \angle16 \) are corresponding angles. By the Corresponding Angles Postulate, this can be used to prove \( p\parallel q \).

Step3: Analyze \( m\angle6 + m\angle7 = 180^{\circ} \) and \( \angle3\cong\angle11 \)

• \( m\angle6 + m\angle7=180^{\circ} \): \( \angle6 \) and \( \angle7 \) are same - side interior angles. By the Same - Side Interior Angles Theorem, this can be used to prove \( m\parallel n \) (since they are formed by a transversal cutting \( m \) and \( n \)).
• \( \angle3\cong\angle11 \): \( \angle3 \) and \( \angle11 \) are corresponding angles. By the Corresponding Angles Postulate, this can be used to prove \( p\parallel q \).

Step4: Analyze \( m\angle9 + m\angle12 = 180^{\circ} \) and \( \angle1\cong\angle3 \)

• \( m\angle9 + m\angle12 = 180^{\circ} \): \( \angle9 \) and \( \angle12 \) are same - side interior angles. By the Same - Side Interior Angles Theorem, this can be used to prove \( p\parallel q \) (formed by a transversal cutting \( p \) and \( q \)).
• \( \angle1\cong\angle3 \): \( \angle1 \) and \( \angle3 \) are corresponding angles. By the Corresponding Angles Postulate, this can be used to prove \( m\parallel n \).

Step5: Analyze \( \angle9\cong\angle16 \) and \( m\angle11 + m\angle7 = 180^{\circ} \)

• \( \angle9\cong\angle16 \): \( \angle9 \) and \( \angle16 \) are corresponding angles. By the Corresponding Angles Postulate, this can be used to prove \( p\parallel q \).
• \( m\angle11 + m\angle7 = 180^{\circ} \): \( \angle11 \) and \( \angle7 \) are same - side interior angles. By the Same - Side Interior Angles Theorem, this can be used to prove \( m\parallel n \).

Step6: Analyze \( m\angle14 + m\angle15 = 180^{\circ} \) and \( \angle1\cong\angle16 \)

• \( m\angle14 + m\angle15 = 180^{\circ} \): \( \angle14 \) and \( \angle15 \) are same - side interior angles. By the Same - Side Interior Angles Theorem, this can be used to prove \( p\parallel q \).
• \( \angle1\cong\angle16 \): \( \angle1 \) and \( \angle16 \) are corresponding angles. By the Corresponding Angles Postulate, this can be used to prove \( m\parallel n \).

Answer:

Statements like \( \boldsymbol{\angle 2\cong\angle5 \text{ and } \angle13\cong\angle15} \), \( \boldsymbol{m\angle1 + m\angle4 = 180^{\circ}\text{ and } \angle8\cong\angle16} \), \( \boldsymbol{m\angle6 + m\angle7=180^{\circ}\text{ and } \angle3\cong\angle11} \), \( \boldsymbol{m\angle9 + m\angle12 = 180^{\circ}\text{ and } \angle1\cong\angle3} \), \( \boldsymbol{\angle9\cong\angle16\text{ and } m\angle11 + m\angle7 = 180^{\circ}} \), \( \boldsymbol{m\angle14 + m\angle15 = 180^{\circ}\text{ and } \angle1\cong\angle16} \) can be used. The choice depends on the specific angle - line relationships, but all these statements use angle - based parallel line theorems (corresponding angles, alternate interior angles, same - side interior angles) to prove \( m\parallel n \) and \( p\parallel q \).