Question

the diagram above shows parallel lines cut by a transversal.
which statements are true and supported by valid reasons about the angles labeled in the diagram? select all that apply.
$m\angle1 + m\angle6 = 180^{\circ}$, because $\angle1$ and $\angle4$ are vertical angles and $\angle4$ and $\angle6$ are consecutive interior angles.
$m\angle4 + m\angle7 = 180^{\circ}$, because $\angle4$ and $\angle6$ are a linear pair and $\angle6$ and $\angle7$ are corresponding angles.
$m\angle2 + m\angle8 = 180^{\circ}$, because $\angle2$ and $\angle6$ are vertical angles and $\angle6$ and $\angle8$ are consecutive interior angles.
$m\angle1 + m\angle7 = 180^{\circ}$, because $\angle1$ and $\angle3$ are a linear pair and $\angle3$ and $\angle7$ are corresponding angles.

Explanation:

Step1: Analyze the first option

• Since \(∠1\) and \(∠4\) are vertical angles, \(m∠1 = m∠4\).
• \(∠4\) and \(∠6\) are consecutive interior angles, so \(m∠4 + m∠6=180^{\circ}\).
• Substituting \(m∠1\) for \(m∠4\), we get \(m∠1 + m∠6 = 180^{\circ}\). This option is correct.

Step2: Analyze the second option

• \(∠4\) and \(∠6\) are a linear pair, so \(m∠4 + m∠6 = 180^{\circ}\), not \(m∠4 + m∠7 = 180^{\circ}\). This option is incorrect.

Step3: Analyze the third option

• \(∠2\) and \(∠4\) are vertical angles, not \(∠2\) and \(∠6\). So this option is incorrect.

Step4: Analyze the fourth option

• \(∠1\) and \(∠3\) are a linear pair, so \(m∠1 + m∠3=180^{\circ}\).
• \(∠3\) and \(∠7\) are corresponding angles, so \(m∠3 = m∠7\).
• Substituting \(m∠7\) for \(m∠3\), we get \(m∠1 + m∠7 = 180^{\circ}\). This option is correct.

Answer:

• \(m∠1 + m∠6 = 180^{\circ}\), because \(∠1\) and \(∠4\) are vertical angles and \(∠4\) and \(∠6\) are consecutive interior angles.
• \(m∠1 + m∠7 = 180^{\circ}\), because \(∠1\) and \(∠3\) are a linear pair and \(∠3\) and \(∠7\) are corresponding angles.