Question

select all the true statements given the figure below:

□ (mangle1 + mangle4=180)
□ (mangle1 = mangle3)
□ (mangle1 + mangle5=180)
□ (mangle1 = mangle4)
□ (mangle1 = mangle5)

Explanation:

Step1: Recall angle - pair relationships

Vertical angles are equal. Adjacent angles on a straight - line sum to 180 degrees. Corresponding angles are equal when two parallel lines are cut by a transversal.

Step2: Analyze \(m\angle1 + m\angle4\)

\(\angle1\) and \(\angle4\) are adjacent angles on a straight - line. So \(m\angle1 + m\angle4=180\) (linear - pair postulate).

Step3: Analyze \(m\angle1\) and \(m\angle3\)

\(\angle1\) and \(\angle3\) are vertical angles. So \(m\angle1 = m\angle3\) (vertical - angles theorem).

Step4: Analyze \(m\angle1 + m\angle5\)

\(\angle1\) and \(\angle5\) are corresponding angles. When the two lines are parallel, \(m\angle1=m\angle5\), not \(m\angle1 + m\angle5 = 180\).

Step5: Analyze \(m\angle1\) and \(m\angle4\)

As mentioned in Step 2, \(m\angle1 + m\angle4 = 180\), not \(m\angle1=m\angle4\).

Step6: Analyze \(m\angle1\) and \(m\angle5\)

\(\angle1\) and \(\angle5\) are corresponding angles. If the two lines are parallel, \(m\angle1 = m\angle5\) (corresponding - angles postulate).

Answer:

\(m\angle1 + m\angle4 = 180\), \(m\angle1 = m\angle3\), \(m\angle1 = m\angle5\)