Question

given: ( y parallel z ) prove: ( mangle5 + mangle2 + mangle6 = 180^circ )

Explanation:

Step1: Use angle - sum property of a straight line

Since \(LAM\) is a straight line, \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) (definition of a straight - line angle).

Step2: Substitute using congruent angles

We know that \(\angle1\cong\angle5\) (alternate interior angles theorem), so \(m\angle1 = m\angle5\) (definition of congruent angles). Also, \(\angle3\cong\angle6\) (alternate interior angles theorem), so \(m\angle3 = m\angle6\) (definition of congruent angles).

Step3: Replace \(m\angle1\) and \(m\angle3\) in the straight - line angle formula

Substitute \(m\angle1\) with \(m\angle5\) and \(m\angle3\) with \(m\angle6\) in the equation \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\). We get \(m\angle5 + m\angle2 + m\angle6=180^{\circ}\).

Answer:

The proof is completed as shown in the steps above.