Question

in the diagram, (overleftrightarrow{de}) is parallel to (overleftrightarrow{ab}). vertex (c) of (\triangle abc) lies on (overleftrightarrow{de}).
noor says that (mangle1 + mangle2 + mangle3 = 180) degree. which angle relationships explain why noor must be correct? select all that apply.
(square mangle4 + mangle2 + mangle5 = 180) degree
(square mangle1 + mangle4 = 180) degree
(square mangle3 + mangle5 = 180) degree
(square mangle1 = mangle4)

Explanation:

Step1: Use the property of parallel lines and alternate - interior angles

Since \(DE\parallel AB\), by the alternate - interior angles theorem, \(\angle1=\angle4\) (because \(DE\) and \(AB\) are parallel and \(AC\) is a transversal). Similarly, \(\angle3 = \angle5\) (because \(DE\) and \(AB\) are parallel and \(BC\) is a transversal).

Step2: Use the linear - pair (straight - line) angle sum property

We know that \(\angle4+\angle2+\angle5 = 180^{\circ}\) (since \(DCE\) is a straight line, and the sum of angles on a straight line is \(180^{\circ}\)).
Substituting \(\angle1\) for \(\angle4\) and \(\angle3\) for \(\angle5\) in the equation \(\angle4+\angle2+\angle5 = 180^{\circ}\), we get \(\angle1+\angle2+\angle3=180^{\circ}\).

Answer:

\(m\angle4 + m\angle2 + m\angle5=180\) degree and \(m\angle1 = m\angle4\)