Question

analyze the diagram to complete the statements. the ( mangle mxn ) is the ( mangle yzx ). the ( mangle lzx ) is the ( mangle zyx + mangle yxz ). the ( mangle myl ) is ( 180^{circ}-mangle zyx )

Explanation:

Step1: Analyze the relationship between ∠MXN and ∠YZX

Since ∠MXN and ∠YZX are vertical angles, vertical angles are equal. So \(m\angle MXN=m\angle YZX\).

Step2: Analyze the relationship between ∠LZX and \(m\angle ZYX + m\angle YXZ\)

By the exterior - angle theorem of a triangle, an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle ZYX\), \(\angle LZX\) is an exterior angle. So \(m\angle LZX=m\angle ZYX + m\angle YXZ\).

Step3: Analyze the relationship between ∠MYL and \(180^{\circ}-m\angle ZYX\)

Since \(\angle MYL\) and \(\angle ZYX\) are adjacent supplementary angles (they form a linear pair), \(m\angle MYL + m\angle ZYX=180^{\circ}\), then \(m\angle MYL=180^{\circ}-m\angle ZYX\).

Answer:

The \(m\angle MXN\) is equal to the \(m\angle YZX\). The \(m\angle LZX\) is equal to the \(m\angle ZYX + m\angle YXZ\). The \(m\angle MYL\) is equal to \(180^{\circ}-m\angle ZYX\).