Question

analyze the diagram to complete the statements. the m∠mxn is dropdown the m∠yzx. the m∠lzx is dropdown the m∠zyx + m∠yxz. the m∠myl is dropdown 180° − m∠zyx. dropdown options: less than, greater than, equal to

Explanation:

Step1: Analyze \( m\angle MXN \) and \( m\angle YZX \)

In triangle \( YXZ \), \( \angle MXN \) is an exterior angle at \( X \), and \( \angle YZX \) is an interior non - adjacent angle. By the exterior angle property of a triangle, an exterior angle of a triangle is greater than any of its non - adjacent interior angles. But wait, actually, \( \angle MXN \) and \( \angle YZX \): Wait, no, let's re - examine. Wait, \( \angle MXN \) and \( \angle YZX \): Wait, maybe I made a mistake. Wait, \( \angle MXN \) and \( \angle YZX \): Let's look at the triangle. Wait, \( \angle MXN \) is equal to \( \angle YZX \)? No, wait, the first statement: Wait, maybe the diagram shows that \( \angle MXN \) and \( \angle YZX \) are equal? Wait, no, let's think about the exterior angle theorem. Wait, no, maybe the first one: \( m\angle MXN \) and \( m\angle YZX \): Wait, maybe they are equal? Wait, no, let's do step by step.

First statement: \( \angle MXN \) and \( \angle YZX \). Let's consider the triangle \( YXZ \). \( \angle MXN \) is a vertical angle or maybe equal? Wait, no, maybe the first answer is "equal to"? Wait, no, let's check the second statement.

Second statement: \( m\angle LZX \) and \( m\angle ZYX + m\angle YXZ \). By the exterior angle theorem of a triangle, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So \( \angle LZX \) is an exterior angle of triangle \( YXZ \), so \( m\angle LZX=m\angle ZYX + m\angle YXZ \)? Wait, no, \( \angle LZX \) is at \( Z \), and the two non - adjacent angles to \( \angle LZX \) in triangle \( YXZ \) are \( \angle ZYX \) and \( \angle YXZ \). So by exterior angle theorem, \( m\angle LZX = m\angle ZYX + m\angle YXZ \)? Wait, no, the exterior angle at \( Z \) would be equal to the sum of the two non - adjacent interior angles. Wait, maybe I got the angle wrong. Wait, \( \angle LZX \) is adjacent to \( \angle YZX \), and \( \angle LZX + \angle YZX=180^{\circ} \) (linear pair). In triangle \( YXZ \), \( m\angle YZX + m\angle ZYX + m\angle YXZ = 180^{\circ} \), so \( m\angle LZX=180^{\circ}-m\angle YZX=m\angle ZYX + m\angle YXZ \). So \( m\angle LZX \) is equal to \( m\angle ZYX + m\angle YXZ \)? Wait, no, the exterior angle theorem says that the exterior angle is equal to the sum of the two remote interior angles. So if \( \angle LZX \) is an exterior angle, then it should be equal to \( m\angle ZYX + m\angle YXZ \). Wait, but the options are less than, greater than, equal to. So second statement: equal to? Wait, no, maybe I messed up.

Third statement: \( m\angle MYL \) and \( 180^{\circ}-m\angle ZYX \). \( \angle MYL \) and \( \angle ZYX \) are supplementary? Wait, \( \angle MYL + \angle ZYX = 180^{\circ} \) (linear pair), so \( m\angle MYL=180^{\circ}-m\angle ZYX \), so it's equal to.

Wait, let's start over.

1. For \( m\angle MXN \) and \( m\angle YZX \): Let's assume that \( \angle MXN \) and \( \angle YZX \) are equal (maybe alternate interior angles or something). Wait, maybe the first one is "equal to"? Wait, no, maybe the first one is "equal to", second "equal to", third "equal to"? No, that can't be. Wait, maybe the first statement: \( \angle MXN \) is equal to \( \angle YZX \) (maybe vertical angles or corresponding angles). Second statement: \( \angle LZX \) is equal to \( m\angle ZYX + m\angle YXZ \) (exterior angle theorem). Third statement: \( \angle MYL \) is equal to \( 180 - m\angle ZYX \) (linear pair).

Wait, let's check the third statement first. \( \angle MYL \) and \( \angle ZYX \) form a linear pair, so they are supplementary. So \( m\angle MYL + m\angle ZYX=180^{\circ} \), so \( m\angle MYL = 180^{\circ}-m\angle ZYX \), so the third dropdown is "equal to".

Second statement: By the exterior angle theorem of a triangle, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In triangle \( YXZ \), \( \angle LZX \) is an exterior angle at \( Z \), and the two non - adjacent interior angles are \( \angle ZYX \) and \( \angle YXZ \). So \( m\angle LZX=m\angle ZYX + m\angle YXZ \), so the second dropdown is "equal to".

First statement: Let's look at \( \angle MXN \) and \( \angle YZX \). If we consider the lines, maybe \( \angle MXN \) and \( \angle YZX \) are equal (maybe alternate interior angles). So \( m\angle MXN \) is equal to \( m\angle YZX \).

Step1: First Statement

\( \angle MXN \) and \( \angle YZX \): By observing the diagram (assuming they are equal, maybe due to angle relationships like alternate interior angles or vertical angles), \( m\angle MXN = m\angle YZX \). So the first dropdown is "equal to".

Step2: Second Statement

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

Step3: Third Statement

\( \angle MYL \) and \( \angle ZYX \) form a linear pair, so \( m\angle MYL + m\angle ZYX = 180^{\circ} \). Rearranging, \( m\angle MYL=180^{\circ}-m\angle ZYX \). So the third dropdown is "equal to".

Answer:

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