Question

in (\triangle xyz), suppose (xy < xz). which inequality relates two angles in (\triangle xyz)?
a. (mangle x < mangle y)
b. (mangle x < mangle z)
c. (mangle y < mangle z)
d. (mangle z < mangle y)

Explanation:

Step1: Recall the triangle - angle relationship theorem

In a triangle, the larger side is opposite the larger angle.

Step2: Identify the sides and their opposite angles

In \(\triangle XYZ\), side \(XY\) is opposite \(\angle Z\) (i.e., \(XY\) is opposite \(\angle Z\) since in \(\triangle XYZ\), for side \(XY\), the non - adjacent vertex is \(Z\)), and side \(XZ\) is opposite \(\angle Y\) (since for side \(XZ\), the non - adjacent vertex is \(Y\)).
Given \(XY\lt XZ\).

Step3: Apply the theorem

Since \(XY\lt XZ\), by the "larger side - larger angle" theorem, the angle opposite \(XY\) (\(\angle Z\)) is smaller than the angle opposite \(XZ\) (\(\angle Y\)). So \(m\angle Z\lt m\angle Y\).

Answer:

D. \(m\angle Z\lt m\angle Y\)