Question

compare the given measures by choosing \\( < \\), \\( > \\), or \\( = \\).
if there is not enough information, select \cannot be determined\.

if necessary, you may learn what the markings on a figure indicate.

first problem:
\\( m\angle wyx \\) select \\( m\angle wyz \\)
cannot be determined

second problem:
\\( uv \\) select \\( df \\)
cannot be determined

Explanation:

Analyze the first pair of triangles

We compare \(\triangle WYX\) and \(\triangle WYZ\).

• Side \(WY\) is shared by both triangles: \(WY = WY\).
• We are given \(XY = 9\) and \(YZ = 9\), so \(XY = YZ\).
• The third sides are \(WX = 19\) and \(WZ = 13\).
• Since \(19 > 13\), we have \(WX > WZ\).
• By the Converse of the Hinge Theorem, the angle opposite the longer third side is larger:

$$m\angle WYX > m\angle WYZ$$

Analyze the second pair of triangles

We compare \(\triangle TUV\) and \(\triangle EDF\).

• The markings indicate congruent sides: \(TV \cong EF\) and \(TU \cong ED\).
• The included angles are \(m\angle UTV = 35^\circ\) and \(m\angle DEF = 81^\circ\).
• However, the angle \(35^\circ\) is \(m\angle UTV\), which is opposite side \(UV\). The included angle between \(TU\) and \(TV\) is \(\angle U\), which is unknown.
• Similarly, the angle \(81^\circ\) is \(m\angle DEF\), which is the included angle between \(ED\) and \(EF\).
• Since we do not know the included angle \(\angle U\) in \(\triangle TUV\), we cannot compare the opposite sides \(UV\) and \(DF\).
• Therefore, the relationship cannot be determined.

Answer:

Question 1

• \(m\angle WYX > m\angle WYZ\) (Correct answer)
• Cannot be determined

Question 2

• \(UV < DF\)
• \(UV > DF\)
• \(UV = DF\)
• Cannot be determined (Correct answer)