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 figure: triangle with points a, y, z, w. ay=16, yz=16, angle at y: 81° and 60°. question: |wa| select |wz|
option: cannot be determined
(second figure: two triangles. first triangle: u, t, f with ut=10, ft=10, marked with tick marks. second triangle: r, d, v with rd and rv marked with tick marks, dv=16. question: m∠t select m∠v
option: cannot be determined)

Explanation:

Step1: Analyze the first triangle (top)

In triangle \( WXY \) and \( WZY \), we have \( WY = WY \) (common side), \( XY = ZY = 16 \) (given), and \( \angle WYX = 81^\circ \), \( \angle WYZ = 81^\circ \)? Wait, no, the angle at \( Y \) for \( \triangle WXY \) is \( 81^\circ \) and for \( \triangle WZY \) is \( 81^\circ \)? Wait, actually, in the first figure, \( XY = ZY = 16 \), \( \angle WYX = 81^\circ \), \( \angle WYZ = 81^\circ \)? Wait, no, the angle at \( Y \) for \( \triangle WXY \) is \( 81^\circ \) and for \( \triangle WZY \) is \( 81^\circ \)? Wait, actually, \( WY \) is common, \( XY = ZY \), and the included angle \( \angle WYX = \angle WYZ \)? Wait, no, the angle at \( Y \) for \( \triangle WXY \) is \( 81^\circ \) and for \( \triangle WZY \) is \( 81^\circ \)? Wait, maybe I misread. Wait, the first comparison is \( |WX| \) vs \( |WZ| \). Using the SAS (Side - Angle - Side) congruence or the Hinge Theorem? Wait, no, if two sides and the included angle are equal, then the triangles are congruent. Wait, \( XY = ZY = 16 \), \( \angle WYX=\angle WYZ = 81^\circ \), and \( WY = WY \). So by SAS, \( \triangle WXY\cong\triangle WZY \), so \( WX = WZ \)? Wait, no, wait the angle: wait the angle at \( Y \) for \( \triangle WXY \) is \( 81^\circ \) and for \( \triangle WZY \) is \( 81^\circ \)? Wait, maybe the angle is the same. Wait, actually, if \( XY = ZY \), \( \angle WYX=\angle WYZ \), and \( WY \) is common, then by SAS, \( \triangle WXY\cong\triangle WZY \), so \( WX = WZ \). Wait, but maybe I made a mistake. Wait, no, let's check again. Wait, the first figure: \( XY = ZY = 16 \), \( \angle WYX = 81^\circ \), \( \angle WYZ = 81^\circ \), and \( WY \) is common. So SAS: two sides (XY = ZY, WY = WY) and included angle ( \( \angle WYX=\angle WYZ \)) are equal, so triangles are congruent, so \( WX = WZ \). Wait, but the options are \( <, >, = \) or cannot be determined. Wait, maybe I misread the angles. Wait, the angle at \( Y \) for \( \triangle WXY \) is \( 81^\circ \) and for \( \triangle WZY \) is \( 81^\circ \)? Wait, the problem says "Compare the given measures by choosing \( <, >, = \) or 'Cannot be determined'". For the first one: \( |WX| \) vs \( |WZ| \). Since \( XY = ZY \), \( \angle WYX=\angle WYZ \), and \( WY \) is common, by SAS congruence, \( \triangle WXY\cong\triangle WZY \), so \( WX = WZ \).

Step2: Analyze the second triangle (bottom)

In triangle \( UVT \) and \( DEF \): \( UV = DE \) (marked with the same tick), \( VT = DF \)? Wait, no, \( UVT \) has sides \( UV \) (tick), \( VT = 10 \), \( UT = 10 \)? Wait, no, \( UVT \): \( UV \) and \( UT \) have ticks? Wait, the second figure: \( \triangle UVT \) has \( UV \) and \( UT \) with ticks? Wait, no, \( \triangle UVT \): side \( UV \) (tick), \( VT = 10 \), \( UT = 10 \)? Wait, \( \triangle DEF \): \( DE \) (tick), \( EF = 16 \), \( DF \) (tick). Wait, in \( \triangle UVT \), the two equal sides are \( UV \) and \( UT \) (ticks), so it's isoceles with \( \angle V=\angle T \). In \( \triangle DEF \), the two equal sides are \( DE \) and \( DF \) (ticks), so it's isoceles with \( \angle E=\angle F \). Now, the side opposite \( \angle T \) in \( \triangle UVT \) is \( UV \), and the side opposite \( \angle F \) in \( \triangle DEF \) is \( DE \) (since \( DE = DF \), so \( \angle E=\angle F \)). Wait, the length of \( VT = 10 \) and \( EF = 16 \). In a triangle, the larger side is opposite the larger angle. In \( \triangle UVT \), the base is \( VT = 10 \), in \( \triangle DEF \), the base is \( EF = 16 \). Since \( VT < EF \), the vertex angle \( \angle U \) (opposite \( VT \)) is less than \( \angle D \) (opposite \( EF \)). Then, in \( \triangle UVT \), \( \angle V=\angle T=\frac{180 - \angle U}{2} \), in \( \triangle DEF \), \( \angle E=\angle F=\frac{180 - \angle D}{2} \). Since \( \angle U < \angle D \), then \( \frac{180 - \angle U}{2}>\frac{180 - \angle D}{2} \), so \( \angle T>\angle F \) (wait, \( \angle T \) is in \( \triangle UVT \), \( \angle F \) is in \( \triangle DEF \)). Wait, the question is \( m\angle T \) vs \( m\angle E \). Since \( \angle E=\angle F \) (in \( \triangle DEF \)) and \( \angle T=\angle V \) (in \( \triangle UVT \)). Since \( VT = 10 < EF = 16 \), the vertex angle \( \angle U \) (opposite \( VT \)) is smaller than \( \angle D \) (opposite \( EF \)). So \( 180 - \angle U>180 - \angle D \), so \( \frac{180 - \angle U}{2}>\frac{180 - \angle D}{2} \), so \( \angle T=\angle V>\angle E=\angle F \). So \( m\angle T>m\angle E \).

But let's focus on the first part (since the user's question is about the first comparison maybe? Wait, the first comparison is \( |WX| \) vs \( |WZ| \). As \( XY = ZY \), \( \angle WYX=\angle WYZ \), \( WY = WY \), by SAS, \( \triangle WXY\cong\triangle WZY \), so \( WX = WZ \). So \( |WX| = |WZ| \).

Step1 (First Comparison)

For \( |WX| \) and \( |WZ| \):
- \( XY = ZY = 16 \) (given).
- \( \angle WYX=\angle WYZ = 81^\circ \) (given, same angle at \( Y \)).
- \( WY \) is common to both \( \triangle WXY \) and \( \triangle WZY \).

By the SAS (Side - Angle - Side) congruence criterion, \( \triangle WXY\cong\triangle WZY \). Therefore, corresponding sides \( WX \) and \( WZ \) are equal, so \( |WX| = |WZ| \).

Step2 (Second Comparison)

For \( m\angle T \) and \( m\angle E \):
- In \( \triangle UVT \), \( UV = UT \) (marked with ticks), so it is isosceles with \( \angle V=\angle T \). The side opposite \( \angle T \) is \( UV \), and the base is \( VT = 10 \).
- In \( \triangle DEF \), \( DE = DF \) (marked with ticks), so it is isosceles with \( \angle E=\angle F \). The side opposite \( \angle F \) is \( DE \), and the base is \( EF = 16 \).

In a triangle, the larger the base, the larger the vertex angle (and vice - versa for the base angles). Since \( VT = 10<EF = 16 \), the vertex angle \( \angle U \) (opposite \( VT \)) in \( \triangle UVT \) is smaller than the vertex angle \( \angle D \) (opposite \( EF \)) in \( \triangle DEF \).

For an isosceles triangle, the base angles are given by \( \angle=\frac{180^\circ - \text{vertex angle}}{2} \). Let \( \angle U = \alpha \) and \( \angle D=\beta \), with \( \alpha<\beta \). Then \( \angle T=\frac{180^\circ - \alpha}{2} \) and \( \angle E=\frac{180^\circ - \beta}{2} \). Since \( \alpha<\beta \), \( 180^\circ - \alpha>180^\circ - \beta \), so \( \frac{180^\circ - \alpha}{2}>\frac{180^\circ - \beta}{2} \), which means \( \angle T>\angle E \), so \( m\angle T>m\angle E \).

Answer:

For \( |WX| \) and \( |WZ| \): \( = \)

For \( m\angle T \) and \( m\angle E \): \( > \)