Question

given: \\( \triangle def \cong \triangle lmn \\). find each value.

1. \\( m \angle l = \\)
2. \\( ef = \\)

hint: first: find the value of \\( x \\)! then, use \\( x \\) to find \\( y \\).

Explanation:

Step1: Find the value of \( x \)

Since \( \triangle DEF\cong\triangle LMN \), corresponding sides are equal. \( DE = LM \), so \( 2x + 3=53 \).
Subtract 3 from both sides: \( 2x=53 - 3=50 \).
Divide both sides by 2: \( x=\frac{50}{2}=25 \).

Step2: Find the value of \( y \)

Corresponding angles are equal. \( \angle F=\angle N \), and in a triangle, the sum of angles is \( 180^{\circ} \).
For \( \triangle LMN \), \( \angle L=(x + 15)^{\circ}=(25+15)^{\circ}=40^{\circ} \), \( \angle N = 120^{\circ} \).
For \( \triangle DEF \), \( \angle F = 5y^{\circ} \). Since \( \angle F=\angle N \) (corresponding angles of congruent triangles), and using the angle - sum property of a triangle (\( \angle E+\angle F+\angle D=\angle M+\angle N+\angle L = 180^{\circ} \)), but more directly, since \( \angle F=\angle N \) (wait, no, actually \( \angle F \) and \( \angle N \) are not corresponding. Wait, \( \triangle DEF\cong\triangle LMN \), so \( \angle D=\angle L=(x + 15)^{\circ} \), \( \angle E=\angle M \), \( \angle F=\angle N \).
Wait, no, congruent triangles: \( \triangle DEF\cong\triangle LMN \) means \( \angle D=\angle L \), \( \angle E=\angle M \), \( \angle F=\angle N \).
In \( \triangle LMN \), \( \angle L=(x + 15)^{\circ} \), \( x = 25 \), so \( \angle L=40^{\circ} \).
For \( EF \): \( EF=LN \), and \( LN = 25.4 \). Also, for \( y \):
In \( \triangle DEF \), \( \angle F = 5y^{\circ} \), in \( \triangle LMN \), \( \angle N=120^{\circ} \), but wait no, \( \triangle DEF\cong\triangle LMN \), so \( \angle D=\angle L=(x + 15)^{\circ}=40^{\circ} \), \( \angle F=\angle N = 120^{\circ} \), then \( 5y=120 \), \( y = 24 \). But \( EF=LN \) (corresponding sides of congruent triangles).

Step3: Find \( m\angle L \)

Since \( x = 25 \), \( m\angle L=(x + 15)^{\circ}=(25+15)^{\circ}=40^{\circ} \).

Step4: Find \( EF \)

Since \( \triangle DEF\cong\triangle LMN \), \( EF = LN \). Given \( LN=25.4 \), so \( EF = 25.4 \).

Answer:

1. \( 40^{\circ} \)
2. \( 25.4 \)