Question

determine if the two triangles are congruent when ( y = 10 ). if so, write the congruence statement along with which congruence theorem proves that they are congruent.
(1 point)
( \triangle jkl cong \triangle mno ) by the aas congruence theorem
( \triangle jkl cong \triangle mno ) by the asa congruence theorem
( \triangle jkl cong \triangle mno ) by the sas congruence theorem
the triangles are not congruent

Explanation:

Step1: Calculate the angles and sides when \(y = 10\)

• For \(\triangle JKL\):
• One side \(JK=25\).
• One angle \(\angle J = 54^{\circ}\).
• Another angle \(\angle L=7y\). Substitute \(y = 10\), then \(\angle L=7\times10 = 70^{\circ}\).
• For \(\triangle MNO\):
• One side \(MN=3y + 5\). Substitute \(y = 10\), then \(MN=3\times10+5=35

eq25\).

• One angle \(\angle O = 70^{\circ}\).
• Another angle \(\angle M=5y + 4\). Substitute \(y = 10\), then \(\angle M=5\times10 + 4=54^{\circ}\).

Step2: Analyze congruence theorems

• AAS (Angle - Angle - Side):
• In \(\triangle JKL\) and \(\triangle MNO\), \(\angle J=\angle M = 54^{\circ}\), \(\angle L=\angle O = 70^{\circ}\), and \(JK\) (side in \(\triangle JKL\)) and \(MO\) (side in \(\triangle MNO\)):
• First, find the third - angle in \(\triangle JKL\): \(\angle K=180^{\circ}-\angle J-\angle L=180 - 54-70=56^{\circ}\).
• In \(\triangle MNO\), \(\angle N=180^{\circ}-\angle M-\angle O=180 - 54-70 = 56^{\circ}\).
• For AAS, we have two angles and a non - included side. \(\angle J=\angle M\), \(\angle L=\angle O\), and \(JK = MO\) (by calculating angles and using the fact that the sum of angles in a triangle is \(180^{\circ}\) and side - angle relationships).

Answer:

\(\triangle JKL\cong\triangle MNO\) by the AAS Congruence Theorem.