Question

use the figure to answer the question.
select all of the statements that are sufficient to determine that ( \triangle jlm ) is isosceles?
a ( j k=k l ) b. ( j m=m l )
c ( j l=m k ) d. ( mathrm{m} angle m j k=mathrm{m} angle m l k )
e ( mathrm{m} angle m j l=mathrm{m} angle j m l ) f. ( mathrm{m} angle j m k=mathrm{m} angle l m k )

Explanation:

Step1: Recall the definition of an isosceles triangle

An isosceles triangle has at least two equal - length sides or two equal - measure angles.

Step2: Analyze each option

• Option A: \(JK = KL\) only tells that \(K\) is the mid - point of \(JL\). It does not give information about the sides \(JM\) and \(ML\) or the angles of \(\triangle JLM\). So, it is not sufficient.
• Option B: If \(JM=ML\), by the definition of an isosceles triangle (a triangle with two equal sides), \(\triangle JLM\) is isosceles.
• Option C: \(JL = MK\) gives the relationship between a side of \(\triangle JLM\) (\(JL\)) and a line segment \(MK\) (not a side of \(\triangle JLM\)). It does not help in determining if \(\triangle JLM\) is isosceles.
• Option D: If \(m\angle MJK=m\angle MLK\), then by the converse of the isosceles triangle theorem (if two angles of a triangle are equal, then the sides opposite those angles are equal), the sides \(JM\) and \(ML\) (opposite to \(\angle MLK\) and \(\angle MJK\) respectively) are equal. So, \(\triangle JLM\) is isosceles.
• Option E: \(m\angle MJL=m\angle JML\) gives the relationship between two angles of \(\triangle JML\) (not \(\triangle JLM\)). It does not help in determining if \(\triangle JLM\) is isosceles.
• Option F: \(m\angle JMK=m\angle LMK\) only tells that \(MK\) is the angle - bisector of \(\angle JML\). It does not give information about the sides \(JM\) and \(ML\) or the angles \(\angle MJK\) and \(\angle MLK\) of \(\triangle JLM\). So, it is not sufficient.

Answer:

B. \(JM = ML\), D. \(m\angle MJK=m\angle MLK\)