Question

which similarity statements are true? check all that apply. □ △jkl ~ △kml □ △jkm ~ △jkl □ △jkm ~ △kml □ △jmk ~ △jkl □ △jmk ~ △kml

Explanation:

Step1: Recall Similarity Criteria

For triangles, similarity can be established by AA (Angle - Angle) criterion, i.e., if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. In right - triangle \( \triangle JKL \) with right angle at \( K \), and \( KM\perp JL \), \( \angle JKM = \angle KML=\angle JKL = 90^{\circ}\).

Step2: Analyze \( \triangle JKL\sim\triangle KML \)

• In \( \triangle JKL \) and \( \triangle KML \):
• \( \angle JKL=\angle KML = 90^{\circ}\) (right angles).
• \( \angle L=\angle L\) (common angle).
• By AA criterion, \( \triangle JKL\sim\triangle KML \).

Step3: Analyze \( \triangle JKM\sim\triangle JKL \)

• In \( \triangle JKM \) and \( \triangle JKL \):
• \( \angle JKM=\angle JKL = 90^{\circ}\) (right angles).
• \( \angle J=\angle J\) (common angle).
• By AA criterion, \( \triangle JKM\sim\triangle JKL \).

Step4: Analyze \( \triangle JMK\sim\triangle JKL \)

• In \( \triangle JMK \) and \( \triangle JKL \):
• \( \angle JMK=\angle JKL = 90^{\circ}\) (right angles).
• \( \angle J=\angle J\) (common angle).
• By AA criterion, \( \triangle JMK\sim\triangle JKL \).

Step5: Analyze \( \triangle JMK\sim\triangle KML \)

• In \( \triangle JMK \) and \( \triangle KML \):
• \( \angle JMK=\angle KML = 90^{\circ}\) (right angles).
• \( \angle J=\angle KML\) (since \( \triangle JKL\sim\triangle KML \) and \( \triangle JKM\sim\triangle JKL \), the angles are equal). Also, \( \angle JKM=\angle L\) (from similar triangles). So, by AA criterion, \( \triangle JMK\sim\triangle KML \).

Step6: Analyze \( \triangle JKM\sim\triangle KML \)

• In \( \triangle JKM \) and \( \triangle KML \):
• \( \angle JKM=\angle KML = 90^{\circ}\) (right angles).
• \( \angle J=\angle LKM\) (from the similarity of other triangles). So, by AA criterion, \( \triangle JKM\sim\triangle KML \). Wait, actually, we can also see that:
• Since \( \triangle JKL\sim\triangle KML \) and \( \triangle JKM\sim\triangle JKL \), transitivity of similarity (if \( A\sim B \) and \( B\sim C \), then \( A\sim C \)) also holds. But let's re - check:
• For \( \triangle JKM \) and \( \triangle KML \):
• \( \angle JKM=\angle KML = 90^{\circ}\).
• \( \angle J=\angle LKM\) (because in \( \triangle JKL \), \( \angle J+\angle L = 90^{\circ}\), and in \( \triangle KML \), \( \angle L+\angle LKM=90^{\circ}\), so \( \angle J=\angle LKM\)). So, by AA criterion, \( \triangle JKM\sim\triangle KML \).

Answer:

\( \triangle JKL\sim\triangle KML \), \( \triangle JKM\sim\triangle JKL \), \( \triangle JMK\sim\triangle JKL \), \( \triangle JMK\sim\triangle KML \), \( \triangle JKM\sim\triangle KML \) (The check - box options that are correct are: \( \triangle JKL\sim\triangle KML \), \( \triangle JKM\sim\triangle JKL \), \( \triangle JMK\sim\triangle JKL \), \( \triangle JMK\sim\triangle KML \), \( \triangle JKM\sim\triangle KML \))