Question

similarity in right triangles quiz
cp geometry a o / trigonometric ratios

△lnm ~ △pno
△mln ~ △pno
△nml ~ △nop
△lnm ~ △onp

Explanation:

To determine similar triangles, we use the AA (Angle - Angle) similarity criterion. In the given diagram, we analyze the angles of the triangles:

1. For \(\triangle LNM\) and \(\triangle PNO\):

• \(\angle LNM=\angle PNO = 90^{\circ}\) (right angles, as indicated by the square symbols at \(N\)).
• Also, \(\angle LMN=\angle PON\) (vertical angles or by the nature of the right - triangle configuration, the non - right angles are equal because the sum of angles in a triangle is \(180^{\circ}\) and one angle is \(90^{\circ}\) for both triangles).
• By the AA similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. So, \(\triangle LNM\sim\triangle PNO\) holds true.
• For the other options:
• For \(\triangle MLN\sim\triangle PNO\), the angle correspondence is incorrect as the vertices are not matched properly to satisfy the AA criterion.
• For \(\triangle NML\sim\triangle NOP\), the angle correspondence and vertex matching do not satisfy the AA similarity condition.
• For \(\triangle LNM\sim\triangle ONP\), the angle correspondence is wrong as the right angles and the other angles do not match in terms of vertex correspondence.

Answer:

A. \(\triangle LNM\sim\triangle PNO\)