Questions: Which method and additional information would prove triangle ONP and triangle MNL similar by the AA similarity postulate? Use a rigid transformation to prove angle NOP congruent to angle NML. Use a rigid transformation to prove angle NOP congruent to angle ONP. Use rigid and nonrigid transformations to prove LN / ON = PN / MN. Use rigid and nonrigid transformations to prove LM / n.1 = PN / ....

Which method and additional information would prove triangle ONP and triangle MNL similar by the AA similarity postulate?

Use a rigid transformation to prove angle NOP congruent to angle NML.
Use a rigid transformation to prove angle NOP congruent to angle ONP.
Use rigid and nonrigid transformations to prove LN / ON = PN / MN.
Use rigid and nonrigid transformations to prove LM / n.1 = PN / ....

Transcript text: Which method and additional information would prove $\triangle O N P$ and $\triangle M N L$ similar by the AA similarity postulate? Ose a rigid transformation to prove $\angle N O P \cong \angle N M L$. Ose a rigid transformation to prove $\angle N O P \cong \angle O N P$. Ose rigid and nonrigid transformations to prove $\frac{\overline{\mathrm{LN}}}{\mathrm{ON}}=\frac{\overline{\mathrm{PN}}}{\mathrm{MN}}$. O Use rigid and nonrigid transformations to prove $\frac{\overline{L M}}{\overline{n .1}}=\frac{\overline{\mathrm{PN}}}{\ldots . .}$.

Solution

Solution Steps

Step 1: Analyze the given triangles and the AA similarity postulate

The AA similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. We are looking for a way to prove that $\triangle ONP$ is similar to $\triangle MNL$. This means we need to show two pairs of corresponding angles are congruent.

Step 2: Identify corresponding angles

The vertices of the triangles are written in corresponding order. This implies that $\angle NOP$ corresponds to $\angle NML$, $\angle ONP$ corresponds to $\angle MNL$, and $\angle OPN$ corresponds to $\angle MLN$.

Step 3: Determine the correct method

The problem states line _k_ is a straight line. Angles $\angle LNM$ and $\angle ONP$ are vertical angles, hence, $\angle LNM \cong \angle ONP$. This gives us one pair of congruent angles. We need another pair.

Since line _k_ is a straight line, we know that $\angle KNO$ and $\angle MNL$ are supplementary as well as $\angle KNO$ and $\angle NOP$. It follows that $\angle MNL \cong \angle NOP$. Therefore, a rigid transformation (specifically, a rotation) could be used to prove $\angle N O P \cong \angle N M L$.