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

A. Use rigid and nonrigid transformations to prove segment PN over segment MN equals segment LN over segment ON.

B. Use a rigid transformation to prove that angle NPO is congruent to angle NLM.

C. Use rigid and nonrigid transformations to prove segment PN over segment ML equals segment LN over segment ON.

D. Use a rigid transformation to prove that angle NLM is congruent to angle LMN.

The AA stands for "angle angle". So we need two pairs of angles to prove the triangles to be similar. The first pair of angles is the vertical angles ONP and MNL, which are congruent. Any pair of vertical angles are always congruent.

The second pair of angles could either be

angle NOP = angle NML
angle NPO = angle NLM

so we have a choice on which to pick. The pairing angle NOP = angle NML is not listed in the answer choices, but angle NPO = angle NLM is listed as choice B.

Saying angle NLM = angle LMN is not useful because those two angles are part of the same triangle. The two angles must be in separate triangles to be able to tie the triangles together.

We would use a rigid transformation to have angle NPO move to angle NLM, or vice versa through the use of a rotation and a translation.

oeerivona oeerivona · Answer 2 · 2021-08-15T13:05:50+02:00

Angle Angle AA similarity postulate states that two triangles are similar if

two angles in one of the triangles are congruent (equal) to two angles in the

other triangle

The correct option for the method and additional information that would

prove triangle ONP and triangle MNL similar by AA similarity postulate is

option B.

B. Use rigid transformation to prove that angle NPO is congruent to angle NLM

The reason for arriving at the above selection is as follows;

The known parameter in triangle ΔONP and ΔMNL are;

Lines MO and LP intersect at vertex point N.

The angles on opposite side of the point N (either side of the X shape

formed by the two lines) which are ∠ONP and ∠LNM are vertically opposite

angles or opposite angles.

According to vertically opposite angles theorem ∠ONP and ∠LNM, which

are vertically opposite angles are congruent, and we can write.

∠ONP ≅ ∠LNM

The required parameter;

To find the additional information that would prove ΔONP and ΔMNL by

AA (Angle-Angle) similarity postulate.

Strategy;

Given that from the drawing, one (vertex) angle in one triangle is congruent

to one (vertex) angle in the other triangle, that is ∠ONP ≅ ∠LNM, it is

required to prove that one other angle in triangle ΔONP is congruent to a

corresponding angle in triangle ΔMNL.

Solution;

Prove that either ∠NPO or ∠NOP in ΔONP are congruent to ∠NLM or

∠NML respectively in ΔMNL by performing rigid transformations

(transformations that does not change shape) on either ΔONP or ΔMNL

including;

Rotating triangle ΔONP by 180° with origin at point N, to form the image ΔO'N'P'.

Translating the vertex point P' in the image ΔO'N'P', to the location of the point L in ΔMNL, and verify that the segments LN and LM in ΔMNL coincide with the segments P'N' and P'O' respectively of triangle ΔO'N'P'.

Therefore, the correct option is option B. Use rigid transformation to prove

that angle NPO is congruent to angle NLM.

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