Question

consider the four triangles.
image of four triangles with angle measures
which triangles are similar?
○ δlmn and δstu
○ δlmn and δpqr
○ δwxv and δstu
○ δwxv and δpqr

Explanation:

Step1: Calculate the third angle of each triangle

• For \(\triangle LMN\):

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(x\). Then \(x = 180-(43 + 73)=180 - 116=64^{\circ}\)

• For \(\triangle WXV\):

Let the third angle be \(y\). Then \(y=180-(37 + 43)=180 - 80 = 100^{\circ}\)

• For \(\triangle STU\):

Let the third angle be \(z\). Then \(z=180-(73 + 54)=180 - 127 = 53^{\circ}\)

• For \(\triangle PQR\):

Let the third angle be \(w\). Then \(w=180-(73+64)=180 - 137=43^{\circ}\)

Step2: Check for AA (Angle - Angle) similarity criterion

AA similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

• \(\triangle LMN\) has angles \(43^{\circ},73^{\circ},64^{\circ}\)
• \(\triangle PQR\) has angles \(73^{\circ},64^{\circ},43^{\circ}\)

Since two pairs of angles (\(43^{\circ},73^{\circ},64^{\circ}\)) are equal (just in a different order), \(\triangle LMN\) and \(\triangle PQR\) satisfy the AA similarity criterion.

Answer:

\(\triangle LMN\) and \(\triangle PQR\) (i.e., the option \(\triangle LMN\) and \(\triangle PQR\))