Question

6 multiple answer 1 point in triangle abc, angle a is 15°, and angle b is 30°. select all triangles that are similar to triangle abc. triangle def where angle e is 15° and angle f is 45°. triangle ghi where angle g is 15° and angle i is 135°. triangle jkl where angle j is 30° and angle l is 55°. triangle mno where angle m is 45° and angle o is 30°. triangle pqr where angle p is 30° and angle q is 135°.

Explanation:

Step1: Calculate angle C in triangle ABC

The sum of angles in a triangle is \(180^{\circ}\). So, \(\angle C=180^{\circ}-\angle A - \angle B\). Substituting \(\angle A = 15^{\circ}\) and \(\angle B=30^{\circ}\), we get \(\angle C=180^{\circ}-15^{\circ}-30^{\circ}=135^{\circ}\).

Step2: Check triangle DEF

\(\angle D = 180^{\circ}-\angle E-\angle F=180^{\circ}-15^{\circ}-45^{\circ}=120^{\circ}\). Angles are \(15^{\circ},45^{\circ},120^{\circ}\). Not similar to \(15^{\circ},30^{\circ},135^{\circ}\).

Step3: Check triangle GHI

\(\angle H=180^{\circ}-\angle G - \angle I=180^{\circ}-15^{\circ}-135^{\circ}=30^{\circ}\). Angles are \(15^{\circ},30^{\circ},135^{\circ}\). Similar.

Step4: Check triangle JKL

\(\angle K=180^{\circ}-\angle J-\angle L=180^{\circ}-30^{\circ}-55^{\circ}=95^{\circ}\). Angles are \(30^{\circ},55^{\circ},95^{\circ}\). Not similar.

Step5: Check triangle MNO

\(\angle N=180^{\circ}-\angle M-\angle O=180^{\circ}-45^{\circ}-30^{\circ}=105^{\circ}\). Angles are \(30^{\circ},45^{\circ},105^{\circ}\). Not similar.

Step6: Check triangle PQR

\(\angle R=180^{\circ}-\angle P-\angle Q=180^{\circ}-30^{\circ}-135^{\circ}=15^{\circ}\). Angles are \(15^{\circ},30^{\circ},135^{\circ}\). Similar.

Answer:

Triangle GHI, Triangle PQR