Question

1. *

which triangles are similar?
choose
this is a required question

Explanation:

Step1: Calculate the third angle of each triangle

For triangle \(ABC\):
The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(\angle B\).
\(\angle B=180^{\circ}-(85^{\circ} + 53^{\circ})=180^{\circ}-138^{\circ} = 42^{\circ}\)

For triangle \(DEF\):
Let the third angle be \(\angle F\).
\(\angle F=180^{\circ}-(47^{\circ}+64^{\circ})=180^{\circ}-111^{\circ}=69^{\circ}\)

For triangle \(GHI\):
Let the third angle be \(\angle H\).
\(\angle H=180^{\circ}-(47^{\circ}+68^{\circ})=180^{\circ}-115^{\circ}=65^{\circ}\)

For triangle \(JKL\):
Let the third angle be \(\angle K\).
\(\angle K=180^{\circ}-(85^{\circ}+42^{\circ})=180^{\circ}-127^{\circ}=53^{\circ}\)

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

Two triangles are similar if two pairs of corresponding angles are equal.

• Triangle \(ABC\) has angles \(85^{\circ},53^{\circ},42^{\circ}\)
• Triangle \(JKL\) has angles \(85^{\circ},42^{\circ},53^{\circ}\)

Since two pairs of angles (\(85^{\circ}\) and \(42^{\circ}\), \(53^{\circ}\) and \(85^{\circ}\) - when considering the order of angles in similarity) are equal, by AA similarity criterion, \(\triangle ABC\sim\triangle JKL\)

Answer:

Triangles \(1\) (i.e., \(\triangle ABC\)) and \(4\) (i.e., \(\triangle JKL\)) are similar.