Question

select all triangles that are similar to the triangle shown. note they may not be drawn to scale.
show your work here

Explanation:

Step1: Calculate the third angle of the original triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(x\). Then \(x = 180-(27 + 67)=180 - 94=86^{\circ}\). So the angles of the original triangle are \(27^{\circ},67^{\circ},86^{\circ}\).

Step2: Check the first sub - triangle

For the triangle with angles \(39^{\circ},74^{\circ}\), the third angle is \(180-(39 + 74)=180 - 113 = 67^{\circ}\). The angles \(39^{\circ},67^{\circ},74^{\circ}\) are not the same as \(27^{\circ},67^{\circ},86^{\circ}\).

Step3: Check the second sub - triangle

For the triangle with angles \(27^{\circ},67^{\circ}\), since the sum of angles in a triangle is \(180^{\circ}\), the third angle is \(180-(27 + 67)=86^{\circ}\). The angles \(27^{\circ},67^{\circ},86^{\circ}\) are the same as the original triangle's angles.

Step4: Check the third sub - triangle

For the triangle with angles \(67^{\circ},86^{\circ}\), the third angle is \(180-(67 + 86)=180 - 153=27^{\circ}\). The angles \(27^{\circ},67^{\circ},86^{\circ}\) are the same as the original triangle's angles.

Step5: Check the fourth sub - triangle

For the triangle with angles \(27^{\circ},86^{\circ}\), the third angle is \(180-(27 + 86)=180 - 113 = 67^{\circ}\). The angles \(27^{\circ},67^{\circ},86^{\circ}\) are the same as the original triangle's angles.

Step6: Check the fifth sub - triangle

For the triangle with angles \(57^{\circ},59^{\circ}\), the third angle is \(180-(57 + 59)=180 - 116 = 64^{\circ}\). The angles \(57^{\circ},59^{\circ},64^{\circ}\) are not the same as \(27^{\circ},67^{\circ},86^{\circ}\).

Answer:

The triangles with angles \(27^{\circ},67^{\circ}\) (second sub - triangle), \(67^{\circ},86^{\circ}\) (third sub - triangle), \(27^{\circ},86^{\circ}\) (fourth sub - triangle) are similar to the given triangle.