Question

each row of the table has two of the interior angle measures of two triangles. which triangles are similar? choose similar or not similar for each pair of triangles. triangles similar not similar triangle 1: 15°, 45° triangle 2: 45°, 120° triangle 1: 90°, 45° triangle 2: 45°, 45° triangle 1: 130°, 35°

Explanation:

Step1: Find the third angle of each triangle

• For Triangle 1 with angles \(15^{\circ},45^{\circ}\):

The third angle \(A_1 = 180^{\circ}-(15^{\circ}+45^{\circ})=120^{\circ}\)

• For Triangle 2 with angles \(45^{\circ},120^{\circ}\):

The third angle \(A_2 = 180^{\circ}-(45^{\circ}+120^{\circ})=15^{\circ}\)
Since all three angles of Triangle 1 (\(15^{\circ},45^{\circ},120^{\circ}\)) are equal to all three angles of Triangle 2 (\(45^{\circ},120^{\circ},15^{\circ}\)), by the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), these two triangles are similar.

• For Triangle 1 with angles \(90^{\circ},45^{\circ}\):

The third angle \(B_1=180^{\circ}-(90^{\circ} + 45^{\circ})=45^{\circ}\)

• For Triangle 2 with angles \(45^{\circ},45^{\circ}\):

The third angle \(B_2=180^{\circ}-(45^{\circ}+45^{\circ})=90^{\circ}\)
But in a triangle, the angles must satisfy the triangle - angle sum property. A triangle with angles \(45^{\circ},45^{\circ},90^{\circ}\) (Triangle 1) and a triangle with angles \(45^{\circ},45^{\circ},90^{\circ}\) (after calculating for Triangle 2) would be similar if the correspondence is correct. However, if we consider the order of angles:
The angles of Triangle 1 are \(90^{\circ},45^{\circ},45^{\circ}\) and of Triangle 2 are \(45^{\circ},45^{\circ},90^{\circ}\). But if we check the AA criterion:
If we assume the correspondence of angles, for two angles of Triangle 1 (\(90^{\circ},45^{\circ}\)) and two angles of Triangle 2 (\(45^{\circ},45^{\circ}\)), since \(90^{\circ}
eq45^{\circ}\), these two triangles are not similar.

• For Triangle 1 with angles \(130^{\circ},35^{\circ}\):

The third angle \(C_1=180^{\circ}-(130^{\circ}+35^{\circ}) = 15^{\circ}\)
Since there is no second triangle given (assuming it's a typo and maybe the intended second triangle for the last row is missing proper data, but based on the given structure of two - triangle comparisons in each row and the previous logic)

Answer:

Triangle 1 (\(15^{\circ},45^{\circ}\)) and Triangle 2 (\(45^{\circ},120^{\circ}\)): Similar
Triangle 1 (\(90^{\circ},45^{\circ}\)) and Triangle 2 (\(45^{\circ},45^{\circ}\)): Not Similar