If two corresponding interior angles of two triangles are congruent, how do you know that the triangles are similar?
1.) Since two corresponding interior angles are congruent, the sides opposite the angles are congruent. The remaining pair of sides must also be congruent. The triangles are similar by SSS criterion.
2.) Since the sum of the angles in a triangle is 360°, the third angle pair must have the same measure and are thus congruent. Because all three corresponding interior angles of the triangle are congruent, the triangles are similar by the definition of similarity.
3.) Since the sum of the angles in a triangle is 180°, the third angle pair must have the same measure and are thus congruent. Because all three corresponding interior angles of the triangle are congruent, the triangles are similar by the definition of similarity.
4.) Since two corresponding interior angles are congruent, the sides opposite the angles are congruent. The triangles are similar by SAS criterion.
1.) Since two corresponding interior angles of two triangles are congruent, the sides opposite the angles are congruent. However, this is not enough to conclude that the triangles are similar. In order to prove similarity, we need to show that all corresponding angles are congruent, not just two.
2.) This statement is not correct. The sum of the angles in a triangle is always 180°, not 360°. Therefore, we cannot conclude that the third angle pair has the same measure.
3.) This statement is correct. Since the sum of the angles in a triangle is always 180°, the third angle pair must have the same measure and be congruent. Therefore, all three corresponding angles of the triangles are congruent, satisfying the definition of similarity.
4.) This statement is also not correct. While two corresponding angles being congruent implies that the sides opposite the angles are congruent, it is not enough to prove similarity by the SAS criterion. The SAS criterion requires that two pairs of corresponding sides are proportional and the included angle is congruent.