^{o}. The difference between similar and congruent triangles is that … 1. [Angle-Angle (AA) Similarity Postulate – if two angles of one trian- gle are congruent to two angles of another, then the triangles must be similar.] When this happens, the opposite sides, namely BC and EF, are parallel lines.. The SAS rule states that, two triangles are similar if the ratio of their corresponding two sides is equal and also, the angle formed by the two sides is equal. Two triangles are similar if one of their angles is congruent and the corresponding sides of the congruent angle are proportional in length. If the triangles △ ABC and △ DEF are similar, we can write this relation as △ ABC ∼ △ DEF. Proving similar triangles refers to a geometric process by which you provide evidence to determine that two triangles have enough in common to be considered similar. – Angle Angle Side (AAS): A pair of corresponding angles and a non-included side are equal. The two triangles are simply called the similar triangles. SAS (side angle side)Two pairs of sides in the same proportion and the included angle equal.See Similar Triangles SAS. alternatives. Example 1 : While playing tennis, David is 12 meters from the net, which is 0.9 meter high. It means that we have 3 similar triangles. Each side of [latex]\Delta ABC[/latex] is four times the length of the corresponding side of [latex]\Delta XYZ[/latex] and their corresponding angles have equal measures. It has been thought that there are no similar triangles on the sphere, but in fact they are not. If two triangles are similar, then the ratio of corresponding sides is equal to the ratio of the angle bisectors, altitudes, and medians of the two triangles. The ratio of side lengths for triangle one is: Thus the ratio of side lengths for the second triangle must following this as well: , because both side lengths in triangle one have been multiplied by a factor of . This is also sometimes called the AAA rule because equality of two corresponding pairs of angles would imply that the third corresponding pair of angles are also equal. In recent lessons, you have learned that similar triangles have equal corresponding angles. Congruent Triangles. Corresponding angles in a triangle have the same measure. Since both ratios equal 2, the two sets of corresponding sides are proportional. In the two triangles, the included angles (the angles between the corresponding sides) are both right angles, therefore they are congruent. If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. – Hypotenuse Leg (HL): Hypotenuse and one leg are equal. Typically, the smaller of the two similar triangles is part of the larger. 2. If two triangles are similar, they remain similar even after rotation or reflection about any axis as these two operations do not alter the shape of the triangle. 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