SBU2 Similarity Theorems and Postulates

Triangle-Angle-Bisector Theorem (7.5)

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Cross Products

If a/b=c/d then ad=cb

Theorem 7-1 Side-Angle-Side Similarity Theorem (SAS~)(7.1)

If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar

Theorem 7-2 Side-Side-Side Similarity Theorem (SSS~)(7.2)

If the corresponding sides of two triangles are proportional, then the triangles are similar.

Corollary to the Side Splitter Theorem (7.4.1)

If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional

Angle-Angle Similarity Postulate (𝑨𝑨~)(7.1)

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Side-Splitter Theorem (7.4)

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally

Altitude to Hypotenuse Theorem (7.3)

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and each other. If ∆𝐴𝐵𝐶 is a right triangle with the right ∟𝐴𝐵𝐶 and (𝐶𝐷) ̅ is the altitude to the hypotenuse.

Right Triangle Leg Rule (7.3.2)

The altitude to the hypotenuse of a right triangle separates the hypotenuse, so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg.

Right Triangle Altitude Rule (7.3.1)

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.