Proofs: Triangle Similarity: Proofs: Triangle Similarity
Three Theorems: Three Theorems There are three theorems we use to prove triangle similarity. These should not be too surprising, believe it or not. Angle-Angle Theorem Side-Side-Side Theorem Side-Angle-Side Theorem
Angle-Angle Theorem: Angle-Angle Theorem What it says : if two angles of one triangle are congruent to the two corresponding angles of another triangle, then the two triangles are similar. What we need to know: the measures of two angles of one triangle and the corresponding angle measures of the other triangle.
Side-Side-Side Theorem: Side-Side-Side Theorem What it says: if you have two triangles, and all corresponding sides have the same scale factor, then the triangles are similar. What we need to know: the lengths of all sides of both triangles.
Side-Angle-Side Theorem: Side-Angle-Side Theorem What it says: if two sides of one triangle and the corresponding sides of a second triangle have the same scale factor, AND the angle that is directly between the two sides on the first triangle is congruent to the angle that is directly between the two sides on the second triangle, then the triangles are similar. What we need to know: the lengths of two sides AND the measure of the angle DIRECTLY between them, for both triangles.
Similar Right Triangles: Similar Right Triangles First, a definition. What would you call a straight line drawn from a vertex and to the opposite side, making a right angle with the side? You should recognize this line as the height of the triangle. Officially, it’s called an altitude . We’re going to learn how to construct an altitude, using a compass and straightedge.
More on the altitude: More on the altitude When an altitude is drawn in a right triangle, going to the hypotenuse, the length of the altitude is referred to as a geometric mean . The definition of geometric mean: when we have geometric mean “x”, and values “a” and “b”, they can be expressed with the following proportion:
Right Triangle Altitude Theorems: Right Triangle Altitude Theorems #1: The measure of the altitude drawn from the vertex of the right angle to the hypotenuse is the geometric mean of the two segments of the hypotenuse. #2: If the altitude is drawn to the hypotenuse of a right triangle, the length of each leg 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 Theorems, in English: Right Triangle Altitude Theorems, in English #1: When the altitude goes to the hypotenuse, the altitude is x , and the two segments created by the altitude are a and b. #2: When the altitude goes to the hypotenuse, the length of each leg is x, the length of the hypotenuse is a and the length of the hypotenuse segment that is touching the given leg is b .