Module 3 Topic 1: Similarity

Angle Bisector/Proportional Theorem

A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle

Dilation

A transformation of a figure in which the figure stretches or shrinks with respect to a fixed point or center of dilation

Similar Figures

Geometric figures where all pairs of corresponding angles are congruent and the lengths of all corresponding sides are proportional Dilations produce similar figures

Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Triangle Proportionality Theorem

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

Side-Side-Side Similarity Theorem

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

Right Triangle/Altitude Similarity Theorem

If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Right Triangle Altitude/Leg Theorem

If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

Proportional Segments Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Angle-Angle Similarity Theorem

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

Side-Angle-Side Similarity Theorem

If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar

Angle of Incidence

The angle formed by the incidence ray and a line perpendicular to the surface of a mirror

geometric mean

The mean of two positive numbers a and b is the positive number x such that a/x = x/b

Right Triangle Altitude/Hypotenuse Theorem

The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

law of reflection

The measure of the angle of incidence equals the measure of the angle of reflection

Triangle Midsegment Theorem

The midsegment of a triangle is parallel to the third side of the triangle and is half the measure of the third side of the triangle

angle bisector

a ray that divides an angle into two congruent angles

Directed line segment

assigned a direction from one endpoint to the other

angle of reflection

the angle formed by the reflected ray and a line perpendicular to the surface of a mirror

indirect measurement

the technique that uses proportions to determine a measurement when direct measurement is not possible