If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
Theorem 7-4-1 Triangle Proportionality Theorem
If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally.
Theorem 7-4-2 Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Theorem 7-4-4 Triangle Angle Bisector Theorem
An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides.
Theorem 7-5-1 Proportional Perimeters and Areas Theorem
If the similarity ratio of two similar figures is a / b, then the ratio of their perimeters is a / b, and the ratio of their areas is a² / b² or ( a / b )².
The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
Theorem 8-5-1 The Law of Sines
For any ΔABC with side lengths a, b, and c, sin A / a = sin B / b = sin C / c.
Theorem 8-5-2 The Law of Cosines
For any ΔABC with sides a, b, and c, a² = b² + c² - 2b cos A, b² = a² + c² - 2ac cos B, and c² = a² + b² - 2ab cos C.
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.
If two segments are tangent to a circle from the same external point, then the segments are congruent.
In a circle or congruent circles: (1) congruent central angles have congruent chords, (2) congruent chords have congruent arcs, and (3) congruent arcs have congruent central angles.
In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc.
In a circle, the perpendicular bisector of a chord is a radius (or diameter).
Theorem 11-4-1 Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
An inscribed angle subtends a semicircle if and only if the angle is a right angle.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc.
If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the difference of the measures of its intercepted arcs.
If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs.
Theorem 11-6-1 Chord-Chord Product Theorem
If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal.
Theorem 11-6-2 Secant-Secant Product Theorem
If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
Theorem 11-6-3 Secant-Tangent Product Theorem
If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.
Theorem 11-7-1 Equation of a Circle
the equation of a circle with center ( h, k ) and radius r is ( x - h )² + ( y - k )² = r².
A composition of two isometries is an isometry.
The composition of two reflections across two parallel lines is equivalent to a translation. The translation vector is perpendicular to the lines. The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. The center of rotation is the intersection of the lines. The angle of rotation is twice the measure of the angle formed by the lines.
Any translation or rotation is equivalent to a composition of two reflection.