Geometry

Similarity of Polygons

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

SSS Similarity

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

SAS Similarity

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

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.

Converse of the Triangle Proportionality Theorem

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

Parallel Lines Theorem

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

Angle Bisector Theorem

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

Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Converse Pythagorean Theorem

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Acute and Obtuse Triangles

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. If greater, the triangle is obtuse.

Similarity of Triangles from Altitude

If the 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.

Geometric Mean: Altitude Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

Geometric Mean: Leg Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. An altitude of a triangle is a line segment that starts at a vertex and extends perpendicularly to the opposite side.

45-45-90 Triangle Theorem

In a 45-45-90 triangle, the hypotenuse is the square root of 2 times as long as each leg.

30-60-90 Triangle Theorem

In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is $ ext{length} imes oot{3}$ as long as the shorter leg.

Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n-gon is (n-2) x 180.

Corollary to Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a quadrilateral is 360.

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.

Properties of Parallelogram

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Congruence and Supplementary Angles of Parallelogram

If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Diagonals of Parallelogram

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Conditions for Parallelogram

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

One Pair of Sides in Parallelogram

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

Region Properties of Quadrilateral

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has four right angles.

Square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Diagonals of Rhombus

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Opposite Angles in Rhombus

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Rectangle Conditions

A parallelogram is a rectangle if and only if its diagonals are congruent.

Isosceles Trapezoid

If a trapezoid is isosceles, then both pairs of base angles are congruent.

Base Angles of Trapezoid

If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

Trapezoid Properties

If a trapezoid has exactly one pair of parallel sides, then it is not a parallelogram.

Isosceles Trapezoid Definition

A trapezoid is isosceles if and only if its diagonals are congruent.

Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to each base and its length is half the sum of the lengths of the bases.

Kite Properties

If a quadrilateral is a kite, then its diagonals are perpendicular.

Kite Angle Properties

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Area of Parallelogram

The area of a parallelogram is the product of a base and its corresponding height.