# Theorems

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### Theorem 1-6-1 Pythagorean Theorem

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

### Theorem 2-6-1 Linear Pair Theorem

If two angles form a linear pair, then they are supplementary.

### Theorem 2-6-2 Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.

### Theorem 2-6-3 RIght Angle Supplements Theorem

All right angles are congruent.

### Theorem 2-6-4 Congruent Complements Theorem

If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent.

### Theorem 2-7-1 Common Segments Theorem

Given collinear points A, B, C, and D arranged as shown, if segment AB is congruent to segment CD, then segment AC is congruent to segment BD.

### Theorem 2-7-2 Vertical Angles Theorem

Vertical angles are congruent.

### Theorem 2-7-3

If two congruent angles are supplementary, then each angle is a right angle.

### Theorem 3-2-2 Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

### Theorem 3-2-3 Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

### Theorem 3-2-4 Same-Side Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.

### Theorem 3-3-3 Converse of the Alternate Interior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.

### Theorem 3-3-4 Converse of the Alternate Exterior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.

### Theorem 3-3-5 Converse of the Same-Side Interior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.

### Theorem 3-4-1

If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.

### Theorem 3-4-2 Perpendicular Transversal Theorem

In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

### Theorem 3-4-3

If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other.

### Theorem 3-5-1 Parallel Lines Theorem

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

### Theorem 3-5-2 Perpendicular Lines Theorem

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

### Theorem 4-2-1 Triangle Sum Theorem

The sum of the angle measures of a triangle is 180°.

### Theorem 4-2-4 Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

### Theorem 4-2-5 Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.

### Theorem 4-5-2 Angle-Angle-Side (AAS) Congruence Theorem

If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.

### Theorem 4-5-3 Hypotenuse-Leg (HL) Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.

### Theorem 4-8-1 Isosceles Triangle Theorem

If two sides of a triangle are congruent, the the angles opposite the sides are congruent.

### Theorem 4-8-2 Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, the the sides opposite those angles are congruent.

### Theorem 5-1-1 Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

### Theorem 5-1-2 Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

### Theorem 5-1-3 Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

### Theorem 5-1-4 Converse of the Angle Bisector Theorem

If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

### Theorem 5-2-1 Circumcenter Theorem

The circumcenter of a triangle is equidistant from the vertices of the triangle.

### Theorem 5-2-2 Incenter Theorem

The incenter of a triangle is equidistant from the sides of the triangle.

### Theorem 5-3-1 Centroid Theorem

The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.

### Theorem 5-4-1 Triangle Midsegment Theorem

A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side.

### Theorem 5-5-1

If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.

### Theorem 5-5-2

If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.

### Theorem 5-5-3 Triangle Inequality Theorem

The sum of any two side lengths of a triange is greater than the third side length.

### Theorem 5-6-1 Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then longer third side is across from the larger included angle.

### Theorem 5-6-2 Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.

### Theorem 5-7-1 Converse of the Pythagorean Theorem

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

### Theorem 5-7-2 Pythagorean Inequalities Theorem

In ΔABC, c is the length of the longest side. If c² > a² + b², then ΔABC is an obtuse triangle. If c² < a² + b², then ΔABC is an acute triangle.

### Theorem 5-8-1 45°-45°-90° Triangle Theorem

In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times √2.

### Theorem 5-8-2 30°-60°-90° Triangle Theorem

In a 30°-60°-90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times √3.

### Theorem 6-1-1 Polygon Angle Sum Theorem

The sum of the interior angle measures of a convex polygon with n sides is ( n - 2 )180°

### Theorem 6-1-2 Polygon Exterior Angle Sum Theorem

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

### Theorem 6-2-1

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

### Theorem 6-2-2

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

### Theorem 6-2-3

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

### Theorem 6-2-4

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

### Theorem 6-3-1

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

### Theorem 6-3-2

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

### Theorem 6-3-3

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

### Theorem 6-3-4

If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

### Theorem 6-3-5

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

### Theorem 6-4-1

If a quadrilateral is a rectangle, then it is a parallelogram.

### Theorem 6-4-2

If a parallelogram is a rectangle, then its diagonals are congruent.

### Theorem 6-4-3

If a quadrilateral is a rhombus, then it is a parallelogram.

### Theorem 6-4-4

If a parallelogram is a rhombus, then its diagonals are perpendicular.

### Theorem 6-4-5

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

### Theorem 6-5-1

If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

### Theorem 6-5-2

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

### Theorem 6-5-3

If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

### Theorem 6-5-4

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

### Theorem 6-5-5

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

### Theorem 6-6-1

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

### Theorem 6-6-2

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

### Theorem 6-6-3

If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

### Theorem 6-6-4

If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.

### Theorem 6-6-5

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

### Theorem 6-6-6 Trapezoid Midsegment Theorem

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

### Theorem 7-3-2 Side-Side-Side (SSS) Similarity Theorem

If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.

### Theorem 7-3-3 Side-Angle-Angle (SAA) Similarity Theorem

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.

### Theorem 8-1-1

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.

### Theorem 11-1-1

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

### Theorem 11-1-2

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.

### Theorem 11-1-3

If two segments are tangent to a circle from the same external point, then the segments are congruent.

### Theorem 11-2-2

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.

### Theorem 11-2-3

In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc.

### Theorem 11-2-4

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.

### Theorem 11-4-3

An inscribed angle subtends a semicircle if and only if the angle is a right angle.

### Theorem 11-4-4

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

### Theorem 11-5-1

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.

### Theorem 11-5-2

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.

### Theorem 11-5-3

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.

### Theorem 12-4-1

A composition of two isometries is an isometry.

### Theorem 12-4-2

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.

### Theorem 12-4-3

Any translation or rotation is equivalent to a composition of two reflection.

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