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Theorem 7.1: Pythagorean Theorem

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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.

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Theorem 7.2: Converse of the Pythagorean Theorem

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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.

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Theorem 7.3

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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 an acute triangle.

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Theorem 7.4

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

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Theorem 7.5

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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.

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Theorem 7.6: Geometric Mean (Altitude) Theorem

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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.

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Theorem 7.7: Geometric Mean (Leg) Theorem

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

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Theorem 7.8: 45º-45º-90º Triangle Theorem

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In a 45º-45º-90º triangle, the hypotenuse is √ 2 times as long as each leg. Hypotenuse = leg x √ 2

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Theorem 7.9: 30º-60º-90º Triangle Theorem

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In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. Hypotenuse = 2 x shorter leg Longer leg = shorter leg x √3

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Theorem 8.1: Polygon Interior Angles Theorem

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

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Corollary to the Theorem 8.1: Interior Angles of a Quadrilateral

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

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Theorem 8.2: Polygon Exterior Angles Theorem

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

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Theorem 8.3

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

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Theorem 8.4

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

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Theorem 8.5

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

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Theorem 8.6

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

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Theorem 8.7

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

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Theorem 8.8

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

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Theorem 8.9

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

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Theorem 8.10

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

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Rhombus Corollary

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

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Rectangle Corollary

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

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Square Corollary

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

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Theorem 8.11

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

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Theorem 8.12

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

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Theorem 8.13

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

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Theorem 8.14

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

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Theorem 8.15

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

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Theorem 8.16

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

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Theorem 8.17

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

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Theorem 8.18

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

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Theorem 8.19

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

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Theorem 10.1

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In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

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Theorem 10.2

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Tangent segments from a common external point are congruent.

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Theorem 10.3

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In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

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Theorem 10.4

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If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

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Theorem 10.5

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If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

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Theorem 10.6

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In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center

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Theorem 10.7: Measure of an Inscribed Angle Theorem

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The measure of an inscribed angle is one half the measure of its intercepted arc.

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Theorem 10.8

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If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

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Theorem 10.9

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If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right triangle.

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Theorem 10.10

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A quadrilateral can be inscribed in a circle if and only of its opposite angles are supplementary.

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Theorem 10.11

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If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

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Theorem 10.12: Angles Inside the Circle Theorem

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If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

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Theorem 10.13 Angles Outside the Circle Theorem

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If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

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Theorem 10.14: Segments of Chords Theorem

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If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

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Theorem 10.15: Segments of Secants Theorem

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If two secant segments share the same endpoint outside of a circle, then the product of the lengths of one secant segment and its external segments equals the product of the lengths of the other secant segment and its external segment.

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Theorem 10.16: Segments of Secants and Tangents Theorem

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If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

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Theorem 11.1: Area of a Rectangle

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The area of a rectangle is the product of its base and height. A = bh

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Theorem 11.2: Area of a Parallelogram

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

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Theorem 11.3: Area of a Triangle

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The area of a triangle is one half the product of a base and its corresponding height. A = 1/2bh

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Theorem 11.4: Area of a Trapezoid

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The area of a trapezoid is one half the product of the height and the sum of the lengths of the bases. A = 1/2h(b1+b2)

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Theorem 11.5: Area of a Rhombus

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The area of a rhombus is one half the product of the lengths of its diagonals. A= 1/2d1d2

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Theorem 11.6: Area of a Kite

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The area of a kite is one half the product of the lengths of its diagonals. A = 1/2d1d2

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Theorem 11.7: Areas of Similar Polygons

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If two polygons are similar with the lengths of corresponding sides in the ratio a:b, then the ratio of their areas is a^2:b^2. Side length of polygon I/Side length of polygon II = a/b Area of polygon I / Area of polygon II = a^2/b^2.

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Theorem 11.8: Circumference of a Circle

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The circumference C of a circle is C = PiD or C = 2PiR, where D is the diameter of the circle and R is the radius of the circle.

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Arc Length Corollary

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In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 degrees.

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Theorem 11.9: Area of a Circle

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The area of a circle is Pi times the square of the radius. A = PiR^2

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Theorem 11.10: Area of a Sector

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The ratio of the area of a sector of a circle to the area of the whole circle (PiR^2) is equal to the ratio of the measure of the intercepted arc to 360 degrees.

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Theorem 11.11: Area of a Regular Polygon

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The area of a regular n-gon with side length s is one half the product of the apothem a and the perimeter P.

Geometry: Theorems and Corollaries Semester 2 – Flashcards

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