Geometry 2/2

Similar Polygons

Similar Polygons

Two triangles that have the same angles and their sides(when they are different sizes) all increase by the same scale factor.

Perimeters of Similar Polygons

If two polygons are similar, then their perimeter ratio will be equal to the ratios of their corresponding side lengths.

Corresponding Lengths in Similar Polygons

If two polygons are similar, then their perimeter ratio will be equal to the ratios of their corresponding side lengths.

Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the scale factor squared.

Angle Angle (AA) Similarity Postulate

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

Side Side Side (SSS) Similarity Theorem

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

Side Angle Side (SAS) Similarity Theorem

If one angle is congruent to another angle in another triangle, and the sides including those angles are proportional, then the two triangles are similar.

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle and intersects the other two sides then the triangle has been divided 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.

Theorem 6.6

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

Theorem 6.7

If a ray bisects an angle of a triangle, then 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 length of its hypotenuse is equal to the sum of the squared lengths of its sides. c2 = a2 + b2

Converse of the Pythagorean Theorem

If the square length of its hypotenuse is equal to the sum of the squared lengths of its sides, then the triangle is a right triangle. c2 = a2 + b2 then ABC = right triangle

Theorem 7.5

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.

The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

45° 45° 90° Triangle Theorem

In a 45 45 90 degree triangle, the hypotenuse is √2 times as long as each leg.

Hypotenuse = leg * √2

30° 60° 90° Triangle Theorem

In a 30 60 90 degree 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 * shorter leg longer leg = shorter leg * √3

Tangent Ratio

Let triangle ABC be a right triangle with acute angle A. The tangent of angle A(written as tan A) is defined a follows:

tan A = length of leg opposite angle A/ length of leg adjacent to angle A = BC/AC

Sine and Cosine Ratios

Let triangle ABC be a right triangle with acute angle A. The sine and cosine of angle A(written as sin A and cos A) is defined a follows:

sin A = length of leg opposite angle A/ length of hypotenuse = BC/AB

cos A = length of leg adjacent angle A/ length of hypotenuse = AC/AB

Inverse Trigonometric Ratios

Let triangle ABC be a right triangle with acute angle A

Inverse Tangent

If tan A = x, then tan-1 x = measure of angle A

Inverse Trigonometric Ratios

Let triangle ABC be a right triangle with acute angle A

Inverse Sine

If tan A = y, then sin-1 y = measure of angle A

Inverse Trigonometric Ratios

Let triangle ABC be a right triangle with acute angle A

Inverse Cosine

If tan A = z, then cos-1 y = measure of angle A

Law of Cosines

If triangle ABC has sides of length a, b, and c then:

a2 = b2 + c2 - 2bc cos A

b2 = a2 + c2 - 2ac cos B

c2 = a2 + b2 - 2ab cos C

Law of Sines

If triangle ABC has sides of length a, b, and c as shown then:

sin A/a = sin B/b = sin C/c

Polygon Interior Angles Theorem

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

Measure 1 + Measure 2 ... + Measure n = (n - 2) * 180 degrees

Interior Angles of a Quadrilateral

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

Polygon Exterior Angles Theorem

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

Theorem 8.7

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

If AB is congruent to CD and BC is congruent to AD then ABCD is a parallelogram.

Theorem 8.8

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

If angle A is congruent to angle C and angle B is congruent to angle D then ABCD is a parallelogram.

Theorem 8.9

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

If BC is parallel to AD and BC is congruent to AD then ABCD is a parallelogram.

Theorem 8.10

If the diagonals of a quadrilateral are congruent and parallel then the quadrilateral is a parallelogram.

If BD and AC bisect each other, then ABCD is a parallelogram

Rhombus

A parallelogram with 4 congruent sides.

Rectangle

A parallelogram with 4 right angles.

Square

A parallelogram with 4 congruent sides and 4 right angles

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has 4 congruent sides

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has 4 right angles.

Square Corollary

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

Theorem 8.11

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

Theorem 8.12

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

Theorem 8.13

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

Theorem 8.14

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

Theorem 8.15

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

Theorem 8.16

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

Midsegment of a Trapezoid

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

½(AB + CD)

Theorem 8.18

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

Theorem 8.19

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

Theorem 10.1

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 an endpoint on the circle.

Theorem 10.2

Tangent segments from a common external point are congruent.

Central Angle

An angle whose vertex is at the center of the circle.

Minor Arc

Formed when the central angle is less than 180°.

Named by its two endpoints, using the arc symbol, AB(with a sort of downward facing parentheses above it).

Major Arc

Formed when the central angle is greater than 180°.

Named by using three letters, the first and last are endpoints, ADB(with a sort of downward facing parentheses above it).

Semicircle

Formed when the central angle is exactly 180°.

Named using three letters.

Postulate 23 - Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs.

Theorem 10.3

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

Theorem 10.4

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

Theorem 10.5

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

Measure of an Inscribed Angle Theorem

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

Theorem 10.8

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

Theorem 10.9

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 angle.

Theorem 10.10

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

Theorem 10.11

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.

Theorem 10.12 - Angles Inside the Circle Theorem

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

Theorem 10.13 - Angles Outside the Circle Theorem

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.

Theorem 10.14 - Segments of Chords Theorem

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.

Theorem 10.15 - Segments of Secants Theorem

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

Theorem 10.16 - Segments of Secants and Tangents Theorem

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.

Standard Equation of a Circle

(x - h)2 + (y - k)2 = r2

Arc Length Corollary

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°.

central angle/360° = arc length/2πr

Theorem 11.3 - Area of a Sector

The ratio of the area of a sector of a circle to the area of the whole circle (πr2) is equal to the ratio of the measure of the intercepted arc to 360°.

Area of sector/ πr2 = measure of intercepted arc/360

OR

Area of sector = measure of intercepted arc/360 times πr2