## Geometry – Theorems

*January 7, 2008 at 12:26 am* *
31 comments *

Theorems

2.1 Properties of Segment Congruence Segment

congruence is reflexive, symmetric, and transitive.

2.2 Properties of Angle Congruence Angle

congruence is reflexive, symmetric, and transitive.

Reflexive: For any angle A, ™A £™A.

Symmetric: If ™A £ ™B, then ™B £™A.

Transitive: If ™A £ ™B and ™B £™C, then

™A £™C. (p. 109)

2.3 Right Angle Congruence Theorem All right

angles are congruent. (p. 110)

2.4 Congruent Supplements Theorem If two angles

are supplementary to the same angle (or to

congruent angles) then they are congruent. (p. 111)

2.5 Congruent Complements Theorem If two angles

are complementary to the same angle (or to

congruent angles) then the two angles are congruent.

(p. 111)

2.6 Vertical Angles Theorem Vertical angles are

congruent. (p. 112)

3.1 If two lines intersect to form a linear pair of

congruent angles, then the lines are perpendicular.

(p. 137)

3.2 If two sides of two adjacent acute angles are

perpendicular, then the angles are complementary.

(p. 137)

3.3 If two lines are perpendicular, then they intersect to

form four right angles. (p. 137)

3.4 Alternate Interior Angles If two parallel lines are

cut by a transversal, then the pairs of alternate

interior angles are congruent. (p. 143)

3.5 Consecutive Interior Angles If two parallel lines

are cut by a transversal, then the pairs of

consecutive interior angles are supplementary.

(p. 143)

3.6 Alternate Exterior Angles If two parallel lines are

cut by a transversal, then the pairs of alternate

exterior angles are congruent. (p. 143)

3.7 Perpendicular Transversal If a transversal is

perpendicular to one of two parallel lines, then it is

perpendicular to the other. (p. 143)

3.8 Alternate Interior Angles Converse If two lines

are cut by a transversal so that alternate interior

angles are congruent, then the lines are parallel.

(p. 150)

3.9 Consecutive Interior Angles Converse If two

lines are cut by a transversal so that consecutive

interior angles are supplementary, then the lines are

parallel. (p. 150)

3.10 Alternate Exterior Angles Converse If two lines

are cut by a transversal so that alternate exterior

angles are congruent, then the lines are parallel.

(p. 150)

3.11 If two lines are parallel to the same line, then they

are parallel to each other. (p. 157)

3.12 In a plane, if two lines are perpendicular to the same

line, then they are parallel to each other. (p. 157)

4.1 Triangle Sum Theorem The sum of the measures

of the interior angles of a triangle is 180°. (p. 196)

Corollary The acute angles of a right triangle are

complementary. (p. 197)

4.2 Exterior Angle Theorem The measure of an

exterior angle of a triangle is equal to the sum of the

measures of the two nonadjacent interior angles.

(p. 197)

4.3 Third Angles Theorem If two angles of one

triangle are congruent to two angles of another

triangle, then the third angles are also congruent.

(p. 203)

4.4 Reflexive Property of Congruent Triangles

Every triangle is congruent to itself.

Symmetric Property of Congruent Triangles

If ¤ABC £¤DEF, then ¤DEF £¤ABC.

Transitive Property of Congruent Triangles

If ¤ABC £¤DEF and ¤DEF £¤JKL, then

¤ABC £¤JKL. (p. 205)

4.5 Angle-Angle-Side (AAS) Congruence Theorem If

two angles and a nonincluded side of one triangle

are congruent to two angles and the corresponding

nonincluded side of a second triangle, then the two

triangles are congruent. (p. 220)

4.6 Base Angles Theorem If two sides of a triangle are

congruent, then the angles opposite them are

congruent. (p. 236)

Corollary If a triangle is equilateral, then it is

equiangular. (p. 237)

4.7 Converse of the Base Angles Theorem If two

angles of a triangle are congruent, then the sides

opposite them are congruent. (p. 236)

Corollary If a triangle is equiangular, then it is

equilateral. (p. 237)

4.8 Hypotenuse-Leg (HL) Congruence Theorem If

the hypotenuse and a leg of a right triangle are

congruent to the hypotenuse and a leg of a second

right triangle, then the two triangles are congruent.

(p. 238)

5.1 Perpendicular Bisector Theorem If a point is on a

perpendicular bisector of a segment, then it is

equidistant from the endpoints of the segment.

(p. 265)

5.2 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. (p. 265)

5.3 Angle Bisector Theorem If a point is on the

bisector of an angle, then it is equidistant from the

two sides of the angle. (p. 266)

5.4 Converse of the Angle Bisector Theorem If a

point is in the interior of an angle and is equidistant

from the sides of the angle, then it lies on the

bisector of the angle. (p. 266)

5.5 Concurrency of Perpendicular Bisectors of a

Triangle The perpendicular bisectors of a triangle

intersect at a point that is equidistant from the

vertices of the triangle. (p. 273)

5.6 Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point

that is equidistant from the sides of the triangle.

(p. 274)

5.7 Concurrency of Medians of a Triangle The

medians of a triangle intersect at a point that is two

thirds of the distance from each vertex to the

midpoint of the opposite side. (p. 279)

5.8 Concurrency of Altitudes of a Triangle The lines

containing the altitudes of a triangle are concurrent.

(p. 281)

5.9 Midsegment Theorem The segment connecting the

midpoints of two sides of a triangle is parallel to the

third side and is half as long. (p. 288)

5.10 If one side of a triangle is longer than another side,

then the angle opposite the longer side is larger than

the angle opposite the shorter side. (p. 295)

5.11 If one angle of a triangle is larger than another

angle, then the side opposite the larger angle is

longer than the side opposite the smaller angle.

(p. 295)

5.12 Exterior Angle Inequality The measure of an

exterior angle of a triangle is greater than the

measure of either of the two nonadjacent interior

angles. (p. 296)

5.13 Triangle Inequality The sum of the lengths of any

two sides of a triangle is greater than the length of

the third side. (p. 297)

5.14 Hinge Theorem If two sides of one triangle are

congruent to two sides of another triangle, and the

included angle of the first is larger than the included

angle of the second, then the third side of the first is

longer than the third side of the second. (p. 303)

5.15 Converse of the Hinge Theorem If two sides of

one triangle are congruent to two sides of another

triangle, and the third side of the first is longer than

the third side of the second, then the included angle

of the first is larger than the included angle of the

second. (p. 303)

6.1 Interior Angles of a Quadrilateral The sum of th

measures of the interior angles of a quadrilateral is

360°. (p. 324)

6.2 If a quadrilateral is a parallelogram, then its

opposite sides are congruent. (p. 330)

6.3 If a quadrilateral is a parallelogram, then its

opposite angles are congruent. (p. 330)

6.4 If a quadrilateral is a parallelogram, then its

consecutive angles are supplementary. (p. 330)

6.5 If a quadrilateral is a parallelogram, then its

diagonals bisect each other. (p. 330)

6.6 If both pairs of opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

(p. 338)

6.7 If both pairs of opposite angles of a quadrilateral ar

congruent, then the quadrilateral is a parallelogram.

(p. 338)

6.8 If an angle of a quadrilateral is supplementary to

both of its consecutive angles, then the quadrilatera

is a parallelogram. (p. 338)

6.9 If the diagonals of a quadrilateral bisect each other,

then the quadrilateral is a parallelogram. (p. 338)

6.10 If one pair of opposite sides of a quadrilateral are

congruent and parallel, then the quadrilateral is a

parallelogram. (p. 340)

Rhombus Corollary A quadrilateral is a rhombus

if and only if it has four congruent sides. (p. 348)

Rectangle Corollary A quadrilateral is a rectangle

if and only if it has four right angles. (p. 348)

Square Corollary A quadrilateral is a square if and

only if it is a rhombus and a rectangle. (p. 348)

6.11 A parallelogram is a rhombus if and only if its

diagonals are perpendicular. (p. 349)

6.12 A parallelogram is a rhombus if and only if each

diagonal bisects a pair of opposite angles. (p. 349)

6.13 A parallelogram is a rectangle if and only if its

diagonals are congruent. (p. 349)

6.14 If a trapezoid is isosceles, then each pair of base

angles is congruent. (p. 356)

6.15 If a trapezoid has a pair of congruent base angles,

then it is an isosceles trapezoid. (p. 356)

6.16 A trapezoid is isosceles if and only if its diagonals

are congruent. (p. 356)

6.17 Midsegment Theorem for Trapezoids The

midsegment of a trapezoid is parallel to each base

and its length is one half the sum of the lengths of

the bases. (p. 357)

6.18 If a quadrilateral is a kite, then its diagonals are

perpendicular. (p. 358)

6.19 If a quadrilateral is a kite, then exactly one pair of

opposite angles are congruent. (p. 358)

6.20 Area of a Rectangle The area of a rectangle is the

product of its base and height. A = bh (p. 372)

6.21 Area of a Parallelogram The area of a

parallelogram is the product of a base and its

corresponding height. A = bh (p. 372)

6.22 Area of a Triangle The area of a triangle is one

half the product of a base and its corresponding

height.

A = 1

2

bh (p. 372)

6.23 Area of a Trapezoid The area of a trapezoid is one

half the product of the height and the sum of the

bases.

A = 1

2

h(b1 + b2 ) (p. 374)

6.24 Area of a Kite The area of a kite is one half the

product of the lengths of its diagonals. A = 1

2

d1d2

(p. 374)

6.25 Area of a Rhombus The area of a rhombus is equal

to one half the product of the lengths of the

diagonals. A = 1

2

d1d2 (p. 374)

7.1 Reflection Theorem A reflection is an isometry.

(p. 404)

7.2 Rotation Theorem A rotation is an isometry.

(p. 412)

7.3 If lines k and m intersect at point P, then a reflection

in k followed by a reflection in m is a rotation about

point P. The angle of rotation is 2x°, where x° is the

measure of the acute or right angle formed by k and

m. (p. 414)

7.4 Translation Theorem A translation is an isometry.

(p. 421)

7.5 If lines k and m are parallel, then a reflection in line

k followed by a reflection in line m is a translation.

If Pﬂ is the image of P, then the following is true:

(1) PPﬂ

¯ ˘is perpendicular to k and m. (2) PPﬂ=2d,

where d is the distance between k and m. (p. 421)

7.6 Composition Theorem The composition of two

(or more) isometries is an isometry. (p. 431)

8.1 If two polygons are similar, then the ratio of their

perimeters is equal to the ratios of their

corresponding side lengths. (p. 475)

8.2 Side-Side-Side (SSS) Similarity Theorem If the

lengths of the corresponding sides of two triangles

are proportional, then the triangles are similar.

(p. 488)

8.3 Side-Angle-Side (SAS) Similarity Theorem 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. (p. 488)

8.4 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. (p. 498)

8.5 Converse of the Triangle Proportionality

Theorem If a line divides two sides of a triangle

proportionally, then it is parallel to the third side.

(p. 498)

8.6 If three parallel lines intersect two transversals, then

they divide the transversals proportionally. (p. 499)

8.7 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.

(p. 499)

9.1 If an 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. (p. 527)

9.2 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. (p. 529)

9.3 In a right triangle, the altitude from the right angle

to the hypotenuse divides the hypotenuse into two

segments. Each leg of the right triangle is the

geometric mean of the hypotenuse and the segment

of the hypotenuse that is adjacent to the leg.

(p. 529)

9.4 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.

(p. 535)

9.5 Converse of the 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. (p. 543)

9.6 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. (p. 544)

9.7 If the square of the length of the longest side of a

triangle is greater than the sum of the squares of the

length of the other two sides, then the triangle is

obtuse. (p. 544)

9.8 45°-45°-90° Triangle Theorem In a 45°-45°-90°

triangle, the hypotenuse is 2 times as long as each

leg. (p. 551)

9.9 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 3 times as long

as the shorter leg. (p. 551)

10.1 If a line is tangent to a circle, then it is

perpendicular to the radius drawn to the point of

tangency. (p. 597)

10.2 In a plane, if a line is perpendicular to a radius of a

circle at its endpoint on the circle, then the line is

tangent to the circle. (p. 597)

10.3 If two segments from the same exterior point are

tangent to a circle, then they are congruent. (p. 598)

10.4 In the same circle, or in congruent circles, two

minor arcs are congruent if and only if their

corresponding chords are congruent. (p. 605)

10.5 If a diameter of a circle is perpendicular to a chord,

then the diameter bisects the chord and its arc.

(p. 605)

10.6 If one chord is a perpendicular bisector of another

chord, then the first chord is a diameter. (p. 605)

10.7 In the same circle or in congruent circles, two

chords are congruent if and only if they are

equidistant from the center. (p. 606)

10.8 If an angle is inscribed in a circle, then its measure

is half the measure of its intercepted arc. (p. 613)

10.9 If two inscribed angles of a circle intercept the same

arc, then the angles are congruent. (p. 614)

10.10 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.

(p. 615)

10.11 A quadrilateral can be inscribed in a circle if and

only if its opposite angles are supplementary.

(p. 615)

10.12 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. (p. 621)

10.13 If two chords intersect in the interior of 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. (p. 622)

10.14 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 one half the

difference of the measures of the intercepted arcs.

(p. 622)

10.15 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. (p. 629)

10.16 If two secant segments share the same endpoint

outside a circle, then the product of the length of

one secant segment and the length of its external

segment equals the product of the length of the other

secant segment and the length of its external

segment. (p. 630)

10.17 If a secant segment and a tangent segment share an

endpoint outside a circle, then the product of the

length of the secant segment and the length of its

external segment equals the square of the length of

the tangent segment. (p. 630)

Entry filed under: Geometry, Personal Notes. Tags: .

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