2.1 Properties of Segment 2.1 Properties of Segment Congruence: Reflexive, Congruence: Reflexive, symmetric, and transitivesymmetric, and transitiveReflexive example:Reflexive example:AB = ABAB = AB

Symmetric example:Symmetric example:AB = BAAB = BA

Transitive example:Transitive example:

AB = CD; CD= BC therefore AB = CD; CD= BC therefore

AB = BCAB = BC

2.2 Properties of Angle 2.2 Properties of Angle Congruence: Angle congruence is Congruence: Angle congruence is reflexive, symmetric, and transitive.reflexive, symmetric, and transitive. Reflexive example:Reflexive example:

m<A = m< Am<A = m< A

Symmetric example:Symmetric example:m<A = m< B; then m<B = m<Am<A = m< B; then m<B = m<A

Transitive example:Transitive example: m<A = m<B; m<B = m<C; thereforem<A = m<B; m<B = m<C; therefore m<A = m<Cm<A = m<C

2.3 Right Angle Congruence 2.3 Right Angle Congruence Theorem: All right angles are Theorem: All right angles are

congruent congruent

Ex:Ex:

2.4 Congruent Supplements 2.4 Congruent Supplements Theorem: If 2 angles are Theorem: If 2 angles are

supplementary to the same supplementary to the same angle (or to congruent angles) angle (or to congruent angles)

then they are congruent.then they are congruent.

EX: EX:

m<1 + m<2 = 180m<1 + m<2 = 180

m<2 + m<3 =180m<2 + m<3 =180

Therefore… m<1= m<3Therefore… m<1= m<3

2.5 Congruent Complements Theorem: 2.5 Congruent Complements Theorem: If 2 angles are complementary to the If 2 angles are complementary to the same angle (or to congruent angles) same angle (or to congruent angles)

then the 2 angles are congruent.then the 2 angles are congruent.EX: EX:

m<4 + m<5 = 90m<4 + m<5 = 90

M<5 + m<6 = 90 M<5 + m<6 = 90

Therefore…Therefore…

m<4 = m< 6m<4 = m< 6

2.6 Vertical angles theorem: 2.6 Vertical angles theorem: Vertical angles are Vertical angles are

congruent.congruent.

EX:EX:

m<1 = m<3m<1 = m<3

M<2 = m<4M<2 = m<4

3.13.1 If 2 lines intersect to form a If 2 lines intersect to form a linear pair of congruent angles, linear pair of congruent angles,

then the lines are perpendicular.then the lines are perpendicular.

EX:EX:

3.2 3.2 If 2 sides of 2 adjacent acute If 2 sides of 2 adjacent acute angles are perpendicular, then angles are perpendicular, then

they are complementary.they are complementary.

EX:EX:

3.3 3.3 If 2 sides of 2 adjacent acute If 2 sides of 2 adjacent acute angles are perpendicular, then angles are perpendicular, then they intersect to form 4 right they intersect to form 4 right

angles.angles.

EX:EX:

3.4 Alternate Interior Angles: If 2 3.4 Alternate Interior Angles: If 2 parallel lines are cut by a parallel lines are cut by a

transversal, then the pairs of transversal, then the pairs of alternate interior angles are alternate interior angles are

congruent.congruent.

EX:EX:

m<3 = m< 5m<3 = m< 5

m<4 = m< 6m<4 = m< 6

3.5 Consecutive Interior Angles: if 2 3.5 Consecutive Interior Angles: if 2 parallel lines are cut by a parallel lines are cut by a

transversal, the n the pairs of transversal, the n the pairs of consecutive interior angles are consecutive interior angles are

supplementarysupplementary..

EX:EX:m<3 + m<6 =180m<3 + m<6 =180m<4 + m<5 =180m<4 + m<5 =180

3.6 3.6 Alternate Exterior Angles: If 2 Alternate Exterior Angles: If 2 parallel lines are cut by a parallel lines are cut by a

transversal, then the pairs of transversal, then the pairs of alternate exterior angles are alternate exterior angles are

congruent.congruent.

EX:EX:

m<1 = m< 7m<1 = m< 7

m<2 = m<8m<2 = m<8

3.7 3.7 Perpendicular Transversal: If a Perpendicular Transversal: If a transversal is perpendicular to 1 transversal is perpendicular to 1

of 2 parallel lines, then it is of 2 parallel lines, then it is perpendicular to the others.perpendicular to the others.

EX:EX:

3.8 3.8 Alternate Interior Angles Alternate Interior Angles Converse: If 2 lines are cut by a Converse: If 2 lines are cut by a transversal so that the alternate transversal so that the alternate

interior angles are congruent, then interior angles are congruent, then the lines are parallel the lines are parallel

EX: EX:

If m<3 = m<5 orIf m<3 = m<5 or

m<4 = m<6 m<4 = m<6

then r ll sthen r ll s

3.9 3.9 Consecutive Interior Angles Consecutive Interior Angles Converse: If 2 lines are cut by Converse: If 2 lines are cut by

a transversal so that the a transversal so that the consecutive interior angles are consecutive interior angles are supplementary, then the lines supplementary, then the lines

are parallel.are parallel.EX:EX:

m<3 + m<6 = 180m<3 + m<6 = 180

m<4 + m<5 = 180m<4 + m<5 = 180

Then r ll sThen r ll s

3.10 Alternate Exterior Angles 3.10 Alternate Exterior Angles Converse: If 2 lines are cut by a Converse: If 2 lines are cut by a transversal so that the alternate transversal so that the alternate

exterior angles are supplementary, exterior angles are supplementary, then the lines are parallel then the lines are parallel

EX:EX:m<1 = m<7 orm<1 = m<7 orm<2 = m< 8 m<2 = m< 8 then r ll sthen r ll s

3.11 3.11 If 2 lines are parallel to the If 2 lines are parallel to the same line, then they are parallel same line, then they are parallel

to each other. to each other.

EX:EX:If p ll q andIf p ll q andq ll rq ll rThen p ll rThen p ll r

3.12 In a plane, if 2 lines are 3.12 In a plane, if 2 lines are perpendicular to the same perpendicular to the same

line, then they are parallel to line, then they are parallel to each other.each other.

EX:EX:

4.1 Triangle Sum Theorem: the 4.1 Triangle Sum Theorem: the sum of the measures of the sum of the measures of the

interior angles of a triangle is interior angles of a triangle is 180 degrees.180 degrees.

EX: EX:

m<A + m<B + m<C = 180m<A + m<B + m<C = 180

4.2 Exterior Angle Theorem: The 4.2 Exterior Angle Theorem: The measure of an exterior angle of measure of an exterior angle of a triangle is equal to the sum of a triangle is equal to the sum of

the measures of the 2 non the measures of the 2 non adjacent interior angles.adjacent interior angles.

EX: m<1 = m<A + m<BEX: m<1 = m<A + m<B

4.3 Third Angle Theorem: If 2 4.3 Third Angle Theorem: If 2 angles of one triangle are angles of one triangle are

congruent to the 2 angles of congruent to the 2 angles of another triangle, then the third another triangle, then the third

angles are also congruent. angles are also congruent.

EX:EX:If m<A = m<D and A DIf m<A = m<D and A Dm<B = m<E thenm<B = m<E thenm<C = m<F B C E Fm<C = m<F B C E F

4.4 Reflexive Property of Congruent 4.4 Reflexive Property of Congruent Triangles: Every triangle is congruent to Triangles: Every triangle is congruent to

itselfitselfSymmetric Property of congruent triangles Symmetric Property of congruent triangles

examplesexamplesTransitive property of congruent triangles Transitive property of congruent triangles

examplesexamplesEX: EX: ABC = ABCABC = ABC ABC = DEF, then DEF = ABCABC = DEF, then DEF = ABC ABC = DEF, and DEF= JKL, thenABC = DEF, and DEF= JKL, then ABC = JKLABC = JKL

4.5 Angle-Angle-Side (AAS): 4.5 Angle-Angle-Side (AAS): Congruence Theorem: If 2 angles Congruence Theorem: If 2 angles

and a nonincluded side of 1 and a nonincluded side of 1 triangle are congruent to 2 angles triangle are congruent to 2 angles

and the corresponding nonincluded and the corresponding nonincluded side of a second triangle, then the side of a second triangle, then the

2 triangles are congruent.2 triangles are congruent. Angle m<A = m<D B EAngle m<A = m<D B E Angle m<B = m< EAngle m<B = m< E Side BC = EF A C D F Side BC = EF A C D F

4.6 Base Angles Theorem: If 2 4.6 Base Angles Theorem: If 2 sides of a triangle are sides of a triangle are

congruent, then the angles congruent, then the angles opposite them are congruent.opposite them are congruent.

If AB = CB, then m<A = m<CIf AB = CB, then m<A = m<C BB

A CA C

4.7 Converse of the Base Angles 4.7 Converse of the Base Angles Theorem: If 2 angles of a Theorem: If 2 angles of a

triangle are congruent, then the triangle are congruent, then the sides opposite them are sides opposite them are

congruent.congruent.

If m<A = m< C, then AB = CBIf m<A = m< C, then AB = CB BB

AA CC

4.8 Hypotenuse-Leg (HL) congruence 4.8 Hypotenuse-Leg (HL) congruence theorem: If the hypotenuse and a leg of theorem: If the hypotenuse and a leg of

a right triangle are congruent to the a right triangle are congruent to the hypotenuse and a leg of a second hypotenuse and a leg of a second triangle, then the 2 triangles are triangle, then the 2 triangles are

congruentcongruent..

If BC = EF, and AC = DF, then If BC = EF, and AC = DF, then

ABC = DEF A DABC = DEF A D

***2 RIGHT TRIANGLES******2 RIGHT TRIANGLES***

B C E FB C E F

5.1 Perpendicular Bisector 5.1 Perpendicular Bisector Theorem: If a point is on a Theorem: If a point is on a perpendicular bisector of a perpendicular bisector of a

segment, then it is equidistant from segment, then it is equidistant from the end points of the segment.the end points of the segment.

If CP is a perpendicular bisector of AB, If CP is a perpendicular bisector of AB, then CA = CBthen CA = CB

5.2 Converse of the Perpendicular 5.2 Converse of the Perpendicular Bisector Theorem: If a point is Bisector Theorem: If a point is

equidistant from the endpoints of a equidistant from the endpoints of a segment, then it is on the segment, then it is on the

perpendicular bisector of the perpendicular bisector of the segment.segment.

If DA = DB, then D lies on the If DA = DB, then D lies on the perpendicular bisector of AB.perpendicular bisector of AB.

5.3 Angle Bisector Theorem: If a 5.3 Angle Bisector Theorem: If a point is on the bisector of an point is on the bisector of an

angle, then it is equidistant from angle, then it is equidistant from the two sides of the angle.the two sides of the angle.

If m<BAD = m<CAD, then DB = DCIf m<BAD = m<CAD, then DB = DC

5.4 Converse of the Angle Bisector 5.4 Converse of the Angle Bisector Theorem: If a point is in the interior Theorem: If a point is in the interior of an angle and is equidistant from of an angle and is equidistant from the sides of the angle, then it lies the sides of the angle, then it lies

on the bisector of the angle. on the bisector of the angle.

If DB = DC, then m<BAD = m<CADIf DB = DC, then m<BAD = m<CAD

5.5 Concurrency if the 5.5 Concurrency if the Perpendicular Bisectors of a Perpendicular Bisectors of a Triangle: The perpendicular Triangle: The perpendicular

bisectors of a triangle intersect at a bisectors of a triangle intersect at a point that is equidistant from the point that is equidistant from the

vertices of the triangle. vertices of the triangle. PA = PB = PC BPA = PB = PC B

A CA C

5.6 Concurrency of Angle Bisectors 5.6 Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a Triangle: The angle bisectors

of a triangle intersect at a point that of a triangle intersect at a point that is equidistant from the sides of the is equidistant from the sides of the

triangle. triangle.

PD = PE = PF B DPD = PE = PF B D FF A P A P EE CC

5.7 Concurrency of the Medians of 5.7 Concurrency of the Medians of a Triangle: The medians of a a Triangle: The medians of a

triangle intersect at a point that is triangle intersect at a point that is two thirds of the distance from each two thirds of the distance from each

vertex to the midpoint of the vertex to the midpoint of the opposite side opposite side

If P is the centroid of triangle ABC, then If P is the centroid of triangle ABC, then AP= 2/3 AD, BP= 2/3 BF,and CE= 2/3 CPAP= 2/3 AD, BP= 2/3 BF,and CE= 2/3 CP

5.8 Concurrency of Altitudes of 5.8 Concurrency of Altitudes of a Triangle: The lies containing a Triangle: The lies containing the altitudes of a triangle are the altitudes of a triangle are

congruent. congruent. If AE, BF, and CF are the altitudes of If AE, BF, and CF are the altitudes of

triangle ABC, then the lines AE, BF, and triangle ABC, then the lines AE, BF, and CD intersect at some point. CD intersect at some point.

5.9 Midsegment Theorem: The 5.9 Midsegment Theorem: The segment connecting the segment connecting the

midpoints of two sides of a midpoints of two sides of a triangle is parallel to the third triangle is parallel to the third

side and is half as long.side and is half as long.

DE ll AB, and DE = ½ ABDE ll AB, and DE = ½ AB

5.10 If one side of a triangle is 5.10 If one side of a triangle is longer than another side, then longer than another side, then the angle opposite the longer the angle opposite the longer side is larger than the angle side is larger than the angle opposite the shorter side.opposite the shorter side.

m<A > m<C Bm<A > m<C B

3 53 5

A CA C

5.11 If one angle of a triangle is 5.11 If one angle of a triangle is larger than another angle, then larger than another angle, then the side opposite the larger the side opposite the larger angle is longer than the side angle is longer than the side opposite the smaller angle. opposite the smaller angle.

EF > DFEF > DF

5.12 Exterior Angle Inequality: The 5.12 Exterior Angle Inequality: The measure of an exterior angle of measure of an exterior angle of

a triangle is greater than the a triangle is greater than the measure of either of the two measure of either of the two nonadjacent interior angles.nonadjacent interior angles.

m<1 > m<A and m<1 > m<Bm<1 > m<A and m<1 > m<B

5.13 Triangle Inequality: The 5.13 Triangle Inequality: The sum of the lengths of any two sum of the lengths of any two sides of a triangle is greater sides of a triangle is greater

than the length of the third side.than the length of the third side.

AB + BC > ACAB + BC > AC A A

AC + BC > ABAC + BC > AB

AB + AC > BCAB + AC > BC

C BC B

5.14 Hinge Theorem: If two sides of one 5.14 Hinge Theorem: If two sides of one triangle are congruent to two sides of triangle are congruent to two sides of

another triangle, and the included angle of another triangle, and the included angle of the first is larger than the included angle of the first is larger than the included angle of the second, then the third side of the first is the second, then the third side of the first is

longer than the third side of the secondlonger than the third side of the second

RT > VX R VRT > VX R V 100 S T 80 W X100 S T 80 W X

5.15 Converse of the Hinge Theorem: If two 5.15 Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of one triangle are congruent to two sides of another triangle, and the third side sides of another triangle, and the third side

of the first is longer than the third side of the of the first is longer than the third side of the second, then the included angle of the first is second, then the included angle of the first is larger than the included angle of the second.larger than the included angle of the second.

m<A > m<D B Em<A > m<D B E A DA D

C FC F

6.1 Interior Angles of 6.1 Interior Angles of Quadrilateral: The sum of the Quadrilateral: The sum of the

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

m<1 + m<2 + m<3 + m<4 = 360 degreesm<1 + m<2 + m<3 + m<4 = 360 degrees

6.2 If a quadrilateral is a 6.2 If a quadrilateral is a parallelogram, then its opposite parallelogram, then its opposite

sides are congruent sides are congruent

PQ = RS and SP = QRPQ = RS and SP = QR Q RQ R

P SP S

6.3 If a quadrilateral is a 6.3 If a quadrilateral is a parallelogram, then its opposite parallelogram, then its opposite

angles are congruent. angles are congruent.

m<P = m<R and m<Q = m<Sm<P = m<R and m<Q = m<S

Q RQ R

P SP S

6.4 If a quadrilateral is a 6.4 If a quadrilateral is a parallelogram, then its parallelogram, then its consecutive angles are consecutive angles are

supplementary.supplementary.m<P + m<Q = 180, m<Q + m<R = 180m<P + m<Q = 180, m<Q + m<R = 180M<R + m<S = 180, m<S + m<P = 180M<R + m<S = 180, m<S + m<P = 180

Q RQ R

P SP S

6.5 If a quadrilateral is a 6.5 If a quadrilateral is a parallelogram, then its parallelogram, then its

diagonals bisect each other diagonals bisect each other QM = SM and PM = RMQM = SM and PM = RM

Q RQ R

P SP S

6.6. 6.6. If both pairs of opposite If both pairs of opposite sides of a quadrilateral are sides of a quadrilateral are

congruent, then the congruent, then the quadrilateral is parallelogram.quadrilateral is parallelogram.

ABCD is a parallelogramABCD is a parallelogram

A BA B

C DC D

6.7 If both pairs of opposite 6.7 If both pairs of opposite angles of a quadrilateral are angles of a quadrilateral are

congruent, then the congruent, then the quadrilateral is a parallelogram.quadrilateral is a parallelogram.

ABCD is a parallelogramABCD is a parallelogram A BA B

C DC D

6.8 6.8 If an angle of a If an angle of a quadrilateral is supplementary quadrilateral is supplementary

to both of its consecutive to both of its consecutive angles, then the quadrilateral is angles, then the quadrilateral is

a parallelogram.a parallelogram.

ABCD is a parallelogramABCD is a parallelogram A BA B C DC D

6.9 6.9 If the diagonals of a If the diagonals of a quadrilateral bisect each other, quadrilateral bisect each other,

then the quadrilateral is then the quadrilateral is parallelogram.parallelogram.

ABCD is a parallelogramABCD is a parallelogram A BA B

C DC D

6.10 6.10 If one pair of opposite sides of a quadrilateral If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is are congruent and parallel, then the quadrilateral is

a parallelogram.a parallelogram.Rhombus Corollary:Rhombus Corollary: A quadrilateral is a rhombus A quadrilateral is a rhombus

if and only if it has four congruent sides.if and only if it has four congruent sides.Rectangle Rectangle Corollary: A quadrilateral is a rectangle Corollary: A quadrilateral is a rectangle

if and only if it has four right anglesif and only if it has four right angles. . Square Corollary: Square Corollary: A quadrilateral is a square if A quadrilateral is a square if

and only if it is a rhombus and a rectangle.and only if it is a rhombus and a rectangle.

ABCD is a parallelogramABCD is a parallelogram

A BA B

C DC D

6.11 A parallelogram is a 6.11 A parallelogram is a rhombus if and only if its rhombus if and only if its

diagonals are perpendicular.diagonals are perpendicular.

ABCD is a rhombus iff AC perpendicular to ABCD is a rhombus iff AC perpendicular to BD BD

B CB C A DA D

6.12 6.12 A parallelogram is a A parallelogram is a rhombus if and only if each rhombus if and only if each diagonal bisects a pair of diagonal bisects a pair of

opposite angles.opposite angles. ABCD is a rhombus iff AC bisects <DAB and <BCD and ABCD is a rhombus iff AC bisects <DAB and <BCD and

BD bisects <ADC and <CBABD bisects <ADC and <CBA

B CB C

A DA D

6.13 A parallelogram is a 6.13 A parallelogram is a rectangle if and only if its rectangle if and only if its diagonals are congruent.diagonals are congruent.

ABCD is a rectangle iff AC = BD ABCD is a rectangle iff AC = BD A BA B

C DC D

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

congruent.congruent.

m<A = m<B, m<C =m<Dm<A = m<B, m<C =m<D

A BA B

C DC D

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

is an isosceles trapezoid is an isosceles trapezoid ABCD is an isosceles trapezoidABCD is an isosceles trapezoid A BA B

C DC D

6.16 A trapezoid is isosceles if 6.16 A trapezoid is isosceles if and only if its diagonals are and only if its diagonals are

congruent.congruent.ABCD is isosceles iff AC = BDABCD is isosceles iff AC = BD

A BA B

C DC D

6.17 Midsegment Theorem for 6.17 Midsegment Theorem for Trapezoids: The midsegment of a Trapezoids: The midsegment of a trapezoid is parallel to each base trapezoid is parallel to each base

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

MN ll AD, MN ll BC, MN = ½ (AD + BC)MN ll AD, MN ll BC, MN = ½ (AD + BC)

B CB C

M NM N

A DA D

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

perpendicular.perpendicular.AC perpendicular BDAC perpendicular BD

CC

B DB D

AA

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

opposite angles are congruent.opposite angles are congruent.

m<A = m<C, m<B is not equal m<Dm<A = m<C, m<B is not equal m<D

CC

B DB D

AA

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

product of its base and height.product of its base and height.

A = bhA = bh

hh bb

6.21 Area of a Parallelogram: 6.21 Area of a Parallelogram: The area of a parallelogram is The area of a parallelogram is the product of a base and its the product of a base and its

corresponding height.corresponding height.

A = bhA = bh

hh

bb

6.22 Area of a Triangle: The 6.22 Area of a Triangle: The area of a triangle is one half area of a triangle is one half the product of a base and its the product of a base and its

corresponding height.corresponding height.

A = ½ bhA = ½ bh

hh

bb

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

half the product of the height half the product of the height and the sum of the bases. and the sum of the bases.

A = ½ h(b 1 + b 2 ) b1A = ½ h(b 1 + b 2 ) b1 hh

b2b2

6.24 Area of a Kite: the area of 6.24 Area of a Kite: the area of a kite is one half the product of a kite is one half the product of

the lengths of its diagonals.the lengths of its diagonals.

A = ½ d1 *d2A = ½ d1 *d2

d1d1

d2d2

6.25 Area of a Rhombus: the 6.25 Area of a Rhombus: the area of a rhombus is equal to area of a rhombus is equal to

one half the product of the one half the product of the lengths of diagonals. lengths of diagonals.

A = ½ d1 *d2A = ½ d1 *d2

d1d1

d2d2

7.1 Reflection Theorem: A 7.1 Reflection Theorem: A reflection is an isometry reflection is an isometry

Isometry is a transformation that preserves Isometry is a transformation that preserves lengthslengths

line of reflectionline of reflection

7.2 Rotation Theorem: A 7.2 Rotation Theorem: A rotation is an isomentry.rotation is an isomentry.

Point of rotationPoint of rotation

180 degree 180 degree

rotation rotation

7.3 Rotational theorem using 7.3 Rotational theorem using lines and coordinate planes lines and coordinate planes

Rotational pointRotational point

7.4 Translation Theorem: A 7.4 Translation Theorem: A translation is an isometry. translation is an isometry.

P’P’ PP Q’Q’

QQ

7.5 Reflections7.5 Reflections

k mk m

Q Q’ Q’’Q Q’ Q’’

P P’ P”P P’ P”

7.6 Composition Theorem: The 7.6 Composition Theorem: The composition of two (or more) composition of two (or more)

isometrics is an isometry isometrics is an isometry

When 2 or more transformations are When 2 or more transformations are combined to produce a single combined to produce a single transformation transformation

EX: reflection then translationEX: reflection then translation

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

is equal to the ratios of their is equal to the ratios of their corresponding side lengths.corresponding side lengths.

Ratio perimeter = ratio of sidesRatio perimeter = ratio of sides

K L P QK L P Q

N M S RN M S R

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

corresponding sides of two corresponding sides of two triangles are proportional, then triangles are proportional, then

the triangles are similar.the triangles are similar. IF IF ABAB = = BCBC = = CACA

PQ QR RP’PQ QR RP’

Then ABC ~ PQRThen ABC ~ PQR

8.3 Side-Angle-Side (SAS) Similarity 8.3 Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is Theorem: If an angle of one triangle is

congruent to an angle of a second congruent to an angle of a second triangle and the lengths of the sides triangle and the lengths of the sides

including these angles are proportional, including these angles are proportional, then the triangles are similarthen the triangles are similar. .

If m<X = m<M and If m<X = m<M and ZXZX = = XYXY

PM MN’PM MN’

Then XYZ ~ MNPThen XYZ ~ MNP

8.4 Triangle Proportionally 8.4 Triangle Proportionally Theorem: If a line parallel to one Theorem: If a line parallel to one side of a triangle intersects the side of a triangle intersects the other two sides, then it divides other two sides, then it divides the two sides proportionally.the two sides proportionally.

If TU ll QS, then If TU ll QS, then RTRT = = RURU

TQ USTQ US

8.5 Converse of the Triangle 8.5 Converse of the Triangle Proportionally Theorem: If a line Proportionally Theorem: If a line

divides two sides of a triangle divides two sides of a triangle proportionally, then it is parallel proportionally, then it is parallel

to the third side.to the third side.

IF IF RTRT = = RURU, then TU ll QS, then TU ll QS

TQ USTQ US

8.6 If three parallel lines 8.6 If three parallel lines intersect two transversals, then intersect two transversals, then

they divide the transversals they divide the transversals proportionally.proportionally.

If r ll s and s ll t, and l and m intersect r, s, If r ll s and s ll t, and l and m intersect r, s, and t, then and t, then UWUW = = VXVX

WY XZWY XZ

8.7 If a ray bisects an angle of a 8.7 If a ray bisects an angle of a triangle, then it divides the opposite triangle, then it divides the opposite side into segments whose lengths side into segments whose lengths are proportional to the lengths of are proportional to the lengths of

the other two sides.the other two sides.

If CD bisects <ACB, thenIf CD bisects <ACB, then

9.1 If an altitude is drawn to the 9.1 If an altitude is drawn to the hypotenuse of a right triangle, hypotenuse of a right triangle, then the two triangles formed then the two triangles formed

are similar to the original are similar to the original triangle and to each other.triangle and to each other.

9.2 In a right triangle, the altitude from the 9.2 In a right triangle, the altitude from the right angle to the hypotenuse divides the right angle to the hypotenuse divides the hypotenuse divides they hypotenuse into hypotenuse divides they hypotenuse into

two segments. The length of the altitude is two segments. The length of the altitude is the geometric mean of the lengths of the two the geometric mean of the lengths of the two

segmentssegments. .

9.3 In a right triangle, the altitude from the 9.3 In a right triangle, the altitude from the right angle to the hypotenuse divides the right angle to the hypotenuse divides the

hypotenuse into two segments. Each leg of hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the right triangle is the geometric mean of the hypotenuse and the segment of the the hypotenuse and the segment of the

hypotenuse that is adjacent to the leg.hypotenuse that is adjacent to the leg.

9.4 Pythagorean Theorem: In a 9.4 Pythagorean Theorem: In a right triangle, the square of the right triangle, the square of the

length of the hypotenuse is length of the hypotenuse is equal to the sum of the squares equal to the sum of the squares

of the lengths of the legs. of the lengths of the legs.

9.5 Converse of the Pythagorean 9.5 Converse of the Pythagorean Theorem: If the square of the length of Theorem: If the square of the length of the longest sides a triangle is equal to the longest sides a triangle is equal to the sum of the sum of the squares of the sum of the sum of the squares of

the lengths of the other two sides, then the lengths of the other two sides, then the triangle is a right triangle.the triangle is a right triangle.

9.6 If the square of the length of the 9.6 If the square of the length of the longest side of triangle is less than longest side of triangle is less than

the sum of the squares of the the sum of the squares of the lengths of the other two sides, then lengths of the other two sides, then

the triangle is acute.the triangle is acute.

9.7 If the square of the length of the 9.7 If the square of the length of the longest side of a triangle is greater longest side of a triangle is greater than the sum of the squares of the than the sum of the squares of the length of the other two sides, then length of the other two sides, then

the triangle is obtuse.the triangle is obtuse.

9.8 45° -45 ° -90° Triangle 9.8 45° -45 ° -90° Triangle Theorem: In a 45° -45 ° -90° Theorem: In a 45° -45 ° -90° triangle, the hypotenuse is √2triangle, the hypotenuse is √2

times as long as each leg.times as long as each leg.

9.9 30° -60° -90° Triangle 9.9 30° -60° -90° Triangle Theorem: In a 30° -60° -90° Theorem: In a 30° -60° -90°

triangle, the hypotenuse is twice as triangle, the hypotenuse is twice as long as the shorter leg, and the long as the shorter leg, and the

longer leg √3 times as long as the longer leg √3 times as long as the shorter leg. shorter leg.

10.1 If a line is tangent to a 10.1 If a line is tangent to a circle, then it is perpendicular to circle, then it is perpendicular to the radius drawn to the point of the radius drawn to the point of

tangency.tangency.

10.2 In a plane, if a line is 10.2 In a plane, if a line is perpendicular to a radius of a perpendicular to a radius of a circle at its endpoint on the circle at its endpoint on the

circle, then the line is tangent to circle, then the line is tangent to the circle.the circle.

10.3 If two segments from the 10.3 If two segments from the same exterior point are tangent same exterior point are tangent

to a circle, then they are to a circle, then they are congruent.congruent.

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

arcs are congruent if and only if arcs are congruent if and only if their corresponding chords are their corresponding chords are

congruent.congruent.

10.5 If a diameter of a circle is 10.5 If a diameter of a circle is perpendicular to a chord, then perpendicular to a chord, then the diameter bisects the chord the diameter bisects the chord

and its arc.and its arc.

10.6 If one chord is a 10.6 If one chord is a perpendicular bisector of perpendicular bisector of

another chord, then the first another chord, then the first chord is a diameter. chord is a diameter.

10.7 In the same circle or in 10.7 In the same circle or in congruent circles, two chords congruent circles, two chords

are congruent if and only if they are congruent if and only if they are equidistant from the center.are equidistant from the center.

10.8 If an angle is inscribed in a 10.8 If an angle is inscribed in a circle, then its measure is half circle, then its measure is half the measure of its intercepted the measure of its intercepted

arc.arc.

10.9 If two inscribed angles of a 10.9 If two inscribed angles of a circle intercept the same arc, circle intercept the same arc,

then the angles are congruent.then the angles are congruent.

10.10 If a right triangle is inscribed in a 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of circle, then the hypotenuse is a diameter of the circle. Conversely, if one of an inscribed the circle. Conversely, if one of an inscribed triangle is a diameter of the circle, then the triangle is a diameter of the circle, then the

triangle is right triangle and the angle triangle is right triangle and the angle

opposite the diameter is the right angle.opposite the diameter is the right angle.

10.11 A quadrilateral can be 10.11 A quadrilateral can be inscribed in a circle if and only if inscribed in a circle if and only if

its opposite angles are its opposite angles are supplementary.supplementary.

10.12 If a tangent and a chord 10.12 If a tangent and a chord intersect at a point on a circle, intersect at a point on a circle,

then the measure of each angle then the measure of each angle formed is one half the measure formed is one half the measure

of its intercepted arc.of its intercepted arc.

10.13 If two chords intersect in the 10.13 If two chords intersect in the interior of a circle, then the interior of a circle, then the

measure of each angle is one half measure of each angle is one half the sum of the measures of the the sum of the measures of the

arcs intercepted by the angle and arcs intercepted by the angle and its vertical angle.its vertical angle.

10.14 If a tangent and a secant, two 10.14 If a tangent and a secant, two tangents, or two secants intersect in tangents, or two secants intersect in

the exterior of a circle, then the the exterior of a circle, then the measure of the angle formed is one measure of the angle formed is one

half the difference of the measures of half the difference of the measures of the intercepted arcs.the intercepted arcs.

10.15 If two chords intersect in the 10.15 If two chords intersect in the interior of a circle, then the product of interior of a circle, then the product of

the lengths of the segments of one the lengths of the segments of one chord is equal to the product of the chord is equal to the product of the

lengths of the segments of the other lengths of the segments of the other chord.chord.

10.16 If two secant segments share the 10.16 If two secant segments share the same endpoint outside a circle, then the same endpoint outside a circle, then the

product of the length of one secant segment product of the length of one secant segment and the length of its external segments and the length of its external segments

equals the product of the length of the other equals the product of the length of the other secant segment and the length of its secant segment and the length of its

external segment.external segment.

10.17 If a secant segment and a tangent 10.17 If a secant segment and a tangent segment share an endpoint outside a circle, segment share an endpoint outside a circle, then the product of the length of the secant then the product of the length of the secant

segment and the length of its external segment and the length of its external segment equals the square of the length of segment equals the square of the length of

the tangent segment.the tangent segment.

11.1 Polygon Interior Angles 11.1 Polygon Interior Angles Theorem: The sum of the Theorem: The sum of the

measures of the interior angles measures of the interior angles of a convex of a convex nn-gon is (n-2) ∙ -gon is (n-2) ∙

180°.180°.

11.2 Polygon Exterior Angles Theorem: The 11.2 Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles sum of the measures of the exterior angles

of a convex polygon, one angle at each of a convex polygon, one angle at each vertex, is 360°.vertex, is 360°.

Corollary: The measure of each exterior Corollary: The measure of each exterior angle of regular angle of regular nn-gon is 1/n ∙ 360°, or -gon is 1/n ∙ 360°, or

360°/n360°/n

11.3 Area of an Equilateral 11.3 Area of an Equilateral Triangle: The area of an Triangle: The area of an

equilateral triangle is one fourth equilateral triangle is one fourth the square of the length of the the square of the length of the

side times √3.side times √3.

11.4 Area of a Regular Polygon: 11.4 Area of a Regular Polygon: The area of a regular The area of a regular nn-gon with -gon with

side length side length ss is half the product of is half the product of the apothem the apothem aa and the perimeter and the perimeter PP, ,

so A = ½ so A = ½ aP, aP, or A = ½or A = ½aa ∙ ∙ nsns..

11.5 Areas of Similar Polygons: 11.5 Areas of Similar Polygons: If two polygons are similar with If two polygons are similar with the lengths of corresponding the lengths of corresponding

sides in the ratio of sides in the ratio of a:b, a:b, then the then the ratio of their areas is ratio of their areas is a2:b2a2:b2..

11.6 Circumference of a Circle: The 11.6 Circumference of a Circle: The circumference circumference CC of a circle is of a circle is C = pdC = pd or or C = C =

2pr,2pr, where where dd is the diameter of the circle and is the diameter of the circle and rr is the radius of the circle. Arc Length is the radius of the circle. Arc Length

Corollary: In a circle, the ratio of the length Corollary: In a circle, the ratio of the length of a given arc to the circumference is equal of a given arc to the circumference is equal

to the ratio of the measure of the arc to to the ratio of the measure of the arc to 360°.360°.

11.7 Area of a Circle: The area 11.7 Area of a Circle: The area of a circle is p times the square of a circle is p times the square

of the radius, or of the radius, or A = pr2.A = pr2.

11.8 Area of a Sector: The ratio of 11.8 Area of a Sector: The ratio of the area A of a sector of a circle to the area A of a sector of a circle to the area of the circle is equal to the the area of the circle is equal to the

ratio of the measure of the ratio of the measure of the intercepted arc to 360°.intercepted arc to 360°.

12.1 Euler’s Theorem: The 12.1 Euler’s Theorem: The number of faces (F), vertices number of faces (F), vertices

(V), and the edges of a (V), and the edges of a polyhedron are related by the polyhedron are related by the

formula formula F + V= E+2.F + V= E+2.

12.2 Surface Area of a Right Prism: 12.2 Surface Area of a Right Prism: The surface area The surface area SS of a right prism of a right prism can be found using the formula can be found using the formula S = S =

2B+Ph 2B+Ph where where BB is the area of a is the area of a base, base, P P is the perimeter of a base, is the perimeter of a base,

and and h h is the height.is the height.

12.3 Surface Area of Right Cylinder: The 12.3 Surface Area of Right Cylinder: The surface area surface area SS of a right cylinder is of a right cylinder is

S = 2B + Ch = 2pr2 + 2prh2S = 2B + Ch = 2pr2 + 2prh2, where , where BB is the is the area of a base, area of a base, CC is the circumference of a is the circumference of a base, base, rr is the radius of a base and is the radius of a base and hh is the is the

height.height.

12.4 Surface Area of a Regular Pyramid: 12.4 Surface Area of a Regular Pyramid: The surface area The surface area SS of a regular pyramid is of a regular pyramid is SS of a regular pyramid is of a regular pyramid is SS = B + ½ P= B + ½ P l, where l, where BB is the area of a base, is the area of a base, PP is the perimeter of is the perimeter of

the base, and l is the slant height.the base, and l is the slant height.

12.5 Surface Area of a Right 12.5 Surface Area of a Right Cone: The surface areaCone: The surface area S S of a of a right cone right cone S =pr2 +prS =pr2 +prl, where l, where rr

is the radius of the base and l is is the radius of the base and l is the slant height.the slant height.

12.6 Cavalieri’s Principle: IF two 12.6 Cavalieri’s Principle: IF two solids have the same height and solids have the same height and the same cross- sectional area the same cross- sectional area at every level then they have at every level then they have

the same volume.the same volume.

12.7 Volume of a Prism: The 12.7 Volume of a Prism: The volume volume VV of a prism is of a prism is V =Bh,V =Bh, wherewhere B B is the area of a base is the area of a base

andand h h is the height.is the height.

12.8 Volume of a Cylinder: The 12.8 Volume of a Cylinder: The volume ofvolume of V V of a cylinder is of a cylinder is V = V = Bh = pr2h, Bh = pr2h, where where BB is the area is the area of a base, and of a base, and hh is the height, is the height, and and rr is the radius of a base.is the radius of a base.

12.9 Volume of a Pyramid: The 12.9 Volume of a Pyramid: The volume volume VV of a pyramid is of a pyramid is V = V =

1/31/3BhBh, where , where B B is the area of a is the area of a base, base, hh is the height. is the height.

12.10 Volume of a Cone: The 12.10 Volume of a Cone: The volume volume VV of a cone is of a cone is V = 1/3Bh V = 1/3Bh = 1/3pr2h= 1/3pr2h, where , where BB is the area is the area of the base, of the base, hh is the height, and is the height, and

rr is the radius of the base. is the radius of the base.

12.11 Surface Area of a Sphere: 12.11 Surface Area of a Sphere: The surface area The surface area SS of a sphere of a sphere

with radius with radius rr is is S S = 4pr2. = 4pr2.

12.12 Volume of a Sphere: The 12.12 Volume of a Sphere: The volume volume VV of a sphere with the of a sphere with the

radius radius rr is is V = 4/3V = 4/3pr3.pr3.

12.13 Similar Solids Theorem: If 12.13 Similar Solids Theorem: If two similar solids have a scale two similar solids have a scale

factor of factor of a:ba:b, then corresponding , then corresponding areas have a ratio of areas have a ratio of a2:b2, a2:b2, and and

corresponding volumes have a ratio corresponding volumes have a ratio of of a3:b3.a3:b3.