geometry resource book

1

Geometry Resource Book

Name _____________________ Period _____

Room #219

2

Geometry Symbols Geometry Marks

∠ angle

congruent segments

arc (“arc AC”)

congruent angles

≅ congruent

(∠G≅ ∠S or 𝐴𝐵 ≅ 𝐶𝐷)

parallel

= equal (2 + 3 = 5)

perpendicular (or right angle)

𝑃𝑅 line

𝐴𝐵 line segment or segment

m∠R or m𝐴𝐵

measure (m∠R = “measure of angle

R” or m𝐴𝐵 = “measure of

segment AB”)

∥or//parallel

(𝐴𝐵 ∥ 𝐶𝐷)

⊥ perpendicular (𝐴𝐵 ⊥ 𝐶𝐷)

𝑍𝑌 ray ~ similar Δ triangle

Vocabulary: 180º, 360º, a

3

Geometry Vocabulary

Section Word Picture Definition

How to write it

with symbols

How to say it

1.3 180º angle

also called a “straight angle”

1.3 360º angle

a full turn around a circle

1.3 acute angle

an angle with a measure less than 90º

is acute.

“Angle R is acute.”

1.5 acute triangle

a triangle in which all of the angles have a

measure less than 90º

ΔABC is acute.

“Triangle ABC is acute.”

3.3 altitude (of a triangle)

a segment from a vertex of a triangle perpendicular to the opposite side or the line containing the

opposite side

€

∠R

Vocabulary: a

4


How to write it

with symbols

How to say it

1.2 angle

made by two rays that have the same

endpoint

∠1 ∠𝐸

not ∠𝐷 not ∠𝐹

not ∠𝐷𝐸 not ∠𝐷𝐹 ∠𝐷𝐸𝐹 ∠𝐹𝐸𝐷

not ∠𝐸𝐷𝐹 not ∠𝐷𝐹𝐸

“Angle 1” “Angle E”

“Angle DEF”

“Angle FED”

1.2 angle bisector

a ray that divides an angle into two

congruent angles

∠ABD≅∠CBD

“Angle ABD is

congruent to angle CBD.”

1.6 arc (of a circle)

two points on a circle and the part of the

circle between them

“Arc AB”

1.6 arc measure

equal to the measure of the central angle

that goes with the arc m =

85°

“The measure of arc AB is

85 degrees.”

Vocabulary: b

5


How to write it

with symbols

How to say it

1.5

base angles of an

isosceles triangle

• the two angles opposite the two sides that are congruent in an isosceles triangle

• the two angles on both ends of the base of an isosceles triangle

∠A and ∠C are the

base angles of ΔABC.

“Angle A and angle C are the

base angles of triangle ABC.”

10.1 base of a solid

•

1.5 base of an isosceles triangle

the side of an isosceles triangle that

is not congruent to either of the other

sides

𝐴𝐶 is the base of ΔABC.

“Segment AC is the base of ΔABC.”

1.1 bisect

divides into two congruent parts

B is the bisector of

“point B is the

bisector of segment

DC”

€

DC

Vocabulary: c

6


How to write it

with symbols

How to say it

1.6 central angle

an angle with its vertex at

the center of a circle

is a central angle of circle O.

“Angle BOC is a central

angle of circle O.”

1.6 chord

a segment whose endpoints are on a circle

is a chord of circle A.

“Segment DC is a chord of

circle A.”

1.6 circle

• all the points that are the same distance from a point (that point is the center of the circle)

• has an arc measure of 360º

circle O “circle O”

6.5 circum-ference

the distance around the

outside of a circle

1.1 collinear

points that are on the same line

(“co” = together, “linear” = line)

A, B, and C are

collinear

“points A, B and C

are collinear”

1.3 comple-mentary angles

two angles whose sum is 90º

35º + 55º = 90º

“35 plus 55 equals

90”

€

∠BOC

€

DC

Vocabulary: c

7


How to write it

with symbols

How to say it

1.4 concave polygon

• has one or more interior angles greater than 180º

• looks like a vertex has been pushed into the polygon

opposite of convex polygon

1.6 concentric circles

circles with the same center

1.8 cone

1.2 congruent angles

two angles that have the same measure ∠G≅ ∠S

“Angle G is

congruent to angle

S.”

1.4 congruent polygons

two polygons that have the same sides as each other and the same angles as each

other

ABCD ≅ HGFE

Quadri-lateral

ABCD is congruent

to quadrilateral HGFE

1.1 congruent segments

segments that have the same length

𝐴𝐵≅ 𝐶𝐷

“Segment AB is

congruent to segment

CD.”

Vocabulary: c

8


How to write it

with symbols

How to say it

2.1 conjecture

a hypothesis or educated guess, a statement which appears to be true but has not been proven

1.4 consecutive angles

• two angles at the ends of the same side

• two angles that are next to each other in a polygon

• “consecutive” means “in a row” or “one after the other”

∠A and ∠E are

consecu-tive angles

1.4 consecutive sides

• two sides that are the sides of the same angle

• two sides that are next to each other in a polygon


𝐴𝐵 and 𝐴𝐸 are consecu-tive sides

1.4 consecutive vertices

• two vertices that are the ends of the same side

• two vertices that are next to each other in a polygon


A and E are

consecu-tive

vertices

1.4 convex polygon

• all the interior angles are less than 180º

• all of the vertices are pushing out, away from the center

• opposite of concave polygon

Vocabulary: c, d

9


How to write it

with symbols

How to say it

1.1 coplanar

on the same plane (“co” = together, “planar” = plane)

D, E, and F are

coplanar

“Points D, E, and F

are coplanar.”

3.7 concurrent

three or more points that intersect at a

single point

1.3 counter-example

an example that shows that something

is NOT true

1.8 cylinder

1.4 decagon

a polygon with 10 sides

2.2 deductive reasoning

Example: If Iqra is a student at South, then Iqra must be in 9th, 10th, 11th, or 12th grade. (It is a fact that South is a high school and only has students grades 9-

12.)

showing that a statement is true

because of ageed-upon assumptions or

facts

1.2 degree

the unit we use for measuring an angle

155º

155 degrees

Vocabulary: d, e

10


How to write it

with symbols

How to say it

1.4 diagonal

• a segment connecting two non-consecutive vertices of a polygon

• a diagonal CAN be horizontal or vertical

𝐴𝐶 and 𝐵𝐷 are

diagonals of ABCD.

Segment AC and segment BD are

diagonals of

quadrilateral ABCD

1.6 diameter

a line segment going through the center of

a circle with its endpoints on the

circle

𝐴𝐵 is a diameter of circle

C.

“Segment AB is a diameter of circle

C.”

1.4 dodecagon


1.1 endpoint

a point at the end of a line segment or a ray

A and D are the

endpoints of .

“Points A and D are

the endpoints

of segment AD.”

1.4 equiangular polygon

a polygon in which all the angles are

congruent

1.4 equilateral polygon

a polygon in which all the sides are

congruent

€

AD

Vocabulary: e, f, g, h, i

11


How to write it

with symbols

How to say it

1.5 equilateral triangle

a triangle with three congruent sides

𝐴𝐵 ≅ 𝐵𝐶≅ 𝐴𝐶

4.3 exterior angle

an angle on the outside of a polygon, formed by extending a side of the polygon

∠4 is an exterior angle of ∆PQR

“Angle 4 is an

exterior angle of triangle PQR.”

1.8 hemisphere

1.4 hexagon


2.1 inductive reasoning

example: All forms of life that we know of need water to survive. If we discover a new form of life, it will probably need water to survive. example:

The next picture in the sequence is:

making a conclusion based on a pattern

Vocabulary: l

12


How to write it

with symbols

How to say it

1.1 intersect

two lines intersect if meet at a single point

“Line NA and line

FP intersect at point C.”

1.5 isosceles triangle

a triangle with two congruent sides

“Segment AB is


BC.”

1.5 kite

a quadrilateral with two different pairs of congruent sides that

are consecutive

“Segment AB is


BC. Segment

AD is congruent to segment

CD.”

9.1 leg (of a

right triangle)

the two sides of a right triangle that

make the right angle, also the two shorter

sides of a right triangle

“Segment AC and segment

CB are the legs of right

triangle ABC.”

€

AB≅BC

€

AB≅BC

€

AD ≅CD

Vocabulary: l

13


How to write it

with symbols

How to say it

4.2 leg (of a an isosceles triangle)

the two congruent sides of an isosceles

triangle

“Sides AB and BC are the legs of triangle ABC.”

1.1 line

is straight, has no thickness, and goes in

both directions forever

𝑃𝑅 𝑅𝑃 𝑃𝑄 𝑄𝑃 𝑄𝑅 m

not 𝑄 not 𝑃𝑄𝑅

Line PR Line RP Line PQ Line QP Line QR Line m

1.1 line segment (or segment)

straight and has no thickness like a line,

but has two endpoints

𝐴𝐶 𝐶𝐴

not 𝐵 not 𝐴𝐵 not 𝐵𝐴 not 𝐵𝐶

not 𝐴𝐵𝐶

line segment AC (or

segment AC) line

segment CA (or

segment CA)

1.3 linear pair

a pair of angles that share a vertex and a side and their non-shared side makes a

line

∠𝐷𝐴𝐶 and ∠𝐵𝐴𝐶 are

a linear pair.

Vocabulary: m

14


How to write it

with symbols

How to say it

1.6 major arc

• an arc of a circle that is larger than a semicircle

• an arc of a circle with a measure greater than 180°

is a major

arc.

“Arc AKB is a major

arc.”

1.1 measure of a segment

the length of a segment

AB = 5 cm or

m𝐴𝐵 = 5 cm

“AB equals 5 cm” or

“the measure of

segment AB is 5

cm.”

1.2 measure of an angle

the number of degrees needed to

rotate to get from one side of the angle to

the other

m∠R =

155º

“The measure of angle R is

155 degrees.”

3.2 median (of a triangle)

the segment connecting the vertex

of a triangle to the midpoint of the opposite side

𝐽𝐾 is a median of ∆KLM.

“Segment JK is a

median of triangle KLM.”

1.1 midpoint

the point on a segment that is the same distance from

both endpoints

“Segment JK is


KL.”

UYAS 1

midpoint formula

formula: 𝑥! + 𝑥!2 ,

𝑦! + 𝑦!2

=3+ 52 ,

4+−22

=82 ,22

= (4, 1)

The midpoint of 𝑀𝑁 is (4 , 1).

€

JK ≅KL

Vocabulary: m, n, o

15


How to write it

with symbols

How to say it

3.2 midsegment

(of a triangle)

a segment connecting the midpoint of one side of a triangle to

the midpoint of another side of the

triangle

𝐵𝐷 is a midsegme

nt of ∆ACD.

“Segment BD is a

midsegment of

triangle ACD.”

1.6 minor arc

• an arc of a circle that is smaller than a semicircle

• an arc of a circle with a measure less than 180°

is a minor arc.

“Arc AB is a minor

arc.”

1.4 n-gon

• a polygon with n sides

• n is a variable so the polygon can have any number of sides

1.4 nonagon


1.3 obtuse angle

an angle with a measure of more than

90º

1.5 obtuse triangle

a triangle with one obtuse angle

m A > 90°

“the measure of angle A is

greater than 90

degrees” €

∠

Vocabulary: o, p

16


How to write it

with symbols

How to say it

1.4 octagon


1.3 parallel

lines that are always the same distance apart and never

intersect 𝐴𝐵//𝑀𝑁

“Line AB is parallel

to line MN.”

1.5 parallel-ogram

a quadrilateral with two pairs of parallel

sides

Segment AB is

parallel to segment

CD. Segment

AD is parallel to segment

BC.

1.4 pentagon


1.4 perimeter

the total distance around the outside of

a polygon

perimeter of DCAB = 5 + 14 + 11 + 17 =

47 cm

€

AB//CD

€

AD//BC

Vocabulary: p

17


How to write it

with symbols

How to say it

1.3 perpen-dicular

or

lines that intersect at 90 degree angles 𝐴𝐵 ⊥ 𝐶𝐷

Segment AB is

perpendic-ular to line

CD.

3.2 perpen-dicular bisector

a line that is perpendicular to and

bisects a segment

𝐴𝐵 ⊥ 𝐶𝐷 and

𝐴𝐷 ≅ 𝐵𝐷

“Segment AB is

perpendic-ular to line

CD and segment AD is


BD.”

1.1 plane

a flat two-dimensional surface that goes on forever

M (written with an

upper-case cursive letter)

“Plane M”

1.1 point

an exact place or location B “Point B”

1.6 point of tangency

the point where a tangent touches a

circle

is tangent to circle A

and B is a point of

tangency.

“Line CD is tangent to circle A and point

B is a point of

tangency.”

€

CD

Vocabulary: p

18


How to write it

with symbols

How to say it

1.4 polygon

a shape with 3 or more sides

1.8 prism

1.2 protractor

a tool for measuring angles

1.8 pyramid

Vocabulary: q, r

19


How to write it

with symbols

How to say it

1.4 quadrilateral

a polygon with four sides

DGMT DTMG TMBD GMTD

not MTGD not DGTM not TMDG

“quadrilateral

DGMT” “quadrilate

ral DTMG”

1.6 radius

• a segment from the center of a circle to a point on the edge of the circle

• radius has an unusual plural – we say “one radius”, but “three radii”

is a radius of circle O.

“Segment AO is a radius of circle O.”

1.1 ray

a part of a line that starts at a point and goes forever in one

direction

𝑍𝑌 𝑍𝑋

not 𝑋𝑌 not 𝑌𝑍 not 𝑌𝑍

not 𝑍𝑋𝑌

“ray ZY” “ray ZX”

1.5 rectangle

a quadrilateral with four congruent angles

Rectangle ABCD

“Rectangle ABCD”

1.4 regular polygon

a polygon that has all congruent sides

(equilateral) and all congruent angles

(equiangular)

€

AO

Vocabulary: r, s

20


How to write it

with symbols

How to say it

1.5 rhombus

a quadrilateral with all congruent sides

(equilateral)

1.3 right angle

an angle with a measure of 90º

1.5 right triangle

a triangle with one right angle

1.5 scalene triangle

a triangle in which none of the sides are

congruent

1.1 segment same as “line segment”

1.6 semicircle

• half a circle • has an arc

measure of 180º

is a semicircle

.

“Arc ABC is a

semicircle.”

1.4 side of a polygon

a line segment that is part of a polygon

Polygon ABCDE

has 5 sides. For example

one of the sides is 𝐶𝐷.

Vocabulary: s, t

21


How to write it

with symbols

How to say it

1.2 sides (of an angle)

the two rays that make an angle

The sides of

are 𝐵𝐴 and

𝐵𝐶.

“The sides of angle CBA are ray BA

and ray BC.”

1.3 skew lines

lines that are not parallel, but never

intersect either

𝑃𝑆 and 𝑅𝑌 are

skew lines

1.8 space all the points in three dimensions (3-D)

1.8 sphere

1.5 square

an equilateral and equiangular quadrilateral

Square ABCD

“Square ABCD”

1.3 supple-mentary angles

a pair of angles whose sum is 180º

45º + 135º = 180º

120º + 60º = 180º

1.6 tangent

a line that intersects a circle at only one

point

is a tangent of circle O.

“Line EF is a

tangent of circle O.”

€

∠CBA

€

EF

Vocabulary: t, u, v

22


How to write it

with symbols

How to say it

2.6 transversal

a line that intersects two or more other

lines

t is a transversal

.

“Line t is a transversal

.”

1.5 trapezoid

a quadrilateral with two parallel sides

(and not more than two parallel sides)

Trapezoid TRAP

“Trapezoid TRAP”

1.4 triangle

a polygon with three sides

ΔABC ΔCBA ΔBAC

Triangle ABC

12.1 trig-onometry

the study of relationships between the sides and angles

of triangles (“trigono” = triangle, “metry” = measure)

1.4 undecagon


1.4 vertex (of a polygon)

• where the sides of a polygon meet

• the plural of vertex is VERTICES (for example 3 vertices)

the polygon

has 5 vertices,

named A, B, C, D,

and E

23


How to write it

with symbols

How to say it

1.2 vertex (of an angle)

• the point where the two sides of an angle meet

• the plural of vertex is VERTICES (for example, 3 vertices)

B is the vertex of ∠𝐴𝐵𝐶.

“B is the vertex of

angle ABC.”

1.5

vertex angle of an

isosceles triangle

the angle between the two congruent sides

of an isosceles triangle

∠𝐴𝐵𝐶 is the vertex angle of ΔABC.

“Angle ABC is the

vertex angle of triangle ABC.”

1.3 vertical angles

the angles opposite each other when two

lines cross

∠1 and ∠3 are vertical angles. ∠2 and ∠4 are vertical angles.

24

Geometry Conjectures 2.5 C-1 Linear Pair Conjecture: If two angles form

a linear pair, then ______________________.

If then

2.5 C-2 Vertical Angles Conjecture: If two angles are vertical angles, then ______________________.

If then

2.6 C-3a Corresponding Angles Conjecture (CA Conjecture): If two parallel lines are cut by a transversal, then corresponding angles are ______________.

If then

2.6 C-3b Alternate Interior Angles Conjecture (AIA Conjecture): If two parallel lines are cut by a transversal, then alternate interior angles are ______________.

If then

2.6 C-3c Alternate Exterior Angles Conjecture (AEA Conjecture): If two parallel lines are cut by a transversal, then alternate exterior angles are ______________.

If then

2.6 C-3 Parallel Lines Conjecture: If two parallel lines are cut by a transversal, then corresponding angles are ______________, alternate interior angles are ______________, and alternate exterior angles are ______________.

If then

25

2.6 C-4 Converse of the Parallel Lines Conjecture: If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior angles, or congruent alternate exterior angles, then the lines are ______________________.

If then

or

or

3.2 C-5 Perpendicular Bisector Conjecture: If a point ( ) is on the perpendicular bisector of a segment, then it is _______________________ from the endpoints.

If then

3.2 C-6 Converse of the Perpendicular Bisector Conjecture: If a point is equidistant from the endpoints of a segment, then it is on the ______________________ of the segment.

If then

3.3 C-7 Shortest Distance Conjecture: The shortest distance from a point to a line is measured along the ______________________ from the point to the line.

If then

26

3.4 C-8 Angle Bisector Conjecture: If a point is on the bisector of an angle, then it is ______________________ from the sides of the angle.

If then

UYAS 3 Parallel Slope Property: In a coordinate plane, two distinct lines are parallel if and only if ________________________.

If then

UYAS 3 Perpendicular Slope Property: In a coordinate plane, two nonvertical lines are perpendicular if and only if _________________________________________________.

If then

3.7 C-9 Angle Bisector Concurrency Conjecture: The three angle bisectors of a triangle

______________________.

If then

27

3.7 C-10 Perpendicular Bisector Concurrency Conjecture: The three perpendicular bisectors of a triangle ______________________.

If then

3.7 C-11 Altitude Concurrency Conjecture: The three altitudes (or the lines containing the altitudes) of a

triangle ______________________.

If then

3.7 C-12 Circumcenter Conjecture: The circumcenter of a triangle _____________________.

If then

28

3.7 C-13 Incenter Conjecture: The incenter of a triangle ______________________.

If then

3.8 C-14 Median Concurrency Conjecture: The three medians of a triangle _____________________.

If then

3.8 C-15 Centroid Conjecture: The centroid of a triangle divides each median into two parts so that the

distance from the centroid to the vertex is _____________________ the distance from the centroid to the midpoint of the opposite side.

If then

29

3.9 C-16 Center of Gravity Conjecture: The ___________________ of a triangle is the center of gravity of the triangular region.

If then

4.1 C-17 Triangle Sum Conjecture: The sum of the measures of the angles in every triangle is _____________________.

If then

4.1 C-18 Third Angle Conjecture: If two angles of one triangle are equal in measure to two angles of

another triangle, then the third angle in each triangle ______________________.

If then

4.2 C-19 Isosceles Triangle Conjecture: If a triangle

is isosceles, then ______________________.

If then

4.2 C-20 Converse of the Isosceles Triangle Conjecture: If a triangle has two congruent angles, then _____________________.

If then

30

4.3 C-21 Triangle Inequality Conjecture: The sum of the lengths of any two sides of a triangle is _______________________ the length of the third side.

If then If then

4.3 C-22 Side-Angle Inequality Conjecture: In a triangle, if one side is longer than another side, then the

angle opposite the longer side is ______________________. If then

largest middle small

sides

angles

4.3 C-23 Triangle Exterior Angle Conjecture: The

measure of an exterior angle of a triangle ______________________.

If then

4.4 C-24 SSS Congruence Conjecture: If the three sides of one triangle are congruent to the three sides of another triangle, then ______________________.

If then

31

4.4 C-25 SAS Congruence Conjecture: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then ______________________.

If then

4.4 NOT A CONJECTURE!

If then

4.5 NOT A CONJECTURE!

If then

4.5 C-26 ASA Congruence Conjecture: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then ______________________.

If then

32

4.5 C-27 SAA Congruence Conjecture: If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then ______________________.

If then

4.6 Corresponding parts of congruent triangles are congruent. (CPCTC)

If then

4.8 C-28 Vertex Angle Bisector Conjecture: In an

isosceles triangle, the bisector of the vertex angle is also _______________________ and ___________________.

If then

4.8 C-29 Equilateral/Equiangular Triangle Conjecture: Every equilateral triangle is ______________________, and, conversely, every equiangular triangle is ______________________.

If then

If then

≅ ∠_____≅ ∠_____ and _____ ≅______ because _________________.

33

5.1 C-30 Quadrilateral Sum Conjecture: The sum of the measures of the four angles of any quadrilateral is ______________________.

If then

5.1 C-31 Pentagon Sum Conjecture: The sum of the measure of the five angles of any pentagon is ______________________.

If then

5.1 C-32 Polygon Sum Conjecture: The sum of the

measures of the n interior angles of an n-gon is ______________________.

If then

5.2 C-33 Exterior Angle Sum Conjecture: For any polygon, the sum of the measures of a set of exterior angles is ______________________.

If then

5.2 C-34 Equiangular Polygon Conjecture: You can

find the measure of each interior angle of an equiangular n-gon by using either of these formulas:

_______________________ or

____________________.

If then

5.3 Vocabulary for kites

• Vertex angles • Nonvertex angles

34

5.3 C-35 Kite Angles Conjecture: The _______________________ angles of a kite are ______________________.

If then

5.3 C-36 Kite Diagonals Conjecture: The diagonals of a kite are ______________________.

If then

5.3 C-37 Kite Diagonal Bisector Conjecture: The diagonal connecting the vertex angles of a kite is the _______________________ of the other diagonal.

If then

5.3 C-38 Kite Angle Bisector Conjecture: The _______________________ angles of a kite are _______________________ by a __________________.

If then

5.3 Vocabulary for trapezoids

• Bases • Pair of base angles

5.3 C-39 Trapezoid Consecutive Angles Conjecture: The consecutive angles between the bases of a trapezoid are ______________________.

If then

35

5.3 C-40 Isosceles Trapezoid Conjecture: The base angles of an isosceles trapezoid are ______________________.

If then

5.3 C-41 Isosceles Trapezoid Diagonals Conjecture: The diagonals of an isosceles trapezoid are ______________________.

If then

5.4 C-42 Three Midsegments Conjecture: The three midsegments of a triangle divide it into ______________________.

If then

5.4 C-43 Triangle Midsegment Conjecture: A midsegment of a triangle is _______________________ to the third side and _______________________ the length of ______________________.

If then

5.4 C-44 Trapezoid Midsegment Conjecture: The midsegment of a trapezoid is _____________________ to the bases and is equal in length to ______________________.

If then

5.5 C-45 Parallelogram Opposite Angles Conjecture: The opposite angles of a parallelogram are ______________________.

If then

36

5.5 C-46 Parallelogram Consecutive Angles Conjecture: The consecutive angles of a parallelogram are ______________________.

If then

5.5 C-47 Parallelogram Opposite Sides Conjecture: The opposite sides of a parallelogram are ______________________.

If then

5.5 C-48 Parallelogram Diagonals Conjecture: The diagonals of a parallelogram ______________________.

If then

5.6 C-49 Double-Edged Straightedge Conjecture: If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a ______________________.

If then

5.6 C-50 Rhombus Diagonals Conjecture: The diagonals of a rhombus are _____________________, and they _____________________.

If then

5.6 C-51 Rhombus Angles Conjecture: The _______________________ of a rhombus _______________________ the angles of the rhombus.

If then

37

5.6 C-52 Rectangle Diagonals Conjecture: The diagonals of a rectangle are ____________________ and _________________.

If then

5.6 C-53 Square Diagonals Conjecture: The diagonals of a square are _______________________, _______________________, and __________________.

If then

6.1 definitions:

central angle arc measure If then

6.1 C-54 Chord Central Angles Conjecture: If two chords in a circle are congruent, then they determine two central angles that are ______________.

If then

6.1 C-55 Chord Arcs Conjecture: If two chords in a

circle are congruent, then their _________________________ are congruent.

If then

6.1 C-56 Perpendicular to a Chord Conjecture: The perpendicular from the center of a circle to a chord is the __________________ of the chord.

If then

82º

38

6.1 C-57 Chord Distance to Center Conjecture: Two congruent chords in a circle are _______________________ from the center of the circle.

If then

6.1 C-58 Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord ___________________ _________________________.

If then

6.2 C-59 Tangent Conjecture: A tangent to a circle _________________________ the radius drawn to the point of tangency.

If then

6.2 C-60 Tangent Segments Conjecture: Tangent segments to a circle from a point outside the circle are __________________.

If then

6.3 examples of central angles

6.3 examples of inscribed angles

39

6.3 C-61 Inscribed Angle Conjecture: The measure of an angle inscribed in a circle is ________________________.

If then

6.3 C-62 Inscribed Angles Intercepting Arcs Conjecture: Inscribed angles that intercept the same arc ______________________.

If then

6.3 C-63 Angles Inscribed in a Semicircle Conjecture: Angles inscribed in a semicircle ________________________.

If then

6.3 C-64 Cyclic Quadrilateral Conjecture: The ____________ angles of a cyclic quadrilateral are ______________________.

If then

40

6.3 C-65 Parallel Lines Intercepted Arcs Conjecture: Parallel lines intercept _________________ arcs on a circle.

If then

6.5 C-66 Circumference Conjecture: If C is the circumference and d is the diameter of a circle, then there is a number π such that C = _________________. If d = 2r where r is the radius, then C = _________________.

If then

If then

6.6 Vocabulary for arcs

• Arc measure • Arc length

6.6 C-67 Arc Length Conjecture: The length of an arc equals the ________________________.

If then

41

7.1 C-68 Reflection Line Conjecture: The line of reflection is the _________________________ of every segment joining a point in the original figure with its image.

7.2 C-69 Coordinate Transformations Conjecture: The ordered pair rule (x, y) → (−x, y) is a ______________ over the __________. The ordered pair rule (x, y) → (x, −y) is a ______________ over the __________. The ordered pair rule (x, y) → (−x, −y) is a ____________ about _____________. The ordered pair rule (x, y) → (y, x) is a ______________ over _______________.

7.2 C-70 Minimal Path Conjecture: If points A and

B are on one side of line ℓ, then the minimal path from point A to line ℓ to point B is found by ______________ _________________________.

7.3 C-71 Reflections over Parallel Lines Conjecture: A composition of two reflections over two parallel lines is equivalent to a single _________________. In addition, the distance from any point to its second image under the two reflections is ___________ the distance between the parallel lines.

7.3 C-72 Reflections over Intersecting Lines Conjecture: A composition of two reflections over a pair of intersecting lines is equivalent to a single ____________________. The angle of _________________ is ___________ the acute angle between the pair of intersecting reflection lines.

7.5 C-73 Tessellating Triangles Conjecture: ___________ triangle will create a monohedral tessellation.

42

7.5 C-74 Tessellating Quadrilaterals Conjecture: __________________ quadrilateral will create a monohedral tessellation.

8.1 C-75 Rectangle Area Conjecture: The area of a rectangle is given by the formula _________________, where A is the area, b is the length of the base, and h is the height of the rectangle.

If then

If then

8.1 C-76 Parallelogram Area Conjecture: The area of a parallelogram is given by the formula _________________, where A is the area, b is the length of the base, and h is the height of the parallelogram.

If then

8.2 C-77 Triangle Area Conjecture: The area of a triangle is given by the formula ____________________, where A is the area, b is the length of the base, and h is the height of the triangle.

If then

43

8.2 C-78 Trapezoid Area Conjecture: The area of a trapezoid is given by the formula

_________________________, where A is the area, b1 and b2 are the lengths of the two bases, and h is the height of the trapezoid.

If then

8.2 C-79 Kite Area Conjecture: The area of a kite is given by the formula __________________, where d1 and d2 are the lengths of the diagonals.

If then

8.4 C-80 Regular Polygon Area Conjecture: The area of a regular polygon is given by the

formula ________________, where A is the area, a is the apothem, s is the length of each side, and n is the number of sides. The length of each side times the number of sides is the perimeter P, so sn = P. Thus you can also write the formula for area as ________________________.

If then

8.5 C-81 Circle Area Conjecture: The area of a circle is given by the formula __________________, where A is the area and r is the radius of the circle.

If then

If then

44

8.6 Area of a sector of a circle

8.6 Area of a segment of a circle

8.6 Area of an annulus of a circle

8.7

8.7 Surface Area of a Cylinder

If then

45

8.7 Surface Area of a Cone

If then

9.1 Vocabulary for right triangles

9.1 C-82 The Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then _________________________.

If then

If then

9.2 C-83 Converse of the Pythagorean Theorem: If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle ________________________.

If then

and

46

9.3 C-84 Isosceles Right Triangle Conjecture: In an isosceles right triangle, if the legs have length l, then the hypotenuse has length __________. If then

If then

If then

9.3 C-85 30°-60°-90° Triangle Conjecture: In a 30°-60°-90° triangle, if the shorter leg has length a, then the longer leg has length ___________, and the hypotenuse has length _____________.

If then

If then

If then

9.5 C-86 Distance Formula: The distance between points ( )1 1,A x y and ( )2 2,B x y is given by AB = ____________________________.

If then

9.5 C-87: Equation of a Circle The equation of a circle with radius r and center (h, k) is ________________________.

If then

the equation for the circle is

47

10.2 C-88 Prism-Cylinder Volume Conjecture: The volume of a prism or a cylinder is the ______________________ multiplied by the ______________.

If then

If then

10.3 C-89 Pyramid-Cone Volume Conjecture: If B is the area of the base of a pyramid or a cone and H is the height of the solid, then the formula for the volume is V = ____________.

If then

If then

10.6 C-90 Sphere Volume Conjecture: The volume of a sphere with radius r is given by the formula ________________________.

If then

10.7 C-91 Sphere Surface Area Conjecture: The surface area, S, of a sphere with radius r is given by the formula _____________________.

If then

48

11.1 C-92 Dilation Similarity Conjecture: If one polygon is the image of another polygon under a dilation, then _________________________.

If then

11.2 C-93 AA Similarity Conjecture: If ________ angles of one triangle are congruent to _________ angles of another triangle, then _____________ _________________.

If then

11.2 C-94 SSS Similarity Conjecture: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are ________________.

If then

11.2 C-95 SAS Similarity Conjecture: If two sides of one triangle are proportional to two sides of another triangle and _______________________, then the ____________________.

If then

49

11.4 C-96 Proportional Parts Conjecture: If two triangles are similar, then the corresponding __________________, ____________________, and _________________________ are ______________________ to the corresponding sides.

If then

11.4 C-97 Angle Bisector/Opposite Side Conjecture: A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as ________________________.

If then

11.5 C-98 Proportional Areas Conjecture: If corresponding sides of two similar polygons or the radii of two circles

compare in the ratio mn

,then their areas

compare in the ratio __________________.

If then

11.5 C-99 Proportional Volumes Conjecture: If corresponding edges (or radii, or heights) of two similar solids compare in the ratio mn

, then their volumes compare in the

ratio __________________. If then

50

11.6 C-100 Parallel/Proportionality Conjecture: If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides ___________________. Conversely, if a line cuts two sides of a triangle proportionally, then it is ___________ to the third side.

If then

AND If then

11.6 C-101 Extended Parallel/Proportionality Conjecture: If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides ____________________.

If then

12.1 Labeling the Sides a Right Triangle

12.1 Definitions of Trigonometric Ratios

51

12.2 Inverse Trigonometric Ratios

12.3 C-102 SAS Triangle Area Conjecture: The area of a triangle is given by the formula

_________________________, where a and b are the lengths of two sides and C is the angle between them.

If then

12.3 C-103 Law of Sines: For a triangle with angles A, B, and C and sides of lengths a, b, and c (a opposite

A, b opposite B, and c opposite C), ________________________.

If then

52

12.4 C-104 Pythagorean Identity: For any angle A, ______________________________________________.

If then

12.4 C-105 Law of Cosines: For any triangle with sides of lengths a, b, and c, and with C the angle opposite

the side with length c ________________________.

If then

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