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Geometry Definitions & Conjectures
Chapter 0 Vocabulary Terms
Symmetry
when an object can be rotated or reflected and still appear the same
Rotational
rotating an object less than 360⁰ about a point - retains shape and appearance
Reflectional
reflecting an object over a line and it retains original shape and appearance
Bilateral
when there is only 1 line of symmetry possible for a particular shape
Chapter 1 Vocabulary Terms
Note: The Basic Building Blocks of Geometry are points, lines, and planes.
Acute Angle
an angle measurement less than 90 degrees
Adjacent
angles sharing a side or sides sharing a vertex
Altitude
the height of a figure (always perpendicular to the base of the figure)
Angle
a geometric figure formed by two rays that have a common endpoint
Bisect
to divide into two equal parts
Collinear
points on the same line
Complementary
two angles whose measures have a sum of 90°
Concave
hollow; curved inward, creating a hollow space; opposite of convex
Congruent
having the same size and shape
Consecutive
2 sides that share an angle or 2 angles that are the endpoints of 1 side
Convex
A polygon in which all vertices appear to be pushed outward.
Coplanar
lying in the same plane
Diagonal
a line segment connecting any 2 non-consecutive vertices of a polygon
Equal
having the same value
Equiangular
a polygon having all angles equal
Equilateral
a polygon having all sides or faces equal
Exterior Angles
angles on the outside of a polygon
Interior Angles
angles on the inside of a polygon
Isosceles
a polygon with at least two congruent sides
Line Segment
part of a line that is bounded by two end points
Linear Pair
two angles that are adjacent (on the same line) and supplementary
Midpoint
a point equidistant from the endpoints
Obtuse
an angle measurement above 90 but below 180 degrees
Parallel
being everywhere equidistant and not intersecting
Perpendicular
intersecting at or forming right angles
Polygon
closed plane figure having angles and straight sides
Ray
a straight line extending from a point
Reflex Angle
the difference between any angle and 360 degrees (360 – 30 = 330  reflex)
Regular Polygon
a polygon with all sides and all angles equal
Page 1 of 21
Geometry Definitions & Conjectures
Right Angle
an angle with the measure of 90 degrees
Scalene
a polygon with no congruent sides
Supplementary
two angles whose sum measures 180 degrees
Transversal
A line or segment crossing two or more parallel lines
Vertex
the point of intersection of lines or the point opposite the base of a figure
Vertical Angles
a pair of opposite congruent angles formed by intersecting lines
Types of Triangles
Classified by Their Sides
Scalene
triangle with no congruent sides
Isosceles
triangle with at least two congruent sides
Equilateral
a triangle having all sides or faces equal (also, all angles are congruent)
Classified by Their Angles
Acute
a triangle having all angles measuring less than 90 degrees
Right
a triangle with exactly 1 right angle
Obtuse
a triangle with exactly 1 obtuse angle
Special Quadrilaterals
Kite
two pairs of adjacent sides congruent and no opposite sides congruent
Parallelogram
a quadrilateral whose opposite sides are both parallel and equal in length
Rectangle
a parallelogram with four right angles
Rhombus
a parallelogram with 4 congruent sides
Square
all sides are the same and all angles are the same -- a regular quadrilateral
Trapezoid
a quadrilateral with exactly one pair of parallel sides
Circles and Their Terms
Circle
the set of all points equidistant from a central point
Radius
a segment from the center to any point on the circle
Diameter
the longest chord in a circle -- passes through the center of the circle
Circumference
the distance around a circle -- like a perimeter but for a circle
Chord
a line segment with both endpoints on the circle
Tangent
a line that intersects the circle at exactly one point
Pt. of Tangency
the point where a tangent and a circle meet
Concentric
concentric circles are circles that share the same center
Arc
the part of a circle between two points on the circle
Minor Arc
an arc of less than 180 degrees
Major Arc
an arc of more than 180 degrees
Semicircle
an arc of exactly 180 degrees
Central Angle
angle formed by segments from the end pts of the arc & the center of the circle
Page 2 of 21
Geometry Definitions & Conjectures
Common Polygons
Total of
Interior Angles
Each Interior
Angle (if
Regular)
Total of
Exterior Angles
Each Exterior
Angle (if
Regular)
Triangle
3-sided polygon
180
60
360
120
Quadrilateral
4-sided polygon
360
90
360
90
Pentagon
5-sided polygon
540
108
360
72
Hexagon
6-sided polygon
720
120
360
60
Heptagon
7-sided polygon
900
128.57
360
51.43
Octagon
8-sided polygon
1080
135
360
45
Nonagon
9-sided polygon
1260
140
360
40
Decagon
10-sided polygon
1440
144
360
36
Dodecagon
12-sided polygon
1620
150
360
30
Icosagon
20-sided polygon
1800
162
360
18
n-gon
an "n"-sided
polygon
𝟏𝟖𝟎(𝒏 − 𝟐)
𝟏𝟖𝟎(𝒏 − 𝟐)
𝒏
360
Special Quadrilateral  ParallelProperty 
ogram
Isosceles
Trapezoid
Kite
Rectangle
𝟑𝟔𝟎
𝒏
Square
Rhombus
Opposite sides are parallel

X
X



Opposites sides are congruent

X
X



Opposite angles are congruent

X
X



Diagonals bisect each other

X
X



Diagonals are perpendicular
X
X

X


Diagonals are congruent
X

X


X
Exactly one line of symmetry
X


X
X
X
Exactly two lines of symmetry
X
X
X

X

Logic Statements
Statement
If p, then q.
If two angles are congruent, then they have the same measure.
Converse
If q, then p.
If two angles have the same measure, then they are congruent.
Inverse
If not p, then
not q.
If two angles are not congruent, then they do not have the same measure.
Contrapositive
If not q, then
not p.
If two angles do not have the same measure, then they are not congruent.
Page 3 of 21
Geometry Definitions & Conjectures
Point of
Concurrency
Constructed
With The …
Incenter
Angle
Bisectors
Circumcenter
Euler Line Euler Line Euler Line
Center of the Equidistant Location (by Type of Triangle)
(Normal (Isosceles (Equilateral
...
from the …
Acute
Right
Obtuse Triangle) Triangle) Triangle)
Inscribed
Circle
Sides of
Triangle
Inside
Perpendicular Circumscribed Vertices of
Bisectors
Circle
Triangle
Centroid
Medians
Orthocenter
Altitudes
2 to 1 ratio
Inside
Inside
Midpoint of
Outside
Hypotenuse
Inside
Inside
Inside
Inside
Inside
Vertex of
Outside
Right Angle
Page 4 of 21
Off
On
On
On
On
On
On
On
All Come
Together at
One Point
Geometry Definitions & Conjectures
Euler Line
The Orthocenter, Circumcenter, and the Centroid will always lie on the Euler Line. The Incenter will
only be on the line in an Isosceles Triangle. In the case of the Equilateral Triangle all of the points
come together to meet at exactly the same point.
Parallel Lines
Name
Property
Description
Example
Corresponding
Angles
Congruent
angles in the same locations on the
transversal & the parallel lines
(See Below)
Alternate Interior
Angles
Congruent
angles on opposite sides of the transversal &
inside the parallel lines
(See Below)
Alternate Exterior
Angles
Congruent
angles on opposite sides of the transversal &
outside the parallel lines
(See Below)
Same Side Interior
Angles
Supplementary
angles on the same side of the transversal &
inside the parallel lines
(See Below)
Same Side Exterior
Angles
Supplementary
angles on the same side of the transversal &
outside the parallel lines
(See Below)
Page 5 of 21
2&6
3&5
1&7
4&5
1&8
Geometry Definitions & Conjectures
Angle of Elevation and Angle of Depression
Parts of Circles
Coordinate Midpoint Property
If (x1, y1) and (x2, y2) are the coordinates for the endpoints of a segment,
then the coordinates for the midpoints can be for the midpoints can be
found using the following formula:
(
Slope
𝒔𝒍𝒐𝒑𝒆 =
𝑥1 + 𝑥2 𝑦1 + 𝑦2
,
)
2
2
𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆
𝒐𝒓
𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙
𝒐𝒓
∆𝒚
∆𝒙
𝒐𝒓
𝒚𝟐 − 𝒚𝟏
𝒙𝟐 − 𝒙𝟏
Slope-intercept form: 𝒚 = 𝒎𝒙 + 𝒃
Where (x,y) are the coordinates for any point on the circle, m is the slope
and b is the y-intercept (where the line crosses the y axis).
The slopes of parallel lines are the same.
𝟑
𝟒
The slopes of perpendicular lines are negative reciprocals of each other ( → − ).
𝟒
𝟑
Page 6 of 21
Geometry Definitions & Conjectures
Triangle Congruence Shortcuts
Transformations
Isometry - A transformation that keeps the size and shape of geometric figures the same.
Translation is when we slide a figure in any direction.
Reflection is when we flip a figure over a line.
Rotation is when we rotate a figure a certain degree around a point.
Dilation is when we enlarge or reduce a figure.
Page 7 of 21
Geometry Definitions & Conjectures
Page 8 of 21
Geometry Definitions & Conjectures
Coordinate Transformations Conjecture
The ordered pair rule (x, y) → (-x, y) is a reflection over the y-axis.
The ordered pair rule (x, y) → (x, -y) is a reflection over the x-axis.
The ordered pair rule (x, y) → (-x, -y) is a rotation about the origin.
The ordered pair rule (x, y) → (y, x) is a reflection over y = x
Conjectures for Chapter 2
Linear Pair Conjecture
If two angles form a linear pair, then the measures of the angles add up to 180°.
Vertical Angles Conjecture
If two angles are vertical angles, then they are congruent (have equal measures).
Corresponding Angles Conjecture, or CA Conjecture
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Alternate Interior Angles Conjecture or AIA Conjecture
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Alternate Exterior Angles Conjecture or AEA Conjecture
If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Parallel Lines Conjecture
Page 9 of 21
Geometry Definitions & Conjectures
If two parallel lines are cut by a transversal, then corresponding angles are congruent, alternate interior
angles are congruent, and alternate exterior angles are congruent.
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 parallel.
Conjectures for Chapter 3
Perpendicular Bisector Conjecture
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints.
Converse of the Perpendicular Bisector Conjecture
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the
segment.
Shortest Distance Conjecture
The shortest distance from a point to a line is measured along the perpendicular segment from the point to
the line.
Angle Bisector Conjecture
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Angle Bisector Concurrency Conjecture
The three angle bisectors of a triangle are concurrent (meet at a point).
Perpendicular Bisector Concurrency Conjecture
The three perpendicular bisectors of a triangle are concurrent.
Altitude Concurrency Conjecture
The three altitudes (or the lines containing the altitudes) of a triangle are concurrent.
Circumcenter Conjecture
The circumcenter of a triangle is equidistant from the vertices.
Incenter Conjecture
The incenter of a triangle is equidistant from the sides.
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Geometry Definitions & Conjectures
Median Concurrency Conjecture
The three medians of a triangle are concurrent.
Centroid Conjecture
The centroid of a triangle divides each median into two parts so that the distance from the centroid to the
vertex is twice the distance from the centroid to the midpoint of the opposite side.
Center of Gravity Conjecture
The centroid of a triangle is the center of gravity of the triangular region.
Conjectures for Chapter 4
Triangle Sum Conjecture
The sum of the measures of the angles in every triangle is 180°.
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 is equal in measure to the third angle in the other triangle.
Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles are congruent.
Converse of the Isosceles Triangle Conjecture
If a triangle has two congruent angles, then it is an isosceles triangle.
Triangle Inequality Conjecture
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Side-Angle Inequality Conjecture
In a triangle, if one side is longer than another side, then the angle opposite the longer side is larger than
the angle opposite the shorter side.
Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior
angles.
SSS Congruence Conjecture
Page 11 of 21
Geometry Definitions & Conjectures
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are
congruent.
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 the triangles are congruent.
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 the triangles are congruent.
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 the triangles are congruent.
Vertex Angle Bisector Conjecture
In an isosceles triangle, the bisector of the vertex angle is also the altitude and the median to the base.
Equilateral/Equiangular Triangle Conjecture
Every equilateral triangle is equiangular. Conversely, every equiangular triangle is equilateral.
Conjectures for Chapter 5
Quadrilateral Sum Conjecture
The sum of the measures of the four angles of any quadrilateral is 360°.
Pentagon Sum Conjecture
The sum of the measures of the five angles of any pentagon is 540°.
Polygon Sum Conjecture
The sum of the measures of the n interior angles of an n-gon is180°(𝑛 − 2).
Exterior Angle Sum Conjecture
For any polygon, the sum of the measures of a set of exterior angles is 360°.
Equiangular Polygon Conjecture
You can find the measure of each interior angle of an equiangular n-gon by using
Page 12 of 21
180°(n − 2)
𝑛
.
Geometry Definitions & Conjectures
Conjectures for Chapter 6
Chord Central Angles Conjecture
If two chords in a circle are congruent, then they determine two central angles that are congruent.
Chord Arcs Conjecture
If two chords in a circle are congruent, then their intercepted arcs are congruent.
Perpendicular to a Chord Conjecture
The perpendicular from the center of a circle to a chord is the bisector of the chord.
Chord Distance to Center Conjecture
Two congruent chords in a circle are equidistant from the center of the circle.
Perpendicular Bisector of a Chord Conjecture
The perpendicular bisector of a chord passes through the center of the circle.
Tangent Conjecture
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
Tangent Segments Conjecture
Tangent segments to a circle from a point outside the circle are congruent.
Inscribed Angle Conjecture
The measure of an angle inscribed in a circle is one-half the measure of the central angle.
Inscribed Angles Intercepting Arcs Conjecture
Inscribed angles that intercept the same arc are congruent.
Angles Inscribed in a Semicircle Conjecture
Angles inscribed in a semicircle are right angles.
Cyclic Quadrilateral Conjecture
The opposite angles of a cyclic quadrilateral are supplementary.
Parallel Lines Intercepted Arcs Conjecture
Parallel lines intercept congruent arcs on a circle.
Page 13 of 21
Geometry Definitions & Conjectures
Circumference Conjecture
If C is the circumference and d is the diameter of a circle, then there is a number π such that C = πd. If d =
2r where r is the radius, then C = 2πr.
Arc Length Conjecture
The length of an arc = the circumference times the measure of the central angle divided by 360°.
Conjectures for Chapter 7
Reflection Line Conjecture
The line of reflection is the perpendicular bisector of every segment joining a point in the original figure
with its image.
Coordinate Transformations Conjecture
The ordered pair rule (x, y) → (-x, y) is a reflection over the y-axis.
The ordered pair rule (x, y) → (x, -y) is a reflection over the x-axis.
The ordered pair rule (x, y) → (-x, -y) is a rotation about the origin.
The ordered pair rule (x, y) → (y, x) is a reflection over y = x
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 reflecting point B over line l, drawing segment A ′ B , then drawing segments AC and CB where point
C is the point of intersection of segment A ′ B and line l.
Reflections over Parallel Lines Conjecture
A composition of two reflections over two parallel lines is equivalent to a single translation. In addition,
the distance from any point to its second image under the two reflections is twice the distance between the
parallel lines.
Reflections over Intersecting Lines Conjecture
A composition of two reflections over a pair of intersecting lines is equivalent to a single rotation. The
angle of rotation is twice the acute angle between the pair of intersecting reflection lines.
Tessellating Triangles Conjecture
Any triangle will create a monohedral tessellation.
Page 14 of 21
Geometry Definitions & Conjectures
Tessellating Quadrilaterals Conjecture
Any quadrilateral will create a monohedral tessellation.
Conjectures for Chapter 8
Rectangle Area Conjecture
The area of a rectangle is given by the formula 𝐴 = 𝑏ℎ, where A is the area, b is the length of the base,
and h is the height of the rectangle.
Parallelogram Area Conjecture
The area of a parallelogram is given by the formula 𝐴 = 𝑏ℎ, where A is the area, b is the length of the
base, and h is the height of the parallelogram.
Triangle Area Conjecture
The area of a triangle is given by the formula 𝐴 =
1
2
𝑏ℎ, where A is the area, b is the length of the base,
and h is the height of the triangle.
Trapezoid Area Conjecture
1
The area of a trapezoid is given by the formula 𝐴 = (𝑏1 + 𝑏2 )ℎ where A is the area, b1 and b2 are the
2
lengths of the two bases, and h is the height of the trapezoid.
Kite Area Conjecture
The area of a kite is given by the formula 𝐴 = 𝑑2 𝑑2 where d1 and d2 are the lengths of the diagonals
Regular Polygon Area Conjecture
1
The area of a regular polygon is given by the formula 𝐴 = 2 𝑎𝑠𝑛, 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
1
is the perimeter P, so sn = P. So the formula for area is also: 𝐴 = 2 𝑎𝑃.
Circle Area Conjecture
The area of a circle is given by the formula 𝐴 = 𝜋𝑟 2 , where A is the area and r is the radius of the circle.
Conjectures for Chapter 9
The Pythagorean Theorem
Page 15 of 21
Geometry Definitions & Conjectures
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 & b are the lengths of the legs, and c is the length of the hypotenuse, then 𝑎2 + 𝑏 2 = 𝑐 2 .
Converse of the Pythagorean Theorem
If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right
triangle.
Isosceles Right Triangle Conjecture
In an isosceles right triangle, if the legs have length s, then the hypotenuse has length 2𝑠.
30°-60°-90° Triangle Conjecture
In a 30°-60°-90° triangle, if the shorter leg has length a, then the longer leg has length 𝑎√3, and the
hypotenuse has length 2𝑎.
Distance Formula
For 2 Dimensions:
The distance between points A(𝑥1 , 𝑦1 ) and B(𝑥2 , 𝑦2 ) is given by 𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
For 3 Dimensions:
The distance between A(𝑥1 , 𝑦1 , 𝑧1 ), B(𝑥2 , 𝑦2 , 𝑧2 ), is 𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2
Equation of a Circle
The equation of a circle with radius r and center (h, k) is:
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Conjectures for Chapter 10
Conjecture A
If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula
for the volume is 𝑉 = 𝐵𝐻.
Conjecture B
If B is the area of the base of a right prism (or cylinder) and H is the height of the solid, then the formula
for the volume is 𝑉 = 𝐵𝐻.
Conjecture C
Page 16 of 21
Geometry Definitions & Conjectures
The volume of an oblique prism (or cylinder) is the same as the volume of a right prism (or cylinder) that
has the same base area and the same height.
Prism-Cylinder Volume Conjecture
The volume of a prism or a cylinder is the area of the base multiplied by the height, 𝑉 = 𝐵𝐻.
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 =
𝐵𝐻
3
.
Sphere Volume Conjecture
4
The volume of a sphere with radius r is given by the formula = 3 𝜋𝑟 3 .
Sphere Surface Area Conjecture
The surface area, SA, of a sphere with radius r is given by the formula 𝑆𝐴 = 4𝜋𝑟 2
Conjectures for Chapter 11
Dilation Similarity Conjecture
If one polygon is the image of another polygon under a dilation, then the polygons are similar.
AA Similarity Conjecture
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
SSS Similarity Conjecture
If the three sides of one triangle are proportional to the three sides of another triangle, then the two
triangles are similar.
SAS Similarity Conjecture
If two sides of one triangle are proportional to two sides of another triangle and the included angles are
congruent, then the triangles are similar.
Proportional Parts Conjecture
If two triangles are similar, then the corresponding altitudes, medians, and angle bisectors are proportional
to the corresponding sides.
Angle Bisector/Opposite Side Conjecture
Page 17 of 21
Geometry Definitions & Conjectures
A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the
same ratio as the lengths of the two sides forming the angle.
Proportional Areas Conjecture
If corresponding sides of two similar polygons or the radii of two circles compare in the ratio m/n, then
their areas compare in the ratio
𝑚2
𝑛2
Proportional Volumes Conjecture
If corresponding edges (or radii, or heights) of two similar solids compare in the ratio m n, then their
𝑚3
volumes compare in the ratio of
𝑛3
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 proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to
the third side.
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 proportionally.
Conjectures for Chapter 12
Trigonometric Functions
For Sine (sin), Cosine (cos), and Tangent (tan):
SOHCAHTOA →
sin =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒐𝒓
𝒚
𝒓
;
S = O/H,
cos =
C = A/H,
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒐𝒓
𝒙
𝒓
T = O/A
;
tan =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
𝒐𝒓
𝒚
𝒙
SAS Triangle Area Conjecture
1
The area of a triangle is given by the formula 𝐴 = 2 𝑎𝑏 sin 𝐶, where a and b are the lengths of two sides
and C is the angle between them.
Page 18 of 21
Geometry Definitions & Conjectures
Law of Sines
For a triangle with angles A, B, and C and sides of lengths a, b, and c (a is opposite A, b is opposite B,
and c is opposite C):
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
sin 𝐶
𝑐
.
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,
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶.
Pythagorean Identity
For any angle A: 𝑠𝑖𝑛2 𝐴 + 𝑐𝑜𝑠 2 𝐴 = 1.
Radian Measures
A degree is 1/360th of a complete rotation around a circle. Radians are alternate units used to measure
angles in trigonometry. Just as it sounds, a radian is based on the radius of a circle. One radian
(abbreviated rad) is the angle created by bending the radius length around the arc of a circle. Because a
radian is based on an actual part of the circle rather than an arbitrary division, it is a much more natural
unit of angle measure for upper level mathematics.
Page 19 of 21
Geometry Definitions & Conjectures
In the figure above, points connected by horizontal, dashed lines share the same y-cordinates while
points that are connected by vertical dashed lines share the same x-coordinates.
Page 20 of 21
Geometry Definitions & Conjectures
1st Quadrant
Degrees
Radians
(Decimal)
30
0.5236
45
0.7854
60
1.0472
90
1.5708
2nd Quadrant
(√3/2,1/2)
Radians
Unit Circle
(√3/2,1/2)
(p)
Coords.
Degrees
Radians
(Decimal)
𝜋
3 1
,
2 2
120
2.0944
𝜋
4
𝜋
3
𝜋
2
2 2
,
2 2
135
2.3562
1 3
,
2 2
150
2.6180
0 ,1
180
3.1416
3rd Quadrant
Degrees
Radians
(Decimal)
210
3.6652
225
3.9270
240
4.1888
270
4.7124
Radians
(p)
𝜋
𝜋
4
4𝜋
3
3𝜋
2
Radians
(p)
Unit Circle
Coords.
2𝜋
3
3𝜋
4
𝜋
1 3
− ,
2 2
𝜋
2 2
,
2 2
−
−
3 1
,
2 2
−1, 0
4th Quadrant
Unit Circle
Coords.
Degrees
Radians
(Decimal)
−
3 1
,−
2
2
300
5.2360
−
2
2
,−
2
2
315
5.4978
1
3
− ,−
2
2
330
5.7596
0, −1
360
6.2832
Page 21 of 21
Radians
(p)
Unit Circle
Coords.
𝜋
3
𝜋
4
11𝜋
1
3
,−
2
2
2𝜋
1,0
2
2
,−
2
2
3
1
,−
2
2