Download Geometry Guide - Canvas by Instructure

Geometry Guide
Basic Terms and Definitions
Term
Point
Line
Plane
Space
Collinear
Coplanar
Segment
Ray
Opposite rays
Intersection
Congruent
Congruent segments
Segment midpoint
Segment bisector
Angle
Acute angle
Obtuse angle
Right angle
Straight angle
Congruent angles
Angle bisector
Adjacent angles
Vertical angles
Definition
A location or position. A point has no dimension. A point has no
length, width, or height.
An infinite set of points that extend in two opposite directions. A line
has length (infinite), but no width, or thickness.
An infinite set of points that form a flat surface extending in all
directions. A plane has length and width (infinite), but no thickness.
The set of all points.
Collinear points lie on the same line. Noncollinear points do not lie on
the same line.
Coplanar points lie on the same plane. Noncoplanar points do not lie
on the same plane.
A section of a line designated by two endpoints and the set of all points
between them.
A section of a line with one endpoint and extending in one direction.
Two rays with the same endpoint that form a straight line.
The set of points in both objects.
Two objects are congruent if they have the same size and shape.
Two segments are congruent if they have the same length.
A midpoint divides a segment into two ≅ segments.
A line, ray, or segment that intersects a segment at its midpoint.
A figure formed by two rays with the same endpoint. The endpoint is
the vertex. The rays form the sides of the angle.
∠A is acute if 0 < m∠A < 90
∠A is obtuse if 90 < m∠A < 180
∠A is a right angle if m∠A = 90
∠A is a straight angle if m∠A = 180
Two angles are congruent if they have equal measure.
A ray that divides an angle into two congruent angles.
Two angles are adjacent if they have:
1. Common vertex
2. Common side
3. No points in common
The two angles “across” from each other at the intersection of lines.
1
Basic Postulates and Theorems about Points, Lines, and Planes:
•
•
•
A line contains at least 2 points. (P.5)
A plane contains at least 3 noncollinear points. (P.5)
Space contains at least 4 points not all in one plane. (P.5)
•
•
Through any 2 points there is exactly one line. (P.6)
Through any 3 noncollinear points there is exactly one plane. (P.7)
•
If 2 points are in a plane, then the line formed by the points is in the plane. (P.8)
•
If 2 planes intersect, then their intersection is a line. (P.9)
The intersection of two planes is a line.
•
If 2 lines intersect, then they intersect in one point. (1.1)
The intersection of two lines is a point.
•
If 2 lines intersect, then one plane contains the lines. (1.3)
•
Through a line and a point not on the line there is exactly one plane. (1.2)
2
Segment Concepts:
•
•
A
C
SAP: Segment Addition Postulate: (P.2)
AC + CB = AB
“The sum of the parts equals the whole.”
Midpoint Theorem: (2.1)/Definition of Midpoint:
If M is midpoint of AB, then it divides AB into 2 ≅ segments.
If M is midpoint of AB, AM ≅ MB.
If AM ≅ MB, then M is midpoint of AB.
M
B
If M is midpoint of AB, then, AM = ½ AB and MB = ½ AB.
If AM = ½ AB or MB = ½ AB, then M is midpoint of AB.
A
Angle Concepts:
•
•
•
•
X
AAP: Angle Addition Postulate: (P.4)
m∠ABX + m∠XBC = m∠ABC.
B
“The sum of the parts equals the whole.”
C
Bisector Theorem: (2.2)/Definition of Bisector:
If BX bisects ∠ABC, then ∠ABX ≅ ∠XBC.
If ∠ABX ≅ ∠XBC, then BX bisects ∠ABC.
If BX bisects ∠ABC, then m∠ABX = ½ m∠ABC and m∠XBC = ½ m∠ABC.
If m∠ABX = ½ m∠ABC or m∠XBC = ½ m∠ABC, then BX bisects ∠ABC.
Definition of Supplementary ∠s:
If ∠1 and ∠2 supplementary, then m∠1 + m∠2 = 180.
If m∠1 + m∠2 = 180, then ∠1 and ∠2 supplementary.
Definition of Complementary ∠s:
If ∠3 and ∠4 complementary, then m∠3 + m∠4 = 90.
If m∠3 + m∠4 = 90, then ∠3 and ∠4 complementary.
•
Vertical angles are ≅. (2.3)
•
•
If 2 ∠s are supplements to ≅ angles (or to the same angle), then the 2 ∠s are ≅. (2.7)
If 2 ∠s are complements to ≅ angles (or to the same angle), then the 2 ∠s are ≅. (2.8)
3
Properties from Algebra:
•
Addition/subtraction property of equality.
If a = b , then a + c = b + c . (add the same thing to both sides).
If a = b and c = d , then a + c = b + d . (add equals to both sides).
•
Multiplication/division property of equality.
If a = b , then ac = bc . (multiply both sides by the same/equal thing.)
a b
If a = b , then = (divide both sides by the same/equal thing.)
c c
•
Distributive property.
a(b + c) = ab + ac
•
Substitution
If a + b = c and a = d , then d + b = c . (replace d for a)
If a + b = z and x + y = z , then a + b = x + y . (two expressions equal to same thing)
•
Transitive
If a = b and b = c , then a = c . (for equality)
If a ≅ b and b ≅ c , then a ≅ c . (for congruence)
•
Reflexive
a=a
•
Symmetric
a = b and b = a .
4
Perpendicular Concepts:
•
Definition of ⊥ lines:
If 2 lines are ⊥, then they form right ∠s (90 degree ∠s).
If 2 lines form right ∠s (90 degree ∠s), then they are ⊥.
•
•
If two lines ⊥, then they form ≅ adjacent ∠s. (2.4)
If 2 lines form ≅ adjacent ∠s, then they are ⊥. (2.5)
•
If the ext. sides of 2 adjacent acute ∠s are ⊥, then the ∠s are complementary. (2.6)
Two adjacent angles are complementary if their exterior sides are ⊥.
5
Parallel Concepts:
•
Definition of parallel lines:
2 coplanar lines that do not intersect (railroad tracks).
•
Properties of parallel lines:
If 2 lines ||, then …
1. Corresponding ∠s ≅. (P.10)
2. Alternate interior ∠s ≅. (3.2)
3. Same-side interior ∠s are supplementary. (3.3)
4. If one || line is ⊥ to the transversal, then other line is also ⊥ to the transversal.
(3.4)
•
Proving lines parallel:
1. If corresponding ∠s ≅, then lines ||. (P.11)
2. If alternate interior ∠s ≅, then lines ||. (3.5)
3. If same side interior ∠s supplementary, then lines ||. (3.6)
4. If 2 lines ⊥ to the same line, then lines ||. (3.7)
5. If 2 lines || to the same line, then lines ||. (3.10)
•
Through a point outside a line, there is exactly one line || to the line. (3.8)
•
Through a point outside a line, there is exactly one line ⊥ to the line. (3.9)
Basic Terms and Definitions
Term
Parallel lines
Transversal
Corresponding angles
Alternate interior angles
Same side interior angles
Skew lines
Definition
2 coplanar lines that do not intersect (railroad tracks).
A line that intersects 2 parallel lines.
2 angles that have the same relative position to || lines.
2 angles on alternate sides of the transversal and interior to the || lines.
2 angles on the same side of the transversal and interior to the || lines.
2 non-coplanar lines that do not intersect.
6
Triangle Concepts:
Basic Terms and Definitions
Term
Triangle
Vertex
Sides
Exterior ∠ of a Δ
Definition
Figure formed by 3 segments joining 3 noncollinear points.
3 points of a triangle.
3 line segments of a triangle.
Angle formed when side of a triangle is extended.
Scalene Δ
Isosceles Δ
Equilateral Δ
Triangle with 3 different length sides.
Triangle with at least 2 sides congruent.
Triangle with 3 sides congruent.
Acute Δ
Obtuse Δ
Equiangular Δ
Triangle with 3 acute angles (all < 90).
Triangle with one obtuse angle (>90 and <180).
Triangle with 3 congruent angles.
Right Δ
Legs of right Δ
Hypotenuse of right Δ
Triangle with one right angle (=90).
The ⊥ sides of a right Δ. The sides that form the right angle.
The side opposite the right ∠ of a right Δ.
Median of Δ
Altitude of Δ
Perpendicular bisector
Segment from vertex to midpoint of opposite side.
⊥ segment from vertex to opposite side/line.
Line, segment, or ray ⊥ to a segment through its midpoint.
•
Angle Sum Theorem (AST): The sum of the ∠s of a Δ = 180. (3.11)
o Corrollaries:
1. If 2 ∠s of one Δ ≅ to 2 ∠s of another Δ, then third ∠s are ≅.
2. Each ∠ of an equiangular Δ = 60.
3. The acute ∠s of a right Δ are complementary.
4. In a Δ, there can be at most one right ∠ or obtuse ∠.
•
Exterior Angle Theorem (EAT):
An exterior angle of a triangle equals sum of the 2 remote interior ∠s. (3.12)
•
Definition of congruent triangles:
Biconditional definition:
Two triangles are congruent if an only if all the corresponding parts (all 3 angles
and all 3 sides) of the triangles are congruent.
Definition as two statements that are converses:
a. If two triangles are congruent, then all the corresponding parts (all 3 angles
and all 3 sides) of the triangles are congruent.
b. If all the corresponding parts (all 3 angles and all 3 sides) of the triangles are
congruent, then the two triangles are congruent.
7
•
Postulates/theorems for proving triangles congruent:
1. SSS (P.12)
2. SAS (P.13)
3. ASA (P.14)
4. AAS (4.3)
5. HL (4.4) (Right Δs only.)
•
CPCTC: Corresponding parts to congruent triangles are congruent.
Once 2 triangles are proved congruent, all the corresponding parts (angles and sides)
are congruent by CPCTC.
•
Isosceles Triangle Theorem (ITT/BAT) (4.1): Base angles of an isosceles Δ ≅.
If 2 sides of a triangle are ≅, then the angles opposite those sides are congruent.
o Corollaries:
1. An equilateral triangle is also equiangular.
2. An equilateral triangle has three 60 degree angles.
3. A bisector of the vertex angle of an isosceles triangle is ⊥ to the base at its
midpoint.
•
Isosceles Triangle Theorem Converse (ITTC/BATC) (4.2):
If 2 angles of a triangle are congruent, then the opposite sides are congruent.
o Corollary:
1. An equiangular triangle is equilateral.
•
Distance from a point to a line:
The length of the ⊥ segment from the point to the line.
•
Segment perpendicular bisector theorems/properties:
o If a point is on the ⊥ bisector of a segment, then it is equidistant from the segment
endpoints. (4.5)
o If a point is equidistant from the segment endpoints, then it is on the ⊥ bisector.
(4.6)
•
Angle bisector theorems/properties:
o If a point lies on the bisector of an angle, then it is equidistant from the sides of
the angle. (4.7)
o If a point is equidistant from the sides of an angle, then it lies on the bisector of
the angle. (4.8)
8
Parallelograms:
•
Definition of parallelogram:
A quadrilateral with both pairs of opposites sides parallel.
•
Properties of parallelograms:
1. Opposite sides of a parallelogram ≅. (5.1)
2. Opposite angles of a parallelogram ≅. (5.2)
3. Diagonals of a parallelogram bisect each other. (5.3)
4. Adjacent or consecutive angles are supplementary.
5. The sum of the ∠s of a quadrilateral = 360.
A
B
1
4
C
2
3
D
•
Proving quadrilaterals are parallelograms:
A quadrilateral is a parallelogram if …
1. Both pairs of opposite sides are ||. (Definition of parallelogram.)
2. Both pairs of opposite sides are ≅. (5.4)
3. Both pairs of opposite ∠s are ≅. (5.6)
4. The diagonals bisect each other. (5.7)
5. One pair of opposite sides are || and ≅. (5.5)
•
Miscellaneous parallelogram theorems:
o If 2 lines are ||, then all points on one line are equidistant from the other line. (5.8)
All points on one || line are equidistant from the other || line.
o If 3 || lines cut off ≅ segments on one transversal, then they cut off ≅ segments on
every transversal. (5.9)
o A line that bisects one side of a Δ and is || to another side bisects the third side.
(5.10)
o The segment that joins the midpoints of 2 sides of a Δ is (5.11)
1. || to the third side.
2. ½ as long as the third side.
9
•
Special Parallelograms:
o Rectangle: A quadrilateral with 4 right angles. Every rectangle is a parallelogram
since both pairs of opposite angles are ≅.
o Rhombus: A quadrilateral with 4 ≅ sides. Every rhombus is a parallelogram since
both pairs of opposite sides are ≅.
o Square: A quadrilateral with 4 right angles and 4 ≅ sides. Every square is a
parallelogram, a rectangle, and a rhombus.
•
Theorems about special parallelograms:
o Rectangle theorems:
• The diagonals of a rectangle are ≅. (5.12)
• If one angle of a parallelogram is a right angle, it is a rectangle. (5.16)
o Rhombus theorems:
• The diagonals of a rhombus are ⊥. (5.13)
• The diagonals of a rhombus bisect the angles. (5.14)
• If 2 consecutive sides of a parallelogram are ≅, it is a rhombus. (5.17)
o Relating to Δs:
• The midpoint of the hypotenuse of a right Δ is equidistant from the 3
vertices. (5.15)
Trapezoids:
• Definition of Trapezoid:
A quadrilateral with exactly one pair of sides parallel.
•
Terms:
o Bases: The parallel sides.
o Legs: The other sides.
o Median of a trapezoid: The segment joining the midpoints of the legs.
o Isosceles trapezoid: A trapezoid with congruent legs.
•
Trapezoid theorems:
o Base angles of an isosceles trapezoid are ≅. (5.18)
o The median of a trapezoid, EF, is (5.19)
1. || to the bases.
2. length is ½ the sum of the base lengths.
1
EF = ( BC + AD )
2
B
E
A
10
C
F
D
Similarity Concepts:
• Cross products are equal
a c
If = , then ad = bc .
b d
•
Definition of Similar Triangles:
Two triangles are similar if and only if all 3 corresponding ∠s are ≅ and all 3 sides
are in proportion.
D
A
ΔABC~ΔDEF
AB BC AC
=
=
DE EF DF
B
E
C
F
•
Proving triangles are similar:
1. AA (P.15)
2. SSS (7.2)
3. SAS (7.1)
•
Triangle Proportionality Theorem: (7.3)
A line || to one side of a triangle intersects the other two sides and divides the sides
proportionally.
A
E
D
AD AE
=
DB EC
B
•
C
Triangle Proportionality Corollary:
If 3 parallel lines intersect 2 transversals, then they divide the transversals
proportionally.
A
AB DE
=
BC EF
D
B
E
C
•
F
Triangle Angle Bisector Theorem (7.4)
A ray that bisects an angle of a triangle divides the opposite side into segments
proportional to the other 2 sides.
A
BD AB
BD DC
or
=
=
AB AC
DC AC
B
11
D
C
Right Triangle Concepts:
• Simplifying Radicals
o
xy = x ⋅ y
x
x
=
y
y
o Rationalizing the denominator
5
3 5 3
⋅
=
3
3 3
o
•
Pythagorean Theorem: (8.2)
a2 + b2 = c2
c
a
legs: a, b
hypotenuse: c
•
b
Special triangles
45/45/90 triangles: (8.6)
Ratio of sides:
a:a:a 2
1:1: 2
a 2
a
hypotenuse = side ⋅ 2
hypotenuse
side =
2
a
30/60/90 triangles: (8.7)
Ratio of sides:
a : 2a : a 3
1: 2 : 3
hypotenuse = short side ⋅ 2
hypotenuse
short side =
2
long side = short side ⋅ 3
long side
short side =
3
•
Pythagorean Triples
o 3, 4, 5
o 5, 12, 13
o 7, 24, 25
12
2a
a 3
a
•
Right triangle parts
Opposite side
Adjacent side
Hypotenuse
H
O
x
A
•
Trigonometry functions
SOH CAH TOA
•
•
sin x =
O
H
cos x =
A
H
tan x =
O
A
Use the sin, cos, tan buttons when you know the angle involved.
Use the sin-1, cos-1, tan-1 buttons when you need to find an unknown angle.
13
Circle Concepts:
• Terms:
Circle
Radius
Chord
Diameter
Secant
Tangent
Minor arc
Major arc
Central angle
Inscribed angle
•
A tangent line is ⊥ to the radius at the point of tangency. (9.1)
•
Tangent segments to a circle from a point outside the circle are ≅. (9.2)
•
Arc Addition Postulate: (P.16)
The measure of an arc formed by two adjacent arcs is the sum of the measures of the
two arcs.
A
!
m!
AC = m !
AB + mBC
B
C
•
Central Angles:
The measure of an arc is equal to the measure of its central angle.
The measure of a central angle is equal to the measure of the intercepted arc.
•
In the same or ≅ circles, two minor arcs are ≅ if their central angles are ≅. (9.3)
•
Congruent Chords Theorem: (9.4)
In the same circle, congruent arcs have congruent chords.
In the same circle, congruent chords have congruent arcs.
•
If a diameter or radius is ⊥ to a chord, then it bisects the chord and its arc. (9.5)
14
•
Chord Distance Theorem: (9.6)
In the same circle, two chords equal distance from the center are congruent.
In the same circle congruent chords are equal distance from the center.
•
Inscribed Angle Theorem: (9.7)
The measure of an inscribed angle is equal to ½ the intercepted arc.
1
m∠x = m !
AB
2
B
A
x
o Corollaries:
1. Inscribed angles that intercept the same arc are congruent.
2. An angle inscribed in a semicircle is a right angle.
3. Opposite angles of a quadrilateral inscribed in a circle are supplementary.
•
The measure of an angle formed by a chord and a tangent is equal to ½ the
intercepted arc. (9.8)
A
1
m∠x = m !
AB
2
x
B
•
When two chords intersect inside a circle, the measure of the angle formed is ½ the
SUM of the two intercepted arcs. (9.9)
m∠x =
1
(a + b)
2
a
15
x
b
•
When two secants, two tangents, or a secant and a tangent form an angle with the
vertex outside the circle, the measure of the angle formed is ½ the
DIFFERENCE of the two intercepted arcs. (9.10)
m∠x =
1
(a − b)
2
m∠x =
1
(a − b)
2
a
x
b
x
b
a
m∠x =
1
(a − b)
2
a
16
b
x
•
When two chords intersect inside a circle, the product of the segments of one chord
equals the product of the segments of the other chord. (9.11)
a ⋅b = c⋅d
a
d
c
b
•
When two secants are drawn to a circle from an external point, the product of one
secant segment and its external segment equals the product of the other secant
segment and its external segment. (9.12)
a
a ⋅b = c⋅d
b
d
c
•
When a secant segment and a tangent segment are drawn to a circle from an
external point, the product of the secant segment and its external segment is
equal to the square of the tangent segment (9.13).
a
a ⋅ b = c2
b
c
17