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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.
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Basic Postulates and Theorems about Points, Lines, and Planes:
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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)
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Through any 2 points there is exactly one line. (P.6)
Through any 3 noncollinear points there is exactly one plane. (P.7)
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If 2 points are in a plane, then the line formed by the points is in the plane. (P.8)
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If 2 planes intersect, then their intersection is a line. (P.9)
The intersection of two planes is a line.
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If 2 lines intersect, then they intersect in one point. (1.1)
The intersection of two lines is a point.
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If 2 lines intersect, then one plane contains the lines. (1.3)
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Through a line and a point not on the line there is exactly one plane. (1.2)
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Segment Concepts:
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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:
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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.
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Vertical angles are ≅. (2.3)
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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)
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Properties from Algebra:
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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).
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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
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Distributive property.
a(b + c) = ab + ac
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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)
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Transitive
If a = b and b = c , then a = c . (for equality)
If a ≅ b and b ≅ c , then a ≅ c . (for congruence)
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Reflexive
a=a
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Symmetric
a = b and b = a .
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Perpendicular Concepts:
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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 ⊥.
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If two lines ⊥, then they form ≅ adjacent ∠s. (2.4)
If 2 lines form ≅ adjacent ∠s, then they are ⊥. (2.5)
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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 ⊥.
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Parallel Concepts:
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Definition of parallel lines:
2 coplanar lines that do not intersect (railroad tracks).
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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)
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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)
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Through a point outside a line, there is exactly one line || to the line. (3.8)
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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.
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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.
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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 ∠.
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Exterior Angle Theorem (EAT):
An exterior angle of a triangle equals sum of the 2 remote interior ∠s. (3.12)
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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.
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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.)
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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.
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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.
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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.
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Distance from a point to a line:
The length of the ⊥ segment from the point to the line.
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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)
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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)
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