Download Geometry Terminology

1.1 Statements and Reasoning
• Statement – group of words/symbols
which is either true or false.
• Examples of geometric statements:
 mA = 80º
 mB + mC = 80º
ABC is a right triangle
Line l is parallel to line m
1.1 Statements and Reasoning
• Deduction – the truth of the conclusion is
guaranteed. Example:
– If p then q
– p
– Therefore q
• Induction – the truth is not guaranteed. Example:
– 3, 5, and 7 are odd numbers that are prime
– Therefore all odd numbers are prime
• Geometric proofs use deductive logic
1.2 Informal Geometry and
Measurement
 Point – represented by a dot
A
x
B
Line – with arrows on each end
Ray – with an arrow on one end
Collinear – 3 points are collinear
if they are on the same line.
In between – x is in between A
and B
C1.2
Informal Geometry and Measurement
1
B
A
D
Angle – may be referred to as
ABC, B, or 1
Triangle – referred to as DEF
E
F
C
B
A
Line segment – referred to as BC ,
BC = length of the segment
B
C
Midpoint – if AB = BC, since B is
between A and C, B is the midpoint
1.2 Informal Geometry and
Measurement
• Congruence (denoted by )
– Two segments are congruent if they have
the same length
– Two angles are congruent if they have the
same measure
• Bisect – to divide into 2 equal parts
1.2 Informal Geometry and
Measurement
• Bisecting a segment
into 2 congruent
segments
• Bisecting an angle
into 2 congruent
angles
1.3 Early Definitions and Postulates
• Definitions – terminology of the
mathematical system is defined.
Examples:
– Isosceles triangle – a triangle that has 2
congruent sides
– Line segment – consists of the 2 points
(endpoints) and all the points between them
1.3 Early Definitions and Postulates
• Postulates – assumptions necessary to build
the mathematical system.
Examples:
– Postulate 1: Through 2 distinct points,
there is exactly one line.
– Postulate 2:The measure of any line
segment is a unique positive number
1.3 Early Definitions and Postulates
A
B
C
• Segment addition –
AB + BC = AC
• If AC = 8 and
BC = 5, what is
length AB?
1.4 Angles and their relationships
A
D
B
C
• Angle addition mABD + mDBC =
mABC
• If ABC = 130º and
mDBC = 50º , what
is mABD?
1.4 Angles and their relationships
• Acute angle –
0 < x < 90
• Right angle - 90
• Obtuse angle –
90 < x < 180
• Straight angle - 180
1.4 Geometry Terminology –
Pairs of Angles
• Complementary
angles – add up to 90
• Supplementary angles
– add up to 180
• Vertical angles – the
angles opposite each
other are congruent
1.5 Introduction to Geometric Proof
• Form of a geometric proof:
Statements
1. mABC = 80º
Reasons
1. Given
2. ABC and DBE
are vertical angles
3. mDBE = 80º
2. Given
3. Vertical angles have
equal measure
1.5 Introduction to Geometric Proof
• Examples of Reasons:
– Given (use first)
– Definitions (like “definition of bisector”)
– Properties (like “corresponding angles are
congruent”)
– Postulates and theorems (like “segment
addition”)
1.5 Introduction to Geometric Proof
• Properties of equality:
– Reflexive (also referred to as “identity”):
a=a
– Symmetric:
if a = b then b = a
– Transitive:
if a = b and b = c, then a = c
1.5 Introduction to Geometric Proof
• Properties of congruence:
– Reflexive (also referred to as “identity”):
1  1
– Symmetric:
if 1  2 then 2  1
– Transitive:
if 1  2 and 2  3 , then 1  3
2.1 Parallel Lines – Special Angles
• Intersection – 2 lines
intersect if they have one
point in common.
• Perpendicular – 2 lines are
perpendicular if they
intersect and form right
angles
• Parallel – 2 lines are parallel
if they are in the same plane
but do not intersect
2.1 Parallel Lines – Special Angles
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following congruent pairs of angles are formed:
– Corresponding angles:1 & 5, 2 & 6, 3
& 7, 4 & 8
– Alternate interior angles: 4 & 5, 3 & 6
– Alternate exterior angles: 1 & 8, 2 & 7
2.1 Parallel Lines – Special Angles
1 2
3 4
5 6
7 8
• When 2 parallel lines are cut by a transversal the
following supplementary pairs of angles are
formed:
– Same side interior angles: 3 & 5, 4 & 6
– Same side exterior angles: 1 & 7, 2 & 8
2.1 Parallel Lines – Special Angles
• Terminology:
– Corresponding angles – in the same relative
“quadrant” (upper right, lower left, etc.)
– Alternate – on opposite sides of the transversal
– Same side – on the same side of the transversal
– Interior – in between the 2 parallel lines
– Exterior – outside the 2 parallel lines
2.3 Parallel Lines – Review
1 2
3 4
5 6
7 8
• What type of angles are:
– 1 & 8
– 4 & 6
– 4 & 5
– 2 & 6
– 1 & 7
2.3 Parallel Lines – Review
• If 2 lines are parallel and cut by a transversal:
– Corresponding angles, alternate interior angles,
and alternate exterior angles are congruent
– Same-side interior angles and same-side
exterior angles are supplementary
2.3 Proving Lines Are Parallel
• Given two lines cut by a transversal, if any one of
the following are true:
– Corresponding angles, alternate interior angles,
or alternate exterior angles are congruent
– Same-side interior angles or same-side exterior
angles are supplementary
• Then the two lines are parallel
2.3 Proving Lines Are Parallel:
2 More Theorems
• Two lines parallel to the same line must be parallel
if l m and m n then l n
• Two lines perpendicular to the same line must be
parallel
if l  m and m  n then l n
2.4 The Angles of a Triangle
• Triangles classified by number of congruent sides
Types of triangles
# sides congruent
scalene
0
isosceles
2
equilateral
3
2.4 The Angles of a Triangle
• Triangles classified by angles
Types of triangles
Angles
acute
All angles acute
obtuse
One obtuse angle
right
One right angle
equiangular
All angles congruent
2.4 Angles of a Triangle
• In a triangle, the sum of the interior angle
measures is 180º
(mA + mB + mC = 180º)
A
C
B
2.4 The Angles of a Triangle
• The measure of an exterior angle of a triangle
equals the sum of the measures of the 2 nonadjacent interior angles - m1 + m2 = m4
2
1
3
4
• Question: What do you call a parrot who
just died?
2.5 Convex Polygons
• Polygon - a closed plane figure whose sides
are line segments that intersect
only at the endpoints
• Regular Polygon – a polygon with all sides
equal length and all interior angles equal
measure
2.5 Convex Polygons
• Concave polygons: A line segment can be
drawn between 2 points and the segment is
outside the polygon
• Convex polygons: A polygon that is not
concave
2.5 Convex Polygons
• Classified by number of sides
Polygons
triangle
quadrilateral
# of sides
3
4
pentagon
hexagon
heptagon
octagon
5
6
7
8
Polygons
nonagon
decagon
dodecagon
15-gon
n-gon
# of
sides
9
10
12
15
n
2.5 Convex Polygons
• Formulas for polygons
Sides Diagonals
n
n  (n  3)
D
2
Sum of the
measures of
interior angles
Si  (n  2) 180
Sum of the
measures of
exterior angles
Se  360
2.5 Convex Polygons
• Formulas for regular polygons
Sides Measure of interior
angle
(n  2) 180
n
I
n
Si  (n  2) 180
Measure of exterior
angle
360
E
n
Se  360
3.1 Congruent Triangles
• ABC  DEF if all 3 angles are
congruent and all 3 sides are congruent.
• This means
– AB = DE, BC = EF, and AC = DF
– ABC   DEF, BAC   EDF and
ACB   DFE
3.1 Congruent Triangles
• Included/opposite sides and angles for ABC are:
– A is opposite side BC
– A is included by sides AB and AC
– Side AB is opposite C
– Side AB is included by A and B
C
B
A
3.1 Congruent Triangles
• SSS – If the 3 sides of a triangle are
congruent to the 3 sides of a second
triangle, then the triangles are congruent
• SAS – If 2 sides and the included angle of
one triangle are congruent to two sides and
the included angle of a second triangle, then
the triangles are congruent.
3.1 Congruent Triangles
• ASA - If 2 angles and the included side of a
triangle are congruent to the two angles and
included side of a second triangle, then the
triangles are congruent.
• AAS - If two angles and the non-included
side of a triangle are congruent to 2 angles
and the non-included side of another
triangle, the triangles are congruent
3.1 Congruent Triangles
• Right Triangle – In a right triangle, the side
opposite the right angle is the hypotenuse and the
sides of the right angle are the legs of the right
triangle.
• HL (hypotenuse-leg) – If the hypotenuse and a leg
of one right triangle are congruent to the
hypotenuse and leg of another right triangle, then
the triangles are congruent.
Note: In the book this is introduced in section 3.2.
3.1 Congruent Triangles
• To show congruence of triangles:
Valid
SSS
SAS
ASA
AAS
HL
Invalid
AAA
SSA
3.2 Corresponding Parts of
Congruent Triangles are Congruent
• CPCTC – Corresponding Parts of Congruent
Triangles are Congruent
• Proofs using CPCTC:
– Recognize that what you are trying to prove
involves corresponding parts of 2 triangles
– Show the triangles are congruent by SSS, SAS,
ASA, AAS, etc.
– State the conclusion with reason “CPCTC”
3.3 Isosceles Triangles
Vertex
• Parts of the
isosceles triangle:
Vertex Angle
Leg
Leg
Base
Base Angles
3.3 Isosceles Triangles
• 2 sides (legs) of an Isosceles triangle are 
(by definition)
• 2 angles (base angles) of a Isosceles
triangle are 
3.3 Equilateral Triangles
• An equilateral triangle is also equiangular
• An equiangular triangle is also equilateral
• Each angle of an equilateral triangle
measures 60
60
60
60
3.3 Triangle Terminology
• Angle bisector: divides an angle of the
triangle into two equal angles
• Median: segment that connects a vertex of a
triangle to the midpoint of the other side
3.3 Triangle Terminology
• Altitude: line segment drawn from the vertex of a
triangle that is perpendicular to the opposite side
(note: the altitude can be outside the triangle)
• Perpendicular bisector: (of a side of a triangle) is
the line that intersects the midpoint of the side and
is perpendicular to the side
3.4 Three Basic Constructions
• Construct the perpendicular bisector (first half of
problem 15)
• Construct the angle bisector (problem 9 and
second half of problem 15)
• Construct an angle with the same measure with a
given ray/segment as one of the sides (problem 7)
• Note: trick to get a 60 degree angle is to construct
an equilateral triangle
3.5 Inequalities in a Triangle
• The angle opposite the larger side is the
bigger angle.
In ABC, if AB > AC then m C > m B
A
C
B
3.5 Inequalities in a Triangle
• The side opposite the larger angle is the
bigger side.
In ABC, if m C > m B then AB > AC
A
C
B
3.5 Inequalities in a Triangle
• Triangle Inequality: The sum of the lengths of any
two sides of a triangle is greater than the length of
the third side
In ABC, CA + AB > BC
A
C
B