Download 3.3 Parallel Lines and the Triangle Angle

Warm-Up
• Find the domain and range:
Geometry Vocabulary & Notation
Point
• Name: Use only the capital letter, without any symbol.
Line
• Name: Use any two points on the line with a line symbol above.
AB
Line Segment
• Name: Use the endpoints of the line segment
with a line segment symbol above.
AB
Ray
• Name: Use the endpoint and one other point on
the ray with a ray symbol above.
AB
Plane
• Name: Use the word Plane followed by any three
non-collinear points.
PlaneACB
Angles
• Name: Use three letters with the vertex in the
middle preceded by the angle symbol.
B
A
1
C
BAC
A
1
Acute Angle
• An angle that measures less than 90 degrees.
Obtuse Angle
• An angle that measures greater than 90
degrees but less than 180 degrees.
Right Angle
• An angle that measures 90 degrees
Naming Geometric Figures
• An angle can be named using three points. The
middle point must be the vertex. How many
different angles can you name from this diagram?
AEB
AEC
AED
BEC
BED
CED
Measure or Length
• Measure of an angle: 𝑚∠𝐴𝐵𝐶
• Length of a segment: 𝐴𝐵
Congruent Segments
In geometry, two segments with the same length are called
________
congruent
_________
segments
Definition of
Congruent
Segments
Two segments are congruent if and only if
________________________
they have the same length
Congruent Segments
In the figures at the right, AB is
congruent to BC, and PQ is
B
A
congruent to RS.
The symbol  is used to
represent congruence.
AB  BC, and PQ  RS.
R
C
Congruent Segments
Since congruence is related to the equality of segment measures, there are
properties of congruence that are similar to the corresponding properties
of equality.
theorems
These statements are called ________.
Theorems are statements that can be justified by using logical reasoning.
2–1
Congruence of segments is
reflexive.
AB  AB
2–2
Congruence of segments is
symmetric.
If AB  CD, then CD  AB
2–3
Congruence of segments is
transitive.
If AB  CD, and CD  EF
then AB  EF
Congruent Segments
A point M is the midpoint of a segment
between S and T and SM = MT
Definition of
Midpoint
S
M
ST
if and only if M is
T
SM = MT
The midpoint of a segment separates the segment into two segments of
equal _____.
length
_____
congruent
So, by the definition of congruent segments, the two segments are _________.
§3.3 The Angle Addition Postulate
The bisector of an angle is the ray with its endpoint at the
vertex of the angle, extending into the interior of the
angle.
The bisector separates the angle into two angles of equal
measure.
Definition of
an Angle
Bisector
P
QA
1
Q
2
is the bisector of PQR.
A
m1 = m2
R
Types of Triangles
• Classifying by Sides
– Equilateral
• All sides are congruent
– Isosceles
• Two sides congruent
– Scalene
• No sides congruent
Types of Triangles
• Classifying by Angles
– Equiangular
• All angles congruent
– Acute
• All angles less than 90
– Right
• One angle 90
– Obtuse
• One angle greater than 90
Isosceles Triangles
• Isosceles Triangle
– A triangle with two congruent sides
• Legs
– Two congruent sides of an isosceles
triangle
• Base
– Non-congruent side of an isosceles
triangle
• Vertex angle
– Angle across from the base
• Base angles
– Two congruent angles, across from
the legs.
Isosceles Triangle Theorem
• If two sides of a triangle are congruent, then
the angles opposite those sides are congruent.
• Example: If 𝐴𝐶 ≅ 𝐶𝐵, then ∠𝐴 ≅ ∠𝐵
C
A
B
Converse of Isosceles Triangle
Theorem
• If two angles of a triangle are congruent, then
the sides opposite those angles are congruent.
• Example: If ∠𝐴 ≅ ∠𝐵, then 𝐴𝐶 ≅ 𝐶𝐵
C
A
B
A triangle is equilateral if and only if it is equiangular.
Triangle Angle-Sum Theorem
• The sum of the measures of the angles of a
triangle is 180 degrees.
2
1
3
3
3
2
1
1
2
1
2
3
Exterior Angles of a Polygon
• An exterior angle is formed by a side and an
extension of an adjacent side.
3  exterior angle
1
2
Two remote interior angles
• Remote interior angles: for each exterior angle if
a triangle, there are two nonadjacent interior
angles.
Triangle Exterior Angle Theorem
• The measure of each exterior angle of a
triangle equals the sum of the measures of its
two remote interior angles.
𝑚∠1 = 𝑚∠2 + 𝑚∠3
2
1
3