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Section 5: Triangles – Part 1
S
ection
Triangles
Part 1
Topic 1:
Introduction to Triangles
Part 1
Topic 2:
Introduction to Triangles
Part 2
Topic 3:
Triangles
i
n the Coordinate Plane
Topic
Triangle Congruence
SSS and SAS
Part 1
Topic
Triangle Congruence
SSS and SAS
Part 2
Topic
Triangle Congruence
ASA and AAS
Part 1
Topic
Triangle Congruence
ASA and AAS
Part 2
Topic
Base Angle of
Isosceles Triangles
Topic
Using
the
Definition of Triangle
C
ongruence in Terms of Rigid Motions
Topic
Using Triangle Congruency to
F
ind Missing Variables
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Section 5: Triangles – Part 1
The following
Mississippi College
and Career
Readiness Standards for Mathematics
will be covered in this section:
G
CO
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion
G congruent. on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are
CO
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
G corresponding pairs of sides and corresponding pairs of angles are congruent.
CO
Explain how the criteria for tri
angle
congruence (ASA, SAS, and SSS
) follow from the definition of congruence in
G terms of rigid motions.
CO
Prove theorems
about triangles
Theorems include: measures of interior angles of a triangle sum to 180
; base
angles of isosceles
triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third
G side and half the length; the medians of a triangle meet at a point.
GPE.
Prove the slope criteria for parallel and perpendicular lines and use t
hem to solve geometric problems (e.g.,
G find the equation of a line parallel or perpendicular to a given line that passes through a given point).
GPE.
Use coordinates to compute t
he perimeter
s
of polygons and areas of triangles and rectangles
, e.g., using the
G distance formula. *
SRT
Prove theorems about
triangles.
Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and
conversely; the Pythagorean Theorem proved using triangle similarity.
G
SRT
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
- Modeling Standards figures.
Section 5: Triangles – Part 1
- Let’s Practice!
a. Consider the figure below.
After connecting the points on the plane,
Marcos
claims that angle
is a right angle. Is Marcos correct?
Explain your reasoning.
b.
plane byHow can you classify a triangle on the coordinate
its
sides?
A
B
C
- 7r\ ,t!
a. Consider the triangle below.
If
ȟ
ܨܧܦ
is an isosceles triangle with base
, what is
the value of
? Justify your answer.
b.
What is the length of each leg?
c.
What is the length of the base?
plane is a right triangle?How can you determine if a triangle on the coordinate
Section 5: Triangles – Part 1
Section
Topic 2
Introduction to Triangles
Part 2
Formulate how you can prove the sum of measures, if possible. triangle? What is the sum of the measures of the interior angles of a
T Triangle Sum Theorem
he sum of the interior angles in a triangle is
ͳͺͲι
B
A
C
- 7r\ ,t!
Connect the points on the plane and cConsider the figure below.
lassify the
resulting
triangle.
Use two different approaches to justify your
answer.
A
C
Section 5: Triangles – Part 1
diagram belowStephen is fencing in his triangular garden as shown by the
Part A
Write an expression for the measure of angle
Part B
Stephen measured angle
as
ͻͲ
ι
. H
e measures
angle
as
͵ͺι
. Did he measure correctly?
Justify
your answer.
ι
ͷͲ
ι
BEAT THE TEST!
Triangle
has vertices at
ͷ
ǡ ͺ
ሻ
,
ܱ
ǡ ͳͲ
, and
ǡ
ሻ .
Part A
: Determine what type of triangle
is and mark
A the most appropriate answer.
Scalene
B
Isosceles
C
Equilateral
D
Right
Part B:
If you move vertex
four units to the left, will the
classification of triangle
change? If so, what
type of triangle will it be? Justify your answer.
Section 5: Triangles – Part 1
_7r_
t!
guard. Each square on Deena’s plan Deena’s mother is helping her sew a large flag for colorConsider the figure below.
above
represents a
square foot
a.
Determine
the amount of fabric
Deena need
s
in
square feet.
b.
The flag will be
sewn along the edges
tenth of a foot? How much ribbon will be needed to the nearest
ݕ
ݔ
Section
Topic
Triangles
in the Coordinate Plane
- Let’s Practice! coordinate plane? How can we find area and perimeter when a figure is on the
Consider the triangle
below.
a.
Which side should be considered the base?
Justify
your answer.
b.
Find the area and perimeter of the triangle.
A
B
ݕ
ݔ
Section 5: Triangles – Part 1
- Let’s Practice!
If
ο
ܮܭܬ
ο
ܱܶܥ
, finish the following congruence statements
corresponding congruent angles. and mark the corresponding congruent sides and the
______
______
______
______
______
______
ο below.Complete the congruence statements for the triangles
ܶ
ܴ ܫ
ο
̴ ̴ ̴ ̴ ̴ ̴ ̴
______
A
C
Section
Topic
Triangle Congruence
SSS and SAS
Part 1
What information do we need
in order
to determine whether
two
different
triangles are congruent?
the names of the triangles is When we state triangle congruency, the order of the letters in
extremely
important.
How can
this congruency be stated?
A
B
C
Section 5: Triangles – Part 1
We can prove the following triangles are congruent
by the SSS
Write the congruency Congruence Postulate.
statement for the triangles above.
Determine if Angle
Angle
Angle congruence exists and
explain why it does or does not.
S
ide
Side
Side (SSS) Congruence Postulate
If three
sides of one triangle are congruent to
triangles are congruent. three sides of a second triangle, then the two
- 7r\ ,t!
Let’s consider the same triangles where
ο
ܴܶܫ
ο
a.
with marks and the corresponding congruent anglesMark the corresponding congruent sides with hash
arcs.
b.
are congruent. need to know that all three sides and all three anglesTo state that two triangles are congruent, we don’t
Four
postulates help us determine
- triangle congruency.
A
Section 5: Triangles – Part 1
7r\ ,t
Consider ∆
and ∆
in the figure below.
Given:
and
Prove:
complete the following two Based on the above figure and the information below,
column proof.
Statements
Reasons
Given
Given
CReflexive Property of
ongruence
C
A
What information is needed to prove the triangles
below
are congruent using the SSS Congruence Postulate?
C
A
Section 5: Triangles – Part 1
BEAT THE TEST!
Moshi is
making a quilt using
the pattern below and
wants
to be sure her triangles are congruent before cutting
the
fabric. She measures and finds that
and
Can Moshi determine if
the triangles are
congruent
with
the given information
? If not, what other information
would allow her to do so? Justify your answer.
A
Consider
ο
ܴܶܩ
and
ο
in the diagram below.
Given:
is the midpoint of
and
Prove:
ο
ܴܶܩ
ο
ܣ
C
omplete the following two
column proof.
Statements
Reasons
is the midpoint of
and
Given
Definition of Midpoint
Definition of Midpoint
ο
؆ ο ܣ ܴ ܧ 5.
A
Section 5: Triangles – Part 1
Identify the postulate Consider the triangles below.
you could use to prove that the two
triangles are congruent
given each additional congruence
statement
below
Congruency Statement
Postulate
B
A
In the Consider the figures below.
above
diagram
based on
the AAS
Congruence
Postulate.
Name the congruent sides and
angles in these two triangles.
Angle
Angle
Side (AAS)
Congruence Postulate
If two angles and a non
included side of one
triangle are congruent to two angles and a non
triangles are included side of a second triangle, then the two
congruent.
A
Section 5: Triangles – Part 1
- Let’s Practice!
Consider
ο
and
ο
ܰܶ
in the diagram below.
Given:
and
are right angles;
is the midpoint of
Prove:
ο
ܲ ܫܣ
ο
ܰܶ
C
omplete the following two
column proof.
Statements
Reasons
and
are right angles
Given
is the midpoint of
Given
Definition of midpoint
ο
ܲܫܣ
ο
A
Nadia would like to use the A Consider the figure below.
AS
Congruence
Postulate
to
prove
that
ο
ܶܵܫ
ο
ܶܫ
Would knowing that
be
enough information for Nadia to use this
postulate
? If not, find
the missing congruence statement.
Section 5: Triangles – Part 1
BEAT THE TEST!
Consider the diagram below.
Given:
Prove:
ο
ܴܲ (^) ܧ
ο
ܱܴܸ
Select the most appropriate reason for #5.
Statements
Reasons
Given
Given
Alternate Interior Angles
Theorem
Vertical angle theorem
ο
ܴܲܧ
ο
ܱܴܸ
A
AAS
B
ASA
C
SAS
D
SSS
7r\ ,t
How would you prove Consider the figures below.
ο
ܥ𝐵𝐵ܣ
ο
ܧܨܩ
by applying ideas of
transformations?
A
B
C
ݕ
ݔ
Section 5: Triangles – Part 1
Section
Topic
Base Angle of Isosceles Triangle
s
Consider with two equal sides. By definition, an _______________ _______________ is a triangle
below.
Draw the angle bisector
of
, where
is the intersection of
the bisector and
Use paragraph proofs to show
that
in two ways:
by using transformations and triangle congruence postulates.
Transformations
Triangle Congruence
Postulates
Part Consider the figure below.
A:
What transformation(s) will prove
ο
ܧܮ𝐵𝐵
ο
ܷ
Justify your answer.
Part B
If
is the angle bisector of
, w
hat additional
information
is
need
ed
to prove that
ο
ο
ܷ
A using ASA?
B
C
D
B
