Download Answer - Net Start Class

4.1 Apply Triangle Sum
Properties
Objectives
 Identify and classify triangles by angles or
sides
 Apply the Angle Sum Theorem
 Apply the Exterior Angle Theorem
Parts of a Triangle
 A triangle is a 3-sided polygon
 The sides of ∆ABC are




AB, BC, and AC
The vertices of ∆ABC are
A, B, and C
Two sides sharing a common
vertex are adjacent sides
The third side is called the
opposite side
All sides can be adjacent or
opposite (it just depends which
vertex is being used)
A
adjacent
adjacent
B
Side
opposite
A
C
Classifying Triangles by Angles
One way to classify triangles is
by their angles…
Acute
Obtuse
all 3 angles are acute 1 angle is obtuse
(measure < 90°)
(measure > 90°)
Right
1 angle is right
(measure = 90°)
An acute ∆ with all angles  is an equiangular ∆ .
Example 1:
ARCHITECTURE The triangular truss below is
modeled for steel construction. Classify
JMN, JKO, and OLN as acute, equiangular, obtuse,
or right.
60°
60°
Example 1:
Answer:
JMN has one angle with measure greater than 90, so it
is an obtuse triangle.
JKO has one angle with measure equal to 90, so it is a
right triangle.
OLN is an acute triangle with all angles congruent, so it
is an equiangular triangle.
Classifying Triangles by Sides
Another way to classify triangles
is by their sides…
Equilateral
3 congruent sides
Isosceles
Scalene
2 congruent sides no congruent sides
Example 2a:
Identify the isosceles triangles in the figure if
Isosceles triangles have at least two sides congruent.
Answer: UTX and UVX are isosceles.
Example 2b:
Identify the scalene triangles in the figure if
Scalene triangles have no congruent sides.
Answer: VYX, ZTX, VZU, YTU, VWX,
ZUX, and YXU are scalene.
Example 2c:
Identify the indicated triangles in the figure.
a. isosceles triangles
Answer: ADE, ABE
b. scalene triangles
Answer: ABC, EBC, DEB, DCE, ADC, ABD
Example 3:
ALGEBRA Find d and the measure of each side of
equilateral triangle KLM if
and
Since KLM is equilateral,
each side has the same
length. So
5=d
Example 3:
Next, substitute to find the length of each side.
KL = 7
LM = 7
KM = 7
Answer: For KLM,
and the measure of
each side is 7.
Example 4:
COORDINATE GEOMETRY Find the measures of the
sides of RST. Classify the triangle by sides.
Example 4:
Use the distance formula to find the lengths of each side.
Answer:
; since all 3 sides
have different lengths, RST is scalene.
Exterior Angles and Triangles
 An exterior angle is formed by one side of a
triangle and the extension of another side
(i.e. 1 ).
1
2
4
3
 The interior angles of the triangle not adjacent to
a given exterior angle are called the remote
interior angles (i.e. 2 and 3).
Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the interior
angles of a triangle is 180°.
X
mX + mY + mZ = 180°
Y
Z
Example 5:
Find the missing angle measures.
Find
first because
the measure of two
angles of the triangle
are known.
Angle Sum
Theorem
Simplify.
Subtract 117 from each
side.
Example 5:
Angle Sum
Theorem
Simplify.
Subtract 142 from each
side.
Answer:
Your Turn:
Find the missing angle measures.
Answer:
Theorem 4.2 – Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two remote interior
angles.
m 1 = m 2 + m 3
1
2
4
3
Example 6:
Find the measure of each numbered angle in
the figure.
Exterior Angle Theorem
Simplify.
If 2 s form a linear pair,
they are supplementary.
Substitution
Subtract 70 from each
side.
Example 6:
Exterior Angle Theorem
Substitution
Subtract 64 from each side.
If 2 s form a linear pair,
they are supplementary.
Substitution
Simplify.
Subtract 78 from each
side.
Example 6:
Angle Sum Theorem
Substitution
Simplify.
Subtract 143 from each
side.
Answer:
Your Turn:
Find the measure of each numbered angle in
the figure.
Answer:
Corollaries
 A corollary is a statement that can be
easily proven using a theorem.
 Corollary 4.1 – The acute s of a right ∆
are complementary.
 Corollary 4.2 – There can be at most one
right or obtuse  in a ∆.
Example 3:
GARDENING The flower bed shown is in the
shape of a right triangle. Find
if
is
20.
Corollary 4.1
Substitution
Answer:
Subtract 20 from each
side.
Your Turn:
The piece of quilt fabric is in the shape of a
right triangle. Find
if
is 62.
Answer:
Assignment
 Geometry:
pg 221-224 #1 – 10, 14 – 19, 21 - 37