Download 4.1 Apply Triangle Sum Properties

4.1 Apply Triangle Sum
Properties
Hubarth
Geometry
Classification of Triangles By Sides
Equilateral
Triangle
3 congruent sides
Isosceles
Triangle
2 congruent sides
Scalene
Triangle
No congruent sides
Classification of Triangles By Angles
Equiangular
Triangle
3 congruent
angles
Right Triangle
1 right angle
Acute Triangle
3 acute angles
Obtuse Triangle
1 obtuse angle
Ex 1 Classify Triangles by Sides and By Angles
Classify the triangular shape of the support beams in the diagram by its sides and by
measuring its angles.
The triangle has a pair of congruent sides, so it is isosceles. By
measuring, the angles are 55° , 55° , and 70° . It is an acute
isosceles triangle.
55
55
Ex 2 Classify Triangles in a Coordinate Plane
Classify PQO by its sides. Then
determine if the triangle is a right triangle.
=
Use the distance formula to find the side lengths.
( x – x )2 + ( y – y )2
=
( (– 1 )
=
( x – x )2 + ( y – y )2
=
( 6 – 0 )2 +
=
( x – x )2 + ( y – y )2
=
( 6 – (– 1 ) ) 2 +
STEP 1
OP
OQ
PQ
2
2
2
2
1
– 0 )2 +
( 2 – 0 )2 =
2
1
1
5
2.2
=
45
6.7
( 3 – 2 )2 =
50
7.1
1
( 3 – 0 )2
1
2
1
Ex 2 Continued
STEP 2
Check for right angles.
2–0
The slope of OP is
–1–0
3–0
The slope of OQ is
The product of the slopes is
so OP
OQ and
Therefore,
= – 2.
=
1
2
6–0
1
= – 1,
–2
2
POQ is a right angle.
PQO is a right scalene triangle.
.
Angles When the sides of a polygon are extended, other angles are formed. The original
angles are the interior angles. The angles that form linear pairs with the interior angles
are the exterior angles.
Exterior Angles
Interior Angles
Theorem
B
Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle
is 180.
A
C
mA  mB  mC  180
Theorem
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two nonadjacent (remoteinterior) interior angles.
B
1
A
C
m1  mA  mB
Ex 3 Find Angle Measures
Find mJKM
2x – 5 = 70 + x
x = 75
2x – 5 = 2(75) - 5
= 150-5
= 145
A
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
B
C
mA  mB  90
Ex 4 Find angle Measures From a Verbal Description
The tiled staircase shown forms a right triangle. The
measure of one acute angle in the triangle is twice
the measure of the other. Find the measure of each
acute angle.
2x
x + 2x = 90
xx
x
3x = 90
x=30
The two acute angles are 30 and 60
Practice
1. Draw an obtuse isosceles triangle and an acute scalene triangle.
Q
B
R
P acute scalene triangle
A
C
obtuse isosceles triangle
2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its
sides. Then determine if it is a right triangle.
ABC is a right Isosceles triangle.
3. Find the measure of
1 in the diagram shown.
The measure of ∠ 1 in the diagram is 65°.
4. Find the measure of each interior angle of
and m
C = 3x°.
A  B  C  180
x  2 x  3 x  180
6 x  180
x  30
ABC, where m
A  30
B  60
C  90
A=x, m
A
x
C
3x
2x
5. Find the measures of the acute angles of the
right triangle in the diagram shown.
2 x  ( x  6)  90
3 x  6  90
3 x  96
x  32
26° and 64°
B
B = 2x° ,