Transcript 4.1 Triangles and Angles

4.1 Triangles and Angles
Definition of a triangle
A triangle is three segments joined at three
noncollinear end points.
vertex
side
side
adjacent sides
vertex
side
vertex
Types of Triangles by Sides
3 Sides congruent →
Equilateral
Types of Triangles by Sides
2 Sides congruent →
Isosceles
Part of the
Isosceles Triangle
leg
leg
base
Types of Triangles by Sides
No Sides congruent →
Scalene
Types of Triangles by Sides
3 Sides congruent → Equilateral
2 Sides congruent → Isosceles
No Sides congruent → Scalene
Types of Triangle by Angles
All Angles less than 90 degrees → Acute
Types of Triangle by Angles
One Angle greater than 90 degrees, but less than 180° →
Obtuse
Types of Triangle by Angles
One Angle equal to 90 degrees → Right
How to classify a triangle
Choose one from each category
Sides
Scalene
Isosceles
Equilateral
Angles____
Acute
Right
Obtuse
Equiangluar
All the angles are Equal
This will ALWAYS be
paired up with
Equilateral
Parts of the Right Triangle
Across from the right angle is the
hypotenuse.
hypotenuse
leg
leg
Interior Angles vs. Exterior Angles
M
a
N
b
c
P
Interior angles: <a, <b, <c
Exterior angles: <M, <N, <P
Triangle Sum Theorem
The sum of the three interior angles of a triangle
is 180º
a
b
c
m<a + m<b + m<c = 180°
Triangle Sum Theorem
110
x
25
Solve for x
Example 2
Find the measure of each angle.
2x + 10
x
x+2
Exterior Angle Theorem
The measure of an exterior angle equals the
measure of the two nonadjecent interior
angles.
a
a b
b
Example 3
Given that ∠ A is 50º and
∠B is 34º, what is the measure of
∠BCD?
B
A
C
What is the measure of ∠ACB?
D
Solve for x
x
79
14
Corollary for the fact that interior
angles add to 180º
The acute angles of a Right triangle are
complementary.
90  x
x
Example 4
A. Given the following
triangle, what is the length
of the hypotenuse?
B. What are the length of
the legs?
13
12
C. If one of the acute angle
measures is 32°, what is
the other acute angle’s
measurement?
5
Example 6
Find the missing measures
80°
53°
Example 7
Given: ∆ABC with mC = 90°
Prove: mA + mB = 90°
Statement
1. mC = 90°
2. mA + mB + mC = 180°
3. mA + mB + 90° = 180°
4. mA + mB = 90°
Reason