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4.1 Triangles and Angles4.1 Triangles and Angles
Definition of a triangle
A triangle is three segments joined at three noncollinear end points.
vertex
vertex
vertex
side side
side
sidesadjacent
Types of Triangles by Sides
3 Sides congruent → Equilateral
Types of Triangles by Sides
2 Sides congruent → IsoscelesPart of the Isosceles
Triangleleg 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 triangleChoose one from each category
Sides Angles____ Scalene Acute
Isosceles Right
Equilateral 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
Solve for x
25
110
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.
b
a
ba
Example 3
Given that ∠ A is 50º and
∠B is 34º, what is the measure of
∠BCD?
What is the measure of ACB?∠
DA
B
C
Solve for x
14
79
x
Corollary for the fact that interior angles add to 180º
The acute angles of a Right triangle are complementary.
x
x90
Example 4A. Given the following triangle,
what is the length of the hypotenuse?
B. What are the length of the legs?
C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement?
13
12
5
Example 6
Find the missing measures
80°
53°
Example 7
Given: ∆ABC with mC = 90°
Prove: mA + mB = 90°Statement Reason
1. mC = 90°
2. mA + mB + mC = 180°
3. mA + mB + 90° = 180°
4. mA + mB = 90°