High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize High School Geometry Textbook (Common Core) > Triangles

Angles of Triangles

Rotation SymmetryPrevious Section
Classifying TrianglesNext Section
Popular Courses
Find My Course
0:00 / 0:00

Angles of Triangles



Triangles are special polygons that contain a variety of angles. Depending on the type of triangle and type of angle, there are all kinds of interesting properties to discover.

Types of Angles

The interior angle of a triangle are the ones located inside of the triangle. If a side is extended, then the angle that forms a linear pair with an interior angle is considered an exterior angle.


Properties

  • Sum of Angles Theorem (Triangle) - The sum of all the angles inside a triangle is 180 degrees.
  • Exterior Angle Theorem (Triangle) - The measure of an exterior angle is equal to the sum of the two nonadjacent angles inside the triangle.
Example
Draw a triangle such that it has the following
  • An interior right angle.
  • An exterior angle with a measure of 120∘120^\circ.
  • Label all the interior angles with their measure.
ANSWER:
One possible answer would be
0:00 / 0:00

Triangles and Terrariums



A terrarium is a closed glass container for growing plants.
Joyce is currently working on a terrarium that she hopes to grow several cacti in. Its sides are made of glass triangles.


Use the diagram of one of these sides to answer the following questions.

1. What do we call angle ∠PRQ\angle{PRQ}?

ANSWER: ∠PRQ\angle{PRQ} is called an exterior angle.

2. What do we call the pair of angles ∠PRQ and ∠PRS\angle{PRQ} \text{ and } \angle{PRS}?

ANSWER: When talking about the pair ∠PRQ and ∠PRS\angle{PRQ} \text{ and } \angle{PRS}, we say they are a linear pair.

3. If m∠PRQ=113∘m\angle{PRQ} = 113^\circ and m∠RPS=46∘m\angle{RPS} = 46^\circ, what is m∠PSRm\angle{PSR} ?

ANSWER: m∠PSR=67∘m\angle{PSR} = 67^\circ

We can use the fact that a linear pair of angles is supplementary to start
m∠PRQ+m∠PRS=180∘113∘+m∠PRS=180∘m∠PRS=67∘\begin{aligned} m\angle{PRQ} + m\angle{PRS} = 180^\circ \\ 113^\circ + m\angle{PRS} = 180^\circ \\ m\angle{PRS} = 67^\circ \\ \end{aligned}
Now using the sum of all interior angles must be 180 we have
m∠PRS+m∠RPS+m∠PSR=180∘67∘+46∘+m∠PSR=180∘113∘+m∠PSR=180∘m∠PSR=67∘\begin{aligned} m\angle{PRS} + m\angle{RPS} + m\angle{PSR} &= 180^\circ \\ 67^\circ + 46^\circ + m\angle{PSR} &= 180^\circ \\ 113^\circ + m\angle{PSR} &= 180^\circ \\ m\angle{PSR} &= 67^\circ \\ \end{aligned}

Identifying Angles



In the diagram of a triangle, several angles have been marked out.
Use the diagram to match the angles with their best description.
A.
Vertical angles
B.
Exterior angles
C.
Linear pair
D.
Interior angles
Angles ∠2 and ∠4\angle{2} \text{ and } \angle{4}
Angles ∠2 and ∠3\angle{2} \text{ and } \angle{3}
Angles ∠3 and ∠5\angle{3} \text{ and } \angle{5}
Angles ∠6 and ∠3\angle{6} \text{ and } \angle{3}

Using properties of Triangles



Use the diagram and the given information to answer the questions.

1. What is the value of x?x?
2. What is the value of yy?

1. What is the value of x?x?


Triangles and Sidewalks


While repairing a cracked sidewalk a triangular piece was cut out and replaced with a new one.
Use the diagram and the given information to answer the following questions.

Given that
  • m∠ACD=146∘m\angle{ACD} = 146^\circ
  • ∠CAB≅∠CBA\angle{CAB} \cong \angle{CBA}
1. What is the measure of angle ∠CAB\angle{CAB} ?

2. What is the measure of angle ∠ACB\angle{ACB} ?
1. What is the measure of angle ∠CAB\angle{CAB} ?