Introduction

A triangle is a geometric shape that consists of three points connected by three line segments, as shown below.



For this triangle,

  • the line segments \overline{AB} , \overline{BC} and \overline{CA} are the sides of the triangle,

  • the points \color{red}A , \color{red}B and \color{red}C are the vertices of the triangle,

  • the angles \angle{A} , \angle{B} and \angle{C} formed by the sides are called the interior angles of the triangle.

To denote a triangle, we use the symbol \boldsymbol{\triangle} and list its vertices (in any order). So, the above diagram shows us \triangle ABC.

It can be shown that all triangles have the following property:

The sum of the measures of the interior angles of any triangle is always equal to 180^\circ.

For example, in \triangle ABC shown above, the interior angle theorem tells us that m\angle{A} + m\angle{B} + m\angle{C} = 180^\circ .

Therefore, if we know the measures of two angles in a triangle, we can calculate the value of the other.

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Example: Finding a Missing Angle in a Triangle

Find the value of x in the triangle above.

EXPLANATION

Since the sum of the measures of the interior angles in any triangle is equal to 180^\circ , we have

\begin{align*} m\angle{A} + m\angle{B} + m\angle{C} &= 180^\circ \\[5pt] 36^\circ + 112^\circ + x &= 180^\circ \\[5pt] 148 ^\circ+ x &= 180^\circ \\[5pt] x &= 32^\circ. \end{align*}

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Practice: Finding a Missing Angle in a Triangle

In the figure above, $x=$

a
b
c
d
e
Practice: Finding a Missing Angle in a Triangle

Determine the value of $x.$

a
 $30^\circ$
b
 $29^\circ$
c
 $34^\circ$
d
 $32^\circ$
e
 $33^\circ$
Example: Calculating an Unknown Quantity From a Triangle

Find the measure of \angle C.

EXPLANATION

Since the sum of the measures of the interior angles of a triangle is 180^\circ , we have

\begin{align*} 3z +4z+ 40^\circ &= 180^\circ \\[3pt] 7z+40^\circ &= 180^\circ \\[3pt] 7z &= 140^\circ \\[3pt] z &= 20^\circ. \end{align*}

Therefore,

\begin{align*} m\angle C &= 4z \\[5pt] &= 4(20^\circ)\\[5pt] &= 80^\circ. \end{align*}

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Practice: Calculating an Unknown Quantity From a Triangle

In the figure above, $x=$

a
b
c
d
e
Practice: Calculating an Unknown Quantity From a Triangle

Find the measure of angle $ A.$

a
 $20^\circ$
b
 $18^\circ$
c
 $22^\circ$
d
 $24^\circ$
e
 $16^\circ$
Example: Calculating an Unknown Quantity Given a Description of a Triangle

Consider a triangle \triangle ABC , where m\angle{A} = 55^\circ , m\angle B = x + 10^\circ , and m\angle C = 3x - 5^\circ. Find the value of x.

EXPLANATION

The sum of the measures of the interior angles must be 180^\circ. Therefore,

\begin{align*} 55^\circ + (x + 10^\circ) + (3x - 5^\circ) &= 180^\circ \\[2pt] 60^\circ + 4x &= 180^\circ \\[2pt] 4x &= 120^\circ \\[2pt] x &= 30^\circ. \end{align*}

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Practice: Calculating an Unknown Quantity Given a Description of a Triangle

Consider a triangle $\triangle RST,$ where $m\angle R = 2x + 31^\circ,$ $m\angle S = x - 4^\circ$ and $m\angle T = x - 7^\circ.$ Determine the value of $x.$

a
 $60^\circ$
b
 $54^\circ$
c
 $36^\circ$
d
 $20^\circ$
e
 $40^\circ$
Practice: Calculating an Unknown Quantity Given a Description of a Triangle

The measure of an angle in a triangle is three times the measure of another angle, and the measure of the third angle is $64^\circ. $ What is the measure of the smallest angle?

a
b
c
d
e
Practice: Calculating an Unknown Quantity Given a Description of a Triangle

The measure of one angle in a triangle is seven times the measure of another angle, and the measure of the third angle is $20^\circ.$ What is the measure of the largest angle?

a
 $135^\circ$
b
 $147^\circ$
c
 $210^\circ$
d
 $105^\circ$
e
 $140^\circ$
Example: Finding Angles in Triangles With Congruent Angles

Find the measure of angle C given that m\angle{A} = m\angle{B}.

EXPLANATION

Using m\angle{A} = m\angle{B}, we get

\begin{align*} m\angle{A} &= m\angle{B} \\[5pt] 3x-27^\circ &= 2x \\[5pt] x &= 27^\circ . \end{align*}

Consequently, substituting x=27^\circ in the expression for m\angle B, we get m\angle{B}=2(27^\circ)=54^\circ.

Now, we can find m\angle C using the fact that the internal angles of any triangle sum to 180^\circ : \begin{align*} m\angle{A} + m\angle{B} + m\angle{C} &= 180^\circ \\[5pt] 2m\angle{B} + m\angle{C} &= 180^\circ \\[5pt] 2\cdot 54^\circ + m\angle{C} &= 180^\circ \\[5pt] 108^\circ+m\angle{C} &= 180^\circ\\[5pt] m\angle{C} &= 72^\circ \end{align*}

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Practice: Finding Angles in Triangles With Congruent Angles

Find the measure of angle $C$ given that $m\angle{A} = m\angle{B}.$

a
b
c
d
e
Practice: Finding Angles in Triangles With Congruent Angles

Find the measure of angle $C$ given that $m\angle{A} = m\angle{B}.$

a
 $105^\circ$
b
 $98^\circ$
c
 $95^\circ$
d
 $103^\circ$
e
 $100^\circ$
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