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Triangles are polygons with three sides and three angles. Its angles help to classify the triangle, and the relationships between those angles are significant. This lesson explores such angle relationships as well as how they relate to an angle formed outside of the triangle.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Discussion

Discussion

Triangles can be classified into different types depending on the side lengths or angle measures. Despite all this variety in triangles, the relationship between the interior angles of any triangle can be explained with a single equation.

Rule

The sum of the measures of the interior angles of a triangle is $180_{∘}.$

Based on this diagram, the following relation holds true.

$m∠A+m∠B+m∠C=180_{∘}$

Example

For summer vacation, Ali and his parents went on a cruise in the Caribbean Sea. The ship sailed from Miami and headed for San Juan, Puerto Rico. Ali was a little scared because the route went through one side of the Bermuda Triangle 🛳️.

What is the measure of the angle formed at San Juan?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["63"]}}

What is the sum of the measures of the interior angles of a triangle?

Notice that Miami, San Juan, and Bermuda form a triangle for which the measures of two interior angles are given.

Recall that the sum of the measures of the interior angles of a triangle is equal to $180_{∘}.$ Use this information to write an equation for the triangle.$m∠M+m∠S+m∠B=180_{∘} $

Substitute the known measures and solve the equation for $m∠S.$
$m∠M+m∠S+m∠B=180_{∘}$

SubstituteValues

Substitute values

$56_{∘}+m∠S+61_{∘}=180_{∘}$

AddTerms

Add terms

$m∠S+117_{∘}=180_{∘}$

SubEqn

$LHS−117_{∘}=RHS−117_{∘}$

$m∠S=63_{∘}$

Example

It got dark before the cruise ship arrived in San Juan. Ali noticed that the sky looked much more starry than it does in his hometown. His father explained that it was due to the low light pollution. Ali recognized a triangle made up of three glowing stars.

Write the measure of three interior angles of the triangle.{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":false,"numinput":3,"listEditable":false,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Angle Measures <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">{<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">}<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["39","79","62"]}}

Use the Interior Angles Theorem to set and solve an equation for $x.$ Then, use the value of $x$ to determine each angle measure.

The measures of the three interior angles of the triangle depend on $x.$ The first step is finding the value of $x.$ The sum of the measures of the interior angles of a triangle is $180_{∘},$ according to the Interior Angles Theorem. Write an equation using this fact.
The measures of the angles of the triangle can be found by substituting $x=13$ into each of the expressions.

$3x+(6x+1)+(5x−3)=180 $

Solve the equation for $x.$
$3x+(6x+1)+(5x−3)=180$

CommutativePropAdd

Commutative Property of Addition

$3x+6x+5x+1−3=180$

AddSubTerms

Add and subtract terms

$14x−2=180$

AddEqn

$LHS+2=RHS+2$

$14x=182$

DivEqn

$LHS/14=RHS/14$

$x=13$

Angle Measure | Substitute $x=13$ | Simplify |
---|---|---|

$3x_{∘}$ | $3(13)_{∘}$ | $39_{∘}$ |

$(6x+1)_{∘}$ | $(6(13)+1)_{∘}$ | $79_{∘}$ |

$(5x−3)_{∘}$ | $(5(13)−3)_{∘}$ | $62_{∘}$ |

Discussion

When the sides of a triangle are extended beyond each vertex, a few more angles are formed. Some of these angles are called *exterior angles*.

Concept

An exterior angle of a triangle is the angle formed between one side of the triangle and the extension of an adjacent side. Any triangle has six exterior angles, two at each vertex and all formed outside the triangle.

Notice that each exterior angle forms a linear pair with its corresponding interior angle. This means that each exterior angle is supplementary to its interior angle.

Interior Angle | Corresponding Exterior Angles | Sum of Measures |
---|---|---|

$∠7$ | $∠1$ and $∠2$ | $m∠1+m∠7=180_{∘}$ $m∠2+m∠7=180_{∘}$ |

$∠8$ | $∠3$ and $∠4$ | $m∠3+m∠8=180_{∘}$ $m∠4+m∠8=180_{∘}$ |

$∠9$ | $∠5$ and $∠6$ | $m∠5+m∠9=180_{∘}$ $m∠6+m∠9=180_{∘}$ |

Discussion

In a triangle, a remote interior angle is an interior angle that is not adjacent to a specific exterior angle. Every exterior angle has two remote interior angles. In the diagram, click on each exterior angle to highlight its two corresponding remote interior angles.

The remote interior angles do not share the vertex with their corresponding exterior angle.

Discussion

In the same way the three interior angles of a triangle are related, the measure of an exterior angle and its two remote interior angles are related.

Rule

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles, or remote interior angles.

Based on the diagram above, the following relation holds true.

$m∠PCA=m∠A+m∠B$

Example

The day after arriving in Puerto Rico, Ali and his parents took a tour of three of the beautiful Leeward Islands. The captain shared the route plan.

The ship returned to Puerto Rico after visiting Brades. a Determine the value of $x,$ which corresponds to the angle that the ship turned to go to Saint Martin island.

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b Determine the value of $y,$ which is the angle at Brades island.

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a The angle whose measure is $x_{∘}$ is an exterior angle to the triangle formed by the first three islands of the route. Also, the value of $x$ is equal to the sum of the measures of the two remote interior angles.

b The angle at Saint Martin island whose measure is $72_{∘}$ is an exterior angle to the triangle formed by the last three islands in the ship's route. The angles whose measures are $x_{∘}$ and $y_{∘}$ are remote interior angles corresponding to the $72_{∘}$ angle.

a Start by naming the islands so that it is easier to refer to them. Focus on $△IUS$ and ignore the map to have a clearer view.

Notice that $UB$ is an extension of $IU$ which means that $∠BUS$ is an exterior angle to $△IUS.$ Also, $∠I$ and $∠ISU$ are remote interior angles corresponding to $∠BUS.$

The measure of $∠BUS$ is equal to the sum of the measures of the two remote interior angles. This is because the Triangle Exterior Angle Theorem. Write an equation using this information.$m∠BUS=m∠I+m∠ISU $

Finally, substitute the corresponding measures to find the value of $x.$
b Focus on $△SBU$ this time.

In Part A, the measure of $∠BUS$ is $38_{∘}.$ The $72_{∘}$ angle is an exterior angle to $△SBU.$ Additionally, $∠U$ and $∠B$ are remote interior angles corresponding to the $72_{∘}$ exterior angle.

Once again, use the fact that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles to write an equation.$m∠BST=m∠U+m∠B $

Finally, substitute the measures of the angles and solve the equation for $y.$
$m∠BST=m∠U+m∠B$

SubstituteValues

Substitute values

$72_{∘}=38_{∘}+y$

SubEqn

$LHS−38=RHS−38$

$34_{∘}=y$

RearrangeEqn

Rearrange equation

$y=34_{∘}$

Example

It is the last day of the cruise and they are on their final island. Ali climbed up a small cliff to take in the beautiful view of the ocean. From the top of the cliff, he could see the docked ship. It is huge! He is curious about some of its measurements.

Assume that the angle that the bow of the ship makes with the water measures $55_{∘}.$ What is the value of $x?${"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["7"]}}

The angle that is outside the boat is an exterior angle. The angles inside the triangle are remote interior angles corresponding to the exterior angle.

Start by labeling the vertices of the triangle and the left endpoint of the extended side. Ignore the photo of the ship to have a clearer view of the diagram.

Notice that $∠ABD$ is an exterior angle to $△ABC.$ Also, $∠A$ and $∠C$ are remote interior angles corresponding to $∠ABD.$ Recall that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. An equation can be written using this information.$m∠ABD=m∠A+m∠C $

The measure of the exterior angle is $55_{∘}$ and the measures of the two remote interior angles were given in terms of $x.$ Substitute the measure and expressions into the equation and solve it for $x.$
$m∠ABD=m∠A+m∠C$

Substitute

$m∠ABD=55_{∘}$

$55_{∘}=m∠A+m∠C$

SubstituteExpressions

Substitute expressions

$55_{∘}=4x+3(x+2)$

Distr

Distribute $3$

$55_{∘}=4x+3x+6$

AddTerms

Add terms

$55_{∘}=7x+6$

SubEqn

$LHS−6=RHS−6$

$49_{∘}=7x$

DivEqn

$LHS/7=RHS/7$

$7_{∘}=x$

RearrangeEqn

Rearrange equation

$x=7_{∘}$

Pop Quiz

Closure

Consider a triangle $ABC.$ No matter the characteristics of this triangle, the sum of the measures of its interior angles is $180_{∘}.$ This fact leads to the question of whether there is a similar relationship between the exterior angles of a triangle.

Recall that the Triangle Exterior Angle Theorem states that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. As a result, three equations can be written.$m∠4m∠5m∠6 =m∠2+m∠3=m∠1+m∠3=m∠1+m∠2 $

Next, add these three equations. The left-hand side of the resulting equation is the sum of the measures of the exterior angles. The right-hand side can be simplified using the fact that the measures of the interior angles add up to $180_{∘}.$
$m∠4+m∠5+m∠6=(m∠2+m∠3)+(m∠1+m∠3)+(m∠1+m∠2)$

AddTerms

Add terms

$m∠4+m∠5+m∠6=2m∠1+2m∠2+2m∠3$

FactorOut

Factor out $2$

$m∠4+m∠5+m∠6=2(m∠1+m∠2+m∠3)$

Substitute

$m∠1+m∠2+m∠3=180_{∘}$

$m∠4+m∠5+m∠6=2(180_{∘})$

Multiply

Multiply

$m∠4+m∠5+m∠6=360_{∘}$

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