Relationships Between Angles: Interior and Exterior Angles of a Triangle

Angle Measure	Substitute $x = 13$	Simplify
$3 x^{\circ}$	$3 (13)^{\circ}$	$3 9^{\circ}$
$(6 x + 1)^{\circ}$	$(6 (13) + 1)^{\circ}$	$7 9^{\circ}$
$(5 x - 3)^{\circ}$	$(5 (13) - 3)^{\circ}$	$6 2^{\circ}$

Angle Measure

Substitute

x = 13

Simplify

3 x^{\circ}

3 (13)^{\circ}

3 9^{\circ}

(6 x + 1)^{\circ}

(6 (13) + 1)^{\circ}

7 9^{\circ}

(5 x - 3)^{\circ}

(5 (13) - 3)^{\circ}

6 2^{\circ}

Interior Angle	Corresponding Exterior Angles	Sum of Measures
$∠ 7$	$∠ 1$ and $∠ 2$	$m ∠ 1 + m ∠ 7 = 18 0^{\circ}$ $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$
$∠ 8$	$∠ 3$ and $∠ 4$	$m ∠ 3 + m ∠ 8 = 18 0^{\circ}$ $m ∠ 4 + m ∠ 8 = 18 0^{\circ}$
$∠ 9$	$∠ 5$ and $∠ 6$	$m ∠ 5 + m ∠ 9 = 18 0^{\circ}$ $m ∠ 6 + m ∠ 9 = 18 0^{\circ}$

Interior Angle

Corresponding Exterior Angles

Sum of Measures

∠ 7

∠ 1

and

∠ 2

m ∠ 1 + m ∠ 7 = 18 0^{\circ}

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

∠ 8

∠ 3

and

∠ 4

m ∠ 3 + m ∠ 8 = 18 0^{\circ}

m ∠ 4 + m ∠ 8 = 18 0^{\circ}

∠ 9

∠ 5

and

∠ 6

m ∠ 5 + m ∠ 9 = 18 0^{\circ}

m ∠ 6 + m ∠ 9 = 18 0^{\circ}

a Start by naming the islands so that it is easier to refer to them. Focus on

△ I U S

and ignore the map to have a clearer view.

Notice that $\overline{U B}$ is an extension of $\overline{I U}$ which means that $∠ B U S$ is an exterior angle to $△ I U S .$ Also, $∠ I$ and $∠ I S U$ are remote interior angles corresponding to $∠ B U S .$

The measure of

∠ B U S

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 ∠ B U S = m ∠ I + m ∠ I S U

Finally, substitute the corresponding measures to find the value of

x .

m ∠ B U S = m ∠ I + m ∠ I S U

SubstituteValues

Substitute values

x = 2 5^{\circ} + 1 3^{\circ}

AddTerms

Add terms

x = 3 8^{\circ}

b Focus on

△ S B U

this time.

In Part A, the measure of $∠ B U S$ is $3 8^{\circ} .$ The $7 2^{\circ}$ angle is an exterior angle to $△ S B U .$ Additionally, $∠ U$ and $∠ B$ are remote interior angles corresponding to the $7 2^{\circ}$ 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 ∠ B S T = m ∠ U + m ∠ B

Finally, substitute the measures of the angles and solve the equation for

y .

m ∠ B S T = m ∠ U + m ∠ B

SubstituteValues

Substitute values

7 2^{\circ} = 3 8^{\circ} + y

SubEqn

$LHS - 38 = RHS - 38$

3 4^{\circ} = y

RearrangeEqn

Rearrange equation

y = 3 4^{\circ}

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Catch-Up and Review

Investigating the Sum of the Angle Measures

Interior Angles of a Triangle

Relationship Between the Interior Angles of a Triangle

Interior Angles Theorem

Bermuda Triangle

Hint

Solution

A Starry Night

Hint

Solution

Angles Outside a Triangle

Exterior Angles of a Triangle

Remote Interior Angles

Relationship Between an Exterior Angle and Its Remote Interior Angles

Triangle Exterior Angle Theorem

Beach's Day

Hint

Solution

Angles in the Ship

Hint

Solution

Finding the Missing Angle Measure

Relationship Between the Exterior Angles of a Triangle

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