cConjecture: The sum of the measures of the exterior angles of a triangle is 360^(∘).
D
d m∠ 1+m∠ 2+m∠ 3=360^(∘)
E
eExample Solution: See solution.
Practice makes perfect
a We will to draw five different triangles by extending the sides of the triangle. We will start by drawing an acute triangle and labeling the exterior angles.
Now, let's draw another acute triangle by changing the measures of the angles.
An equilateral triangle is a triangle in which all the sides are congruent.
Acute Triangle
An acute triangle is a triangle where all angles are less than 90^(∘) or π2.
Obtuse Triangle
An obtuse triangle is a triangle with exactly one an angle whose measure is greater than 90^(∘) or π2.
Right Triangle
A right triangle is a specific type of triangle that contains one angle of 90^(∘).
b Now we will use a protractor to measure the exterior angles. Be certain to note the measurements accurately.
We will also include the sums of the angle measures in the table, as directed.
m∠ 1
m∠ 2
m∠ 3
Sum of Angle Measures
102
136
122
360
119
139
102
360
132
90
138
360
138
85
137
360
51
158
151
360
c Notice that in the table, the sums of the angle measures in all our example triangles are 360^(∘). Based on this observation, we can make the following conjecture.
The sum of the measures of the
exterior angles of a triangle is360^(∘).
d Using the names of the exterior angles, we can write the following algebraic relationship.
m∠ 1+m∠ 2+m∠ 3=360^(∘)
e We can use the Exterior Angle Theorem to express the measure of the exterior angles as the sum of the measures of the remote interior angles. Let's start by copying the given generic diagram.
We can use the digram to write the expressions in another table.
Measure of Exterior Angle
Expression Using Interior Angle Measures
m∠ 1
m∠ B+m∠ C
m∠ 2
m∠ A+m∠ B
m∠ 3
m∠ C+m∠ A
Using these expressions, we can find the sum of the measures of the exterior angles in terms of the measures of the interior angles.
We can now see that on the right hand side of the equation, we have twice the sum of the measures of the interior angles. According to the Triangle Angle-Sum Theorem, this sum is 180^(∘).
