# 2.2 – The Angles of a Triangle

## Objectives

• Find the sum of a triangle’s internal angles.
• Discover the third angle theorem.
• Identify interior, remote interior, and exterior angles of triangles.
• Discover the exterior angle theorem.
• Solve for unknown angle measures of a triangle.
• List the maximum number of obtuse or right angles that can be found in any triangle.

## Key Terms

• Exterior Angles – Angles on the outside of a triangle that form linear pairs with interior angles of the triangle.
• Vertical angles are NOT exterior angles of a triangle!
• The exterior angles of the triangle below are:
• $\angle2$ and $\angle4$
• $\angle3$ is NOT an exterior angle.  It is a vertical angle with $\angle5$.
• Interior Angles – Angles inside a triangle.
• The interior angles of the triangle below are:
• $\angle1$, $\angle5$, and $\angle6$
• Remote Interior Angles – Angles in a triangle that are not adjacent to a given exterior angle.
• All exterior angles have two remote interior angles.
• The remote exterior angles of the triangle below are:
• $\angle4$ is a remote exterior angle from angles $\angle6$ and $\angle1$
• So, $\angle6$ and $\angle1$ are remote interior angles from angles $\angle4$
• $\angle2$ is a remote exterior angle from angles $\angle6$ and $\angle5$
• So, $\angle6$ and $\angle5$ are remote interior angles from angles $\angle2$

## Notes

Triangle Classification
• If two angles of a triangle are acute, then the third angle could be acute, obtuse, or right.
• One triangle can have 3 acute angles
• One triangle can have 1 right angle and 2 acute angles
• One triangle can have 1 obtuse angle and 2 acute angles
• Triangles cannot have more than one right angle
• The other two angles will be acute and add up to 90°
• Triangles cannot have more than one obtuse angle
• The other two angles will be acute and add up to 89°
• The interior angles of a triangle measure a total of 180°

Exterior Angle Corollary
• $\angle 4$ is an exterior (outside the triangle and a linear pair with an interior angle)
• The degree measure of $\angle 4$ equals the sum of the degree measures of $\angle 1$ and $\angle 2$; so,
• $\angle 4$ is greater than $\angle 2$
• $\angle 4$ is greater than $\angle 1$
• We cannot determine the measure of $\angle 3$

Exterior Angle Theorem
• Remote interior angles add up to their remote exterior angle.
• Think of “remote” as something that is far away!
• Remote exterior angles are remote (far away) from their remote interior angles.
• Remote interior angles are remote (far away) from their remote exterior angles.
• They are NOT linear pairs.  They are NOT adjacent.
• Example 1
• The remote interior angles are  $\angle 55^{\circ}$ and $\angle x$
• The remote exterior angle is $\angle 105^{\circ}$
• Set up the equation and solve for x:
• 55 + x = 105 (subtract 55 on both sides)
• x = 50°
• Test this theorem
• Remember that $\angle 105^{\circ}$ is a linear pair with the interior angle next to it (unlabeled); so, 105 + the missing angle add up to 180.
• So, the missing angle is 75°
• You already know that the interior angles of a triangle add up to 180°
• Does 75 + 55 + 50 = 180?  Yes!

• Example 2
• Find $\angle p$
• You will have to use what you know about linear pairs to find the other interior angles first, then you can find the missing interior angle $\angle p$
• Step 1: Find the missing interior angles using linear pairs (supplementary angles theorem):
• $133^{\circ}+\angle a=180^{\circ}$
• Subtract 133 on both sides and get:  $\angle a=47^{\circ}$
• $90^{\circ}+\angle b=180^{\circ}$
• Subtract 90 on both sides and get: $\angle b=90^{\circ}$
• Step 2: Decide which method you want to use
1. 90 + 47 + p = 180, or…
2. p + 47 = 90, or…
3. p + 90 = 133
• Step 3: Solve from step 2 to get p = 43°

Angle Sum Theorem (Any Triangle)
• Example 1: To find a missing value, set up an equation (equal to 180), then solve for x
• Step 1: Write the equation: x + 65 + 79 = 180
• Step 2: Combine like terms: x + 144 = 180
• Step 3: Isolate the variable by subtracting 144 on both sides: x = 180 – 144
• Step 4: Write the answer: x = 36°
• Note: Use the degree symbol when marking measurement: °

•  Example 2: It doesn’t matter if the angle has an expression or a term instead of a value or variable, all 3 angles still add up to 180°
• Step 1: Write the equation: (y + 10) + (2y) + (50) = 180
• Step 2: Remove parenthesis and combine like terms:  3y + 60 = 180
• Step 3: Isolate the variable term by subtracting 60 on both sides: 3y = 180 – 60
• Step 4: Simplify: 3y = 120
• Step 5: Divide both sides by 3: $\frac{3y}{3} = \frac{120}{3}$
• Step 6: Solve for y = 40°
• Step 7: Substitute the value for y (which is 40°) to find each angle measure:
• y + 10 becomes 40 + 10, which is 50°
• 2y becomes 2(40), which is 80°
• So, the three angles of this triangle are: 50°, 80°, and 50°; therefore, this is an isosceles triangle because there are two congruent angles.

Angle Sum Theorem (Isosceles)
• Remember: Isosceles triangles have two base angles that are congruent
• Example: To find the variable, remember that you can add two variables together if they are the same: y + y = 2y.
• Step 1: Write the equation: 54 + 2y = 180
• Step 2: Isolate the variable term by subtracting 54 on both sides:  2y = 180 – 54
• Step 3: Simplify:  2y = 126
• Step 4: Divide both sides by 2:  $\frac{2y}{2} = \frac{126}{2}$
• Step 5: Solve for y

y = 63°

Third Angle Theorem
• If two angles have the same measure, they are congruent
• Example: What do you know to be true about the values of p and q?
• Calculate each triangle for the missing angle
• Left: 60 + 30 + p = 180
• 90 + p = 180
• p = 90°
• Right: 45 + 45 + q = 180
• 90 + q = 180
• q = 90°

So, p = q because they are both equal to 90°.

• Example: Sometimes, you cannot determine the measures of angles or their congruency.
• When you have 2 variables in one triangle that are different from the two variables in the other triangle, the angle measures cannot be determined.
• It is impossible to know if $\angle a$ in the left triangle is congruent to any of the angles in the right triangle.
• It is impossible to know if $\angle b$ or $\angle y$ in the right triangle is congruent to any of the angles in the left triangle.