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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:
 and
 is NOT an exterior angle. It is a vertical angle with .
 Interior Angles – Angles inside a triangle.
 The interior angles of the triangle below are:
 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:
 is a remote exterior angle from angles and
 So, and are remote interior angles from angles
 is a remote exterior angle from angles and
 So, and are remote interior angles from angles
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°

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:
 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:
 Step 5: Solve for y
y = 63°

Important!
Practice (Apex Study 2.2)
 Practice: Pgs 4, 11, 12, 13, 20, 21
 Watch the animation on Pg 14
 Click through and understand the proofs on Pg 5, 17
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