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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:
      • \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

Geo A 2.02 - Exterior Angles 01

 

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

Geo A 2.02 - Remote Angle Rules

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!

Geo A 2.02 - Exterior Angles 02


  • 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°

Geo A 2.02 - Exterior Angles 03

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: °

Geo A 2.02 - Angles Triangle x 01


  •  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.

Geo A 2.02 - Angles Triangle Substitution x 03

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

Geo A 2.02 - Angles Triangle Isos x 02

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°

Geo A 2.02 - Congruent Angles 04

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.

Geo A 2.02 - Congruent Angles 05

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