Triangle angle relationships are all about understanding how angles work together in triangles. From the Triangle Sum Theorem to exterior angles, these concepts help us solve problems and find missing angles.

Parallel lines and special triangles like equilateral and isosceles add more tools to our geometry toolkit. Mastering these relationships lets us tackle complex triangle problems with confidence and precision.

Triangle Angle Relationships

Triangle Sum Theorem applications

• States the sum of the measures of the three interior angles of a triangle is always 180°
  • In $\triangle ABC$, $m\angle A + m\angle B + m\angle C = 180°$
• Find the measure of a missing angle in a triangle by subtracting the sum of the known angles from 180°
  • $\triangle XYZ$ has $m\angle X = 50°$ and $m\angle Y = 70°$, then $m\angle Z = 180° - (50° + 70°) = 60°$
• Solve problems involving the measures of angles in a triangle
  • $\triangle MNO$ has $m\angle M = 2x°$, $m\angle N = 3x°$, and $m\angle O = 40°$, set up equation $2x + 3x + 40 = 180$ and solve for $x$ to find angle measures
• Determine the measure of an angle in a triangle given the measures of the other two angles
  • $\triangle PQR$ has $m\angle P = 45°$ and $m\angle Q = 60°$, then $m\angle R = 180° - (45° + 60°) = 75°$

Interior and exterior angle measures

• Exterior Angle Theorem states the measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles
  • In $\triangle PQR$, if $\angle S$ is an exterior angle adjacent to $\angle R$, then $m\angle S = m\angle P + m\angle Q$
• The measure of an exterior angle and its adjacent interior angle sum to 180°
  • In $\triangle DEF$, if $\angle G$ is an exterior angle adjacent to $\angle F$, then $m\angle F + m\angle G = 180°$
• Find the measure of an exterior angle given the measures of the interior angles of a triangle
  • $\triangle HIJ$ has $m\angle H = 70°$ and $m\angle I = 50°$, then the measure of the exterior angle adjacent to $\angle J$ is $70° + 50° = 120°$
• Determine the measure of an interior angle given the measure of an exterior angle
  • $\triangle KLM$ has an exterior angle adjacent to $\angle L$ measuring 110°, then $m\angle L = 180° - 110° = 70°$

Parallel lines in triangles

• When parallel lines are cut by a transversal, corresponding angles are congruent
  • In a triangle where one side is a transversal intersecting two parallel lines, the angles formed by the transversal and the parallel lines are congruent to their corresponding angles in the triangle
• Alternate interior angles formed by parallel lines and a transversal are congruent
  • In a triangle where one side is a transversal intersecting two parallel lines, the alternate interior angles formed by the transversal and the parallel lines are congruent to their corresponding angles in the triangle
• Use the properties of parallel lines to solve for angle measures in a triangle
  • $\triangle ABC$ has $\overline{AB} \parallel \overline{DE}$, $m\angle BAC = 60°$, and $m\angle BDE = 120°$, then $m\angle ABC = 60°$ (corresponding angles) and $m\angle ACB = 60°$ (alternate interior angles)

Equilateral and isosceles triangle angles

• In an equilateral triangle, all three angles are congruent and measure 60°
  • The sum of the angles in an equilateral triangle is 180°, and since all angles are congruent, each angle measures $180° \div 3 = 60°$
• In an isosceles triangle, the two base angles are congruent
  • If the vertex angle measures $x°$, then each base angle measures $\frac{180° - x°}{2}$
• The measure of the vertex angle in an isosceles triangle can be found by subtracting the measure of one base angle from 180° and dividing the result by 2
  • If a base angle in an isosceles triangle measures $y°$, then the vertex angle measures $\frac{180° - y°}{2}$
• Solve for angle measures in equilateral and isosceles triangles
  • An isosceles $\triangle XYZ$ has vertex angle $\angle Y$ measuring 40°, then each base angle ($\angle X$ and $\angle Z$) measures $\frac{180° - 40°}{2} = 70°$