The Triangle Sum Theorem is the foundation for nearly every angle problem in this unit: the three interior angles of any triangle always add to $180°$ .

$m\angle A + m\angle B + m\angle C = 180°$

To find a missing angle, subtract the sum of the two known angles from $180°$ .

If $\triangle XYZ$ has $m\angle X = 50°$ and $m\angle Y = 70°$ , then $m\angle Z = 180° - (50° + 70°) = 60°$ .

When angles are given as algebraic expressions, set up an equation using the theorem and solve for the variable:

Write the sum of all three angle expressions equal to $180$ .
Combine like terms and solve for the variable.
Plug the variable back into each expression to find the angle measures.
Verify that all three angles sum to $180°$ .

For example, $\triangle MNO$ has $m\angle M = 2x°$ , $m\angle N = 3x°$ , and $m\angle O = 40°$ :

$2x + 3x + 40 = 180$ $5x = 140$ $x = 28$

So $m\angle M = 56°$ and $m\angle N = 84°$ . Check: $56 + 84 + 40 = 180°$ .

One thing to watch for: after solving, make sure every angle is positive. If you get a negative angle measure, that means something went wrong in your setup or the given information describes an impossible triangle.

Interior and Exterior Angle Measures

An exterior angle of a triangle is formed by extending one side of the triangle beyond a vertex. The two remote interior angles are the interior angles that are not adjacent to that exterior angle.

Exterior Angle Theorem: The measure of an exterior angle equals the sum of the two remote interior angles.

In $\triangle PQR$ , if $\angle S$ is an exterior angle at vertex $R$ , then $m\angle S = m\angle P + m\angle Q$ .
So if $m\angle P = 70°$ and $m\angle Q = 50°$ , the exterior angle at $R$ is $70° + 50° = 120°$ .

Linear Pair Relationship: An exterior angle and its adjacent interior angle form a linear pair, so they're supplementary (they add to $180°$ ).

If the exterior angle adjacent to $\angle L$ in $\triangle KLM$ measures $110°$ , then $m\angle L = 180° - 110° = 70°$ .

These two facts are really two sides of the same coin. The Exterior Angle Theorem follows directly from the Triangle Sum Theorem. If the interior angles are $m\angle P$ , $m\angle Q$ , and $m\angle R$ , and the exterior angle at $R$ is supplementary to $m\angle R$ , then:

$m\angle \text{ext} = 180° - m\angle R = m\angle P + m\angle Q$

So if you ever forget one result, you can derive it from the other.

This theorem is especially useful when you're given an exterior angle as an algebraic expression. Set the expression equal to the sum of the two remote interior angles and solve, just like you would with the Triangle Sum Theorem.

Triangle Sum Theorem applications, 9.2 Solve Applications: Sine, Cosine and Tangent Ratios. – Introductory Algebra

Parallel Lines in Triangles

When a side of a triangle (or an auxiliary line through the triangle) acts as a transversal cutting two parallel lines, you can transfer angle relationships into the triangle using these properties:

Corresponding angles are congruent: angles in the same position relative to the transversal and each parallel line have equal measures.
Alternate interior angles are congruent: angles on opposite sides of the transversal, between the parallel lines, have equal measures.
Co-interior (same-side interior) angles are supplementary: they add to $180°$ .

A common setup in Honors Geometry is a triangle with a midsegment or auxiliary line drawn parallel to one side. The strategy is to identify which angles are corresponding or alternate interior pairs, mark them as congruent, and then use the Triangle Sum Theorem to find every remaining angle.

For instance, if $\overline{AB} \parallel \overline{DE}$ and a transversal crosses both, any angle formed at $\overline{AB}$ has a corresponding congruent angle at $\overline{DE}$ . From there, you can fill in the rest of the triangle's angles using the $180°$ sum.

A useful proof technique: you can actually prove the Triangle Sum Theorem by drawing a line through one vertex parallel to the opposite side, then using alternate interior angles. That connection shows up on honors-level proofs, so it's worth understanding.

Equilateral and Isosceles Triangle Angles

Equilateral triangles have three congruent sides and three congruent angles. Since the angles must sum to $180°$ , each one measures exactly $60°$ . No calculation needed once you identify a triangle as equilateral.

The converse is also true and shows up in proofs: if all three angles of a triangle are congruent, the triangle is equilateral.

Isosceles triangles have at least two congruent sides, and the angles opposite those congruent sides (the base angles) are congruent. This is the Base Angles Theorem (sometimes called the Isosceles Triangle Theorem). The third angle is the vertex angle, located between the two congruent sides.

Given the vertex angle, find each base angle:

$\text{base angle} = \frac{180° - \text{vertex angle}}{2}$

Isosceles $\triangle XYZ$ with vertex angle $m\angle Y = 40°$ : each base angle $= \frac{180° - 40°}{2} = 70°$ .

Given a base angle, find the vertex angle:

$\text{vertex angle} = 180° - 2(\text{base angle})$

If a base angle measures $55°$ , the vertex angle $= 180° - 2(55°) = 70°$ .

The converse of the Base Angles Theorem holds too: if two angles of a triangle are congruent, then the sides opposite those angles are congruent, making the triangle isosceles. This converse is useful when a problem gives you angle information and asks you to conclude something about side lengths.

Watch out for a common mistake: the vertex angle is not always at the top of the triangle in a diagram. It's defined by being between the two congruent sides, regardless of orientation. Always identify the congruent sides first, then locate the vertex angle between them.

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