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Triangle Angle
Relationships

Discover how the angles inside every triangle always add to \(180^\circ\), and learn to find missing angles using the Triangle Angle Sum and Exterior Angle Theorem.

Pre-Algebra

What Is an Angle in a Triangle?

An angle is the amount of turn between two sides that meet at a vertex. Every triangle has exactly three vertices and three interior angles. We measure angles in degrees (\(^\circ\)).

A B C
\(\angle A\)
\(\angle B\)
\(\angle C\)
side c
side a
side b

Interior vs. Exterior Angles

Interior Angle
An angle inside the triangle, formed where two sides meet. Every triangle has three interior angles.
Exterior Angle
An angle formed by one side of the triangle and the extension of an adjacent side. It sits outside the triangle.
A B C
interior
exterior

Why Does This Matter?

Triangle angle relationships are the foundation of geometry. They show up in architecture, engineering, art, and everyday problem-solving. If you know two angles, you can always find the third: a simple but powerful idea.

Architecture
Roof trusses, bridges, and support beams all rely on triangles and their angle properties.
Navigation
Pilots and sailors use triangle angles to calculate distances and plot courses.
Art & Design
Artists use triangular composition to create balance and guide the viewer's eye.

The Triangle Angle Sum Theorem

A B C
\(\angle A\)
\(\angle B\)
\(\angle C\)
\(\angle A + \angle B + \angle C = 180^\circ\)

This works for every triangle: acute, right, or obtuse. No exceptions.

The Exterior Angle Theorem

A B C
\(\angle A\)
\(\angle B\)
\(\angle d\)
\(\angle d = \angle A + \angle B\)

The exterior angle \(\angle d\) is always larger than either remote interior angle on its own.

Angle Sum vs. Exterior Angle

Triangle Angle Sum Exterior Angle Theorem
What it says \(\angle A + \angle B + \angle C = 180^\circ\) \(\text{ext} = \angle_1 + \angle_2\)
Angles used All 3 interior angles 2 remote interior + 1 exterior
When to use Know 2 interior angles, find the 3rd Know an exterior angle, find remote interior (or vice versa)
Key number \(180^\circ\) Sum of two remotes

Why Does the Shortcut Work?

1
Angle Sum: \(\angle A + \angle B + \angle C = 180^\circ\)
2
Supplementary pair: \(\angle C + \angle d = 180^\circ\) (they form a straight line)
3
Set them equal: \(\angle A + \angle B + \angle C = \angle C + \angle d\)
4
Subtract \(\angle C\): \(\angle A + \angle B = \angle d\) ✓

How to Find a Missing Angle

1
Identify what you know. Label the angles you are given and the angle you need to find.
2
Choose a theorem. If you have two interior angles, use the Angle Sum. If you see an exterior angle, use the Exterior Angle Theorem.
3
Write the equation. Plug in the known values and solve for the unknown.
4
Check your answer. Do all three interior angles sum to \(180^\circ\)? Does the exterior angle equal the two remotes?

Example 1: Find the Missing Angle

A B C
\(50^\circ\)
\(70^\circ\)
?

Example 2: Exterior Angle

A B C
\(45^\circ\)
\(65^\circ\)
\(\angle d = ?\)

Example 3: Algebraic Angles

A B C
\((2x)^\circ\)
\((x+10)^\circ\)
\(50^\circ\)

Types of Triangles by Angle

Acute
All three angles are less than \(90^\circ\).
Right
Exactly one angle is exactly \(90^\circ\).
Obtuse
Exactly one angle is greater than \(90^\circ\).

Isosceles & Equilateral Shortcuts

Isosceles Triangle
Two sides are equal, so the two base angles are also equal. If the apex is \(\angle A\), then \(\angle B = \angle C = \frac{180^\circ - \angle A}{2}\).
Equilateral Triangle
All three sides are equal, so all three angles are equal: each is \(\frac{180^\circ}{3} = 60^\circ\).

Common Mistakes to Avoid

✕ Using \(360^\circ\) Instead of \(180^\circ\)
A full circle is \(360^\circ\), but a triangle's angles only sum to \(180^\circ\). Don't confuse them.
✕ Mixing Up Interior and Exterior
An exterior angle is not part of the triangle's angle sum. Make sure you know which angle you are looking at.
✕ Forgetting to Solve for the Variable
Finding \(x\) is only half the job. Plug \(x\) back in to get the actual angle measure.

Quick Reference Table

Rule Formula When to Use
Angle Sum \(\angle A + \angle B + \angle C = 180^\circ\) Finding a missing interior angle
Exterior Angle \(\text{ext} = \angle_1 + \angle_2\) Exterior angle given, or two remotes given
Supplementary \(\text{int} + \text{ext} = 180^\circ\) Converting between interior and exterior
Equilateral \(\text{Each angle} = 60^\circ\) All sides/angles are equal
Isosceles Base angles are equal Two sides are equal

Question 1 of 5

A triangle has angles of \(55^\circ\) and \(80^\circ\). What is the measure of the third angle?

Question 2 of 5

An exterior angle of a triangle measures \(130^\circ\). One of the remote interior angles is \(50^\circ\). What is the other remote interior angle?

Question 3 of 5

In a triangle, one angle is \(3x\), another is \(2x\), and the third is \(40^\circ\). What is the value of \(x\)?

Question 4 of 5

An isosceles triangle has a vertex angle of \(40^\circ\). What is the measure of each base angle?

Question 5 of 5

A triangle has angles \((x + 15)^\circ\), \((2x - 5)^\circ\), and \(60^\circ\). An exterior angle is formed at the \(60^\circ\) vertex. What is the measure of that exterior angle?