Triangle Angle Sum Theorem

📐 Triangle Angle Sum Theorem

Part 1 of 5 — The 180° Rule

Topics in This Part

🔑 Key Concept: In every triangle — no matter its shape or size — the three interior angles add up to exactly $180^\circ$. That single fact is the engine behind nearly all triangle problems.

What the Theorem Says

The Triangle Angle Sum Theorem states:

$\text{The sum of the interior angles of a triangle is } 180^\circ.$

If a triangle has angles $A$, $B$, and $C$, then:

$m\angle A + m\angle B + m\angle C = 180^\circ$

The notation $m\angle A$ just means "the measure of angle $A$" — its size in degrees.

A Few Quick Examples

💡 This holds for any triangle — equilateral, isosceles, scalene, right, or obtuse. The shape changes, but the total never does.

Why It's Always True

Here's the classic "tear and line up" picture. Take any triangle, tear off its three corners, and place them tip-to-tip along a straight line:

$\underbrace{\;\angle A + \angle B + \angle C\;}_{\text{three corners}} = \underbrace{180^\circ}_{\text{a straight line}}$

The three angles fit together perfectly to form a straight angle — and a straight angle measures exactly $180^\circ$.

🔑 The big idea: "Three triangle corners" and "one straight line" are the same total: $180^\circ$. We'll prove this rigorously with parallel lines in Part 4.

Concept Check 🎯

Finding a Missing Angle

If you know two angles, you can always find the third — just subtract from $180^\circ$.

$m\angle C = 180^\circ - m\angle A - m\angle B$

Worked Example

A triangle has angles of $35^\circ$ and $65^\circ$. Find the third angle.

$m\angle C = 180^\circ - 35^\circ - 65^\circ = 180^\circ - 100^\circ = 80^\circ$

✅ Check: $35^\circ + 65^\circ + 80^\circ = 180^\circ$ ✓

Fill in the Steps 🔽

A triangle has two known angles of $70^\circ$ and $60^\circ$. Walk through finding the third.

Find the Third Angle 🧮

Two angles of a triangle are given. Find the missing third angle (in degrees — enter just the number).

1) $40^\circ$ and $75^\circ$ $\;\Rightarrow\;$ third angle $= \,?$ 2) $90^\circ$ and $25^\circ$ $\;\Rightarrow\;$ third angle $= \,?$ 3) $110^\circ$ and $30^\circ$ $\;\Rightarrow\;$ third angle $= \,?$

Part 1 Recap

• Every triangle's interior angles sum to $\mathbf{180^\circ}$.
• The notation $m\angle A$ means "the measure of angle $A$."
• To find a missing angle: $180^\circ - (\text{sum of the other two})$.

Next up: classifying triangles by their angles, and what the theorem forces each type to look like.

📐 Triangle Angle Sum Theorem

Part 2 of 5 — Classifying Triangles by Their Angles

🔑 The Idea: Because all three angles must total $180^\circ$, a triangle can have at most one right or obtuse angle. The $180^\circ$ budget simply isn't big enough for two.

Three Types by Angle

💡 Why only one big angle? Two angles of $90^\circ$ already use the entire $180^\circ$, leaving $0^\circ$ for the third — which is impossible. So a triangle can never have two right angles or two obtuse angles.

The Right-Triangle Shortcut

In a right triangle, one angle is $90^\circ$, so the other two must add to the remaining $90^\circ$:

$m\angle A + m\angle B = 180^\circ - 90^\circ = 90^\circ$

The two non-right angles are called complementary (they sum to $90^\circ$).

Worked Example

A right triangle has one acute angle of $32^\circ$. Find the other acute angle.

$90^\circ - 32^\circ = 58^\circ$

✅ Check: $90^\circ + 32^\circ + 58^\circ = 180^\circ$ ✓

Classify Each Triangle 🔽

First find the missing angle, then pick the triangle's type.

Concept Check 🎯

Right-Triangle Practice 🧮

Each triangle is a right triangle (one $90^\circ$ angle). Given one acute angle, find the other.

1) acute angle $= 41^\circ \;\Rightarrow\;$ other acute angle $= \,?$ 2) acute angle $= 17^\circ \;\Rightarrow\;$ other acute angle $= \,?$ 3) acute angle $= 45^\circ \;\Rightarrow\;$ other acute angle $= \,?$

📐 Triangle Angle Sum Theorem

Part 3 of 5 — Algebra with Triangle Angles

🔑 Why it works: When angles are written with variables, set their sum equal to $180^\circ$, then solve the equation. The theorem turns a geometry problem into ordinary algebra.

Setting Up the Equation

When angles are expressions in $x$, just add them and set the total to $180^\circ$.

Worked Example: $x$, $2x$, $3x$

The three angles of a triangle are $x$, $2x$, and $3x$. Find $x$ and all three angles.

$x + 2x + 3x = 180^\circ$ $6x = 180^\circ$ $x = 30^\circ$

So the angles are $30^\circ$, $60^\circ$, and $90^\circ$.

✅ Check: $30^\circ + 60^\circ + 90^\circ = 180^\circ$ ✓ — and it's a right triangle.

Worked Example: with a Constant

The angles of a triangle are $(x + 10)^\circ$, $(2x)^\circ$, and $(x + 30)^\circ$. Solve for $x$.

Add them and set equal to $180^\circ$:

$(x + 10) + 2x + (x + 30) = 180$

Combine like terms — $x + 2x + x = 4x$ and $10 + 30 = 40$:

$4x + 40 = 180$ $4x = 140$ $x = 35$

The angles are $45^\circ$, $70^\circ$, and $65^\circ$.

✅ Check: $45^\circ + 70^\circ + 65^\circ = 180^\circ$ ✓

Build the Equation 🔽

A triangle has angles $2x$, $3x$, and $4x$. Fill in each step.

Solve for $x$ 🧮

Each set lists a triangle's three angles. Set up "sum $= 180$" and solve for $x$ (enter just the number).

1) $x,\ x,\ x \;\Rightarrow\; x = \,?$ 2) $x,\ 2x,\ 2x \;\Rightarrow\; x = \,?$ 3) $(x+20),\ (x+40),\ x \;\Rightarrow\; x = \,?$

Concept Check 🎯

📐 Triangle Angle Sum Theorem

Part 4 of 5 — The Exterior Angle Theorem

🔑 Big Payoff: Extend one side of a triangle and you create an exterior angle. It equals the sum of the two remote (non-adjacent) interior angles — a direct consequence of the $180^\circ$ rule.

What Is an Exterior Angle?

Extend a side of a triangle past a vertex. The angle between that extension and the next side is an exterior angle.

An exterior angle and the interior angle right next to it form a straight line, so they are supplementary:

$\text{exterior angle} + \text{adjacent interior angle} = 180^\circ$

The Exterior Angle Theorem

$\text{exterior angle} = \text{sum of the two } \textbf{remote} \text{ interior angles}$

The two "remote" interior angles are the ones not touching the exterior angle.

💡 Why this is true: The three interior angles sum to $180^\circ$, and the exterior angle plus its adjacent interior angle also sum to $180^\circ$. Subtracting that shared adjacent angle from both leaves: exterior $=$ the other two interiors.

Worked Examples

Example 1 — Using the theorem directly

A triangle has remote interior angles of $50^\circ$ and $70^\circ$. The exterior angle at the third vertex is:

$50^\circ + 70^\circ = 120^\circ$

✅ Check via supplements: The adjacent interior angle is $180^\circ - 50^\circ - 70^\circ = 60^\circ$, and $180^\circ - 60^\circ = 120^\circ$ ✓ — same answer.

Example 2 — Solving for $x$

An exterior angle measures $(5x)^\circ$, and the two remote interior angles are $(2x)^\circ$ and $40^\circ$.

$5x = 2x + 40$ $3x = 40$ $x = \tfrac{40}{3} \approx 13.3$

(The exterior angle theorem sets exterior equal to the sum of the two remote interiors.)

Exterior Angle Reasoning 🔽

Exterior Angle Practice 🧮

Use exterior angle $=$ sum of the two remote interior angles. Enter the exterior angle in degrees.

1) remote interiors $45^\circ$ and $60^\circ \;\Rightarrow\;$ exterior $= \,?$ 2) remote interiors $90^\circ$ and $35^\circ \;\Rightarrow\;$ exterior $= \,?$ 3) remote interiors $72^\circ$ and $38^\circ \;\Rightarrow\;$ exterior $= \,?$

Concept Check 🎯

📐 Triangle Angle Sum Theorem

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) use the $180^\circ$ rule to find a missing angle, (2) classify triangles by angle, (3) solve angle equations with algebra, and (4) apply the Exterior Angle Theorem. Let's put it all together.

Quick Reference

⚠️ Common traps: Don't confuse the exterior angle with its adjacent interior angle — they're supplements, not equals. And remember a triangle can have at most one angle that is $90^\circ$ or larger.

Mixed Practice 🎯

One More Set 🧮

Enter each answer in degrees (number only).

1) A triangle has angles $55^\circ$ and $80^\circ$. The third angle $= \,?$ 2) A right triangle has one acute angle of $19^\circ$. The other acute angle $= \,?$ 3) Remote interior angles $48^\circ$ and $66^\circ$. The exterior angle $= \,?$

Exit Quiz ✅

Answer all three to finish the lesson.