Lesson 4 ~25 min Unit 3 · Geometry +85 XP

The Angle Sum of Triangles

The three interior angles of any triangle always add to $180^{\circ}$. Use this rule to find unknown angles, including base angles of isosceles triangles and algebraic expressions like $x + 2x + 30^{\circ} = 180^{\circ}$.

Today's hook: Tear the three corners off ANY triangle (try a few different shapes) and line them up edge-to-edge. They always make a perfectly straight line. Why does this work for every triangle in the universe, no matter its shape?

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Think First

warm-up

A triangle has two known angles of $70^{\circ}$ and $50^{\circ}$. Without measuring, can you predict the third angle? Write down your guess and your method — what fact about triangles tells you this?

Record your answer in your workbook.

The Big Idea

+5 XP

The interior angle sum of any triangle is always $180^{\circ}$. This is one of the most powerful rules in geometry — once you know two angles in a triangle, you can ALWAYS find the third by subtraction. The shorthand reason for this fact is "(angle sum of $\triangle$)".

For any triangle with interior angles $a, b$ and $c$, the rule is $a + b + c = 180^{\circ}$. If two angles are known, the third angle equals $180 - (\text{sum of the other two})$. The reason this is true: if you tear off the three corners and lay them along a straight line, they cover the line exactly — a straight angle equals $180^{\circ}$.

$a + b + c = 180^{\circ}$ (angle sum of $\triangle$)

Subtract from $180^{\circ}$

Two angles known? Just do $180 - (\text{sum})$ to find the third.

Give a reason

Write "(angle sum of $\triangle$)" next to your working — markers expect a justification.

It works for EVERY triangle

Big, small, fat, thin — the angles always sum to $180^{\circ}$.

What to write in your book

The interior angles of ANY triangle sum to $180^{\circ}$: $a + b + c = 180^{\circ}$.
Given two angles, the third is $180^{\circ} - (\text{sum of the other two})$.
Always write the reason "(angle sum of $\triangle$)" next to your working.

Quick check — two angles of a triangle are $50^{\circ}$ and $70^{\circ}$. What is the third angle?

What You'll Master

objectives

Know

The interior angles of any triangle sum to $180^{\circ}$
The shorthand reason "(angle sum of $\triangle$)"
In an isosceles triangle, the two base angles are equal
The third angle of a triangle $= 180^{\circ} - $ (sum of the other two)

Understand

Why the tear-and-line-up demonstration always produces a straight line
How to apply the rule to isosceles triangles (with two unknowns equal)
How to set up an algebraic equation when angles are expressions in $x$

Can Do

Find an unknown angle given the other two
Find both base angles of an isosceles triangle given the apex angle
Solve $ax + b = 180$ style equations to find an unknown variable

Words You Need

vocabulary

Interior angleAn angle inside a polygon, formed where two sides meet at a vertex.

Angle sumThe total of all interior angles. For a triangle, this is always $180^{\circ}$.

(angle sum of $\triangle$)The shorthand reason to write next to a calculation that uses the $180^{\circ}$ rule.

Apex angleIn an isosceles triangle, the angle at the top — between the two equal sides.

Base anglesThe two angles opposite the equal sides of an isosceles triangle; they are equal to each other.

Variable / unknownA letter (often $x$) that stands for an unknown angle until we solve for it.

Spot the Trap

heads-up

Wrong: "The angles in a triangle sum to $360^{\circ}$." That's angles at a point or in a quadrilateral — not a triangle.

Right: Triangle interior angle sum is $180^{\circ}$. Quadrilateral is $360^{\circ}$. Don't confuse them.

Wrong: "Both base angles of an isosceles triangle equal the apex angle." No — ONLY the two base angles are equal. The apex is usually different.

Right: If the apex is $40^{\circ}$, the two base angles share the remaining $180 - 40 = 140^{\circ}$, so each is $70^{\circ}$.

Applying to Isosceles Triangles

+5 XP

In an isosceles triangle, the two base angles are equal. So if the apex angle is known, you can find the base angles using:
$\text{base angle} = \dfrac{180^{\circ} - \text{apex}}{2}$. And if a base angle is known, the apex is $180 - 2 \times \text{base}$.

Let the apex angle be $A$ and each base angle be $B$. Then $A + B + B = 180^{\circ}$, i.e. $A + 2B = 180$. From this: if $A$ is given, $B = \dfrac{180 - A}{2}$. If $B$ is given, $A = 180 - 2B$. In an equilateral triangle, all three angles are equal, so each is $60^{\circ}$.

Apex $A$ + 2 base $B$ = $180^{\circ}$

Split what's left

Subtract the apex from $180$, then halve the remainder — that's each base angle.

Base → apex

Each base angle doubles when finding the apex: apex $= 180 - 2 \times \text{base}$.

Equilateral = $60, 60, 60$

All sides equal ⇒ all angles equal ⇒ each angle $= 180 \div 3 = 60^{\circ}$.

What to write in your book

In an isosceles triangle: $A + 2B = 180^{\circ}$ where $A$ is the apex and $B$ is each base angle.
Base angle formula: $B = \dfrac{180 - A}{2}$. Apex formula: $A = 180 - 2B$.
Equilateral → each angle is $60^{\circ}$ (since $180 \div 3 = 60$).

True or false?

An isosceles triangle with an apex angle of $40^{\circ}$ has base angles of $70^{\circ}$ each.

When Angles are Algebraic Expressions

+5 XP

Sometimes the three angles of a triangle are written using a variable, like $x^{\circ}, 2x^{\circ}$ and $30^{\circ}$. The angle-sum rule turns this into an equation: $x + 2x + 30 = 180$. Solve for $x$, then back-substitute to get each angle.

Step 1: write the equation using sum = $180^{\circ}$. Step 2: simplify by combining like terms. Step 3: solve for $x$ by isolating it. Step 4: substitute back to find each angle, and check they sum to $180^{\circ}$.

Form equation → combine → solve $x$ → substitute back

Always show the equation

Write $x + 2x + 30 = 180$ explicitly — markers want to see your set-up.

Don't forget to back-substitute

Once you find $x$, work out each actual angle. The question often wants the angles, not just $x$.

Sanity check the sum

Add the three resulting angles. They MUST equal $180^{\circ}$.

What to write in your book

When angles are written using a variable, sum them and set equal to $180^{\circ}$.
Solve for the variable (combine like terms first), then back-substitute into each expression.
Always check: the three actual angles you find must add to $180^{\circ}$.

Fill in the blank.

A triangle has angles $x^{\circ}, x^{\circ}$ and $80^{\circ}$. Solving $2x + 80 = 180$ gives $x = $ .

Watch Me Solve It · 3 examples

Watch Me Solve It · Find the missing angle

+15 XP per step

PROBLEM

Two angles of a triangle are $48^{\circ}$ and $76^{\circ}$. Find the third angle, giving a reason.

1
State the rule
Sum of interior angles $= 180^{\circ}$
2
Add the two known
$48 + 76 = 124^{\circ}$
3
Subtract from $180$
$180 - 124 = 56^{\circ}$ (angle sum of $\triangle$)
Check: $48 + 76 + 56 = 180$ ✓

AnswerThird angle $= 56^{\circ}$.

Watch Me Solve It · Isosceles base angles

+15 XP per step

PROBLEM

An isosceles triangle has an apex angle of $34^{\circ}$. Find the size of each base angle.

1
Set up
Let each base angle be $B$. Then $34 + 2B = 180$ (angle sum of $\triangle$)
2
Solve for $2B$
$2B = 180 - 34 = 146$
3
Solve for $B$
$B = 146 \div 2 = 73^{\circ}$
Check: $34 + 73 + 73 = 180$ ✓

AnswerEach base angle is $73^{\circ}$.

Watch Me Solve It · Algebraic angles

+15 XP per step

PROBLEM

A triangle has angles $x^{\circ}, 2x^{\circ}$ and $30^{\circ}$. Find $x$ and state the three angles.

1
Form the equation
$x + 2x + 30 = 180$ (angle sum of $\triangle$)
2
Combine like terms
$3x + 30 = 180$, so $3x = 150$
3
Solve and back-substitute
$x = 50$. Angles: $50^{\circ}, 100^{\circ}, 30^{\circ}$.
Check: $50 + 100 + 30 = 180$ ✓

Answer$x = 50$; angles $50^{\circ}, 100^{\circ}, 30^{\circ}$.

Common Pitfalls

heads-up

Using $360^{\circ}$ instead of $180^{\circ}$

A common slip is to use the quadrilateral or "angles at a point" sum instead of the triangle sum.

Fix: Triangle = $180^{\circ}$. Quadrilateral = $360^{\circ}$. Three sides ↔ three corners ↔ $180^{\circ}$.

Forgetting that base angles are equal

Students sometimes treat both unknowns as different in an isosceles triangle.

Fix: In isosceles, the two base angles are equal — use the SAME letter for both.

Solving for $x$ but forgetting the actual angles

If the question asks for the angles, you must substitute $x$ back into each expression.

Fix: After finding $x$, write each angle out: $x = 50 \Rightarrow$ angles $50, 100, 30$.

Copy Into Your Books

The Rule

$a + b + c = 180^{\circ}$
True for every triangle
Reason: (angle sum of $\triangle$)

Find third angle

Third $= 180 - (\text{sum of others})$
Always cite the reason

Isosceles

Base angles equal
$A + 2B = 180^{\circ}$
Apex known → $B = (180 - A) \div 2$

Algebra

Write sum = $180^{\circ}$
Combine like terms
Solve for $x$
Substitute back; check

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Angle Sum

4 problems

Four quick drills on the $180^{\circ}$ angle sum. Solve first, then reveal.

1 Two angles of a triangle are $35^{\circ}$ and $85^{\circ}$. Find the third.

Angle sum of $\triangle$ is $180^{\circ}$.$180 - 35 - 85 = 60^{\circ}$
2 An isosceles triangle has apex angle $80^{\circ}$. Find each base angle.

Base angles equal: $(180-80) \div 2$.$50^{\circ}$ each
3 A triangle has angles $90^{\circ}, x^{\circ}, x^{\circ}$. Find $x$.

$90 + 2x = 180$, so $2x = 90$.$x = 45^{\circ}$
4 A triangle has angles $3x^{\circ}, 4x^{\circ}, 5x^{\circ}$. Find $x$ and list the three angles.

$3x + 4x + 5x = 180 \Rightarrow 12x = 180$.$x = 15$; angles $45^{\circ}, 60^{\circ}, 75^{\circ}$

Complete in your workbook.

Quick Check · 5 questions

Two angles of a triangle are $70^{\circ}$ and $50^{\circ}$. Find the third.

+10 XP

An isosceles triangle has apex $50^{\circ}$. Each base angle is:

+10 XP

A triangle has angles $x^{\circ}, 2x^{\circ}, 30^{\circ}$. Find $x$.

+10 XP

A right triangle has one acute angle of $35^{\circ}$. The other acute angle is:

+10 XP

Which statement is TRUE?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Find the missing angle in each triangle. Show working and give a reason.
(a) angles $40^{\circ}, 60^{\circ}, x^{\circ}$
(b) angles $90^{\circ}, 25^{\circ}, x^{\circ}$
(c) angles $110^{\circ}, 35^{\circ}, x^{\circ}$

Answer in your workbook.

Apply Medium 3 MARKS

Q7. An isosceles triangle has apex angle of $20^{\circ}$.
(a) Find the size of each base angle.
(b) Classify the triangle by its angles (acute, right, or obtuse).
(c) Could this triangle be equilateral? Justify briefly.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. The angles of a triangle are $2x^{\circ}, (3x + 10)^{\circ}$ and $(x - 10)^{\circ}$. Find $x$, state the three angles, and classify the triangle by its angles.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $60^{\circ}$. $180 - 70 - 50$.

2. A — $65^{\circ}$. $(180 - 50) \div 2$.

3. D — $x = 50$. From $3x + 30 = 180$.

4. B — $55^{\circ}$. $180 - 90 - 35$.

5. A — The interior angles of every triangle add to $180^{\circ}$.

Show Your Working Model Answers

Q6 (3 marks): (a) $x = 180 - 40 - 60 = 80^{\circ}$ (angle sum of $\triangle$) [1]. (b) $x = 180 - 90 - 25 = 65^{\circ}$ (angle sum of $\triangle$) [1]. (c) $x = 180 - 110 - 35 = 35^{\circ}$ (angle sum of $\triangle$) [1].

Q7 (3 marks): (a) $20 + 2B = 180 \Rightarrow B = 80^{\circ}$ [1]. (b) Every angle ($20, 80, 80$) is less than $90^{\circ}$, so it's acute-angled [1]. (c) No — an equilateral has every angle $60^{\circ}$, but here two are $80^{\circ}$ [1].

Q8 (3 marks): $2x + (3x + 10) + (x - 10) = 180 \Rightarrow 6x = 180 \Rightarrow x = 30$ [1]. Angles: $2(30) = 60^{\circ}, 3(30) + 10 = 100^{\circ}, 30 - 10 = 20^{\circ}$ [1]. $100^{\circ} > 90^{\circ}$, so the triangle is obtuse-angled [1].

Stretch Challenge · +25 XP, +10 coins

Why Does Tear-and-Line-Up Always Work?

A triangle is drawn between two parallel lines so one side lies along the lower line. Use parallel-line angle rules (alternate angles and co-interior angles) to explain why the three angles of any triangle add to $180^{\circ}$ — the same fact that the tear-and-line-up demonstration shows physically. Sketch your reasoning.

Reveal solution

Draw a line through the apex of the triangle parallel to the base. The two base angles of the triangle reappear at the apex as alternate angles (equal). Together with the apex angle, the three angles at the apex sit along the upper parallel line and so form a straight angle ($180^{\circ}$). Therefore the three interior angles of the triangle sum to $180^{\circ}$.

Quick Review

Angle sum

Any triangle: $a + b + c = 180^{\circ}$.

Reason

Write "(angle sum of $\triangle$)" next to working.

Third angle

$=180 - $ (sum of other two).

Isosceles

$A + 2B = 180$; base angles equal.

Algebra

Equation → combine → solve $x$ → substitute back.

Check

Final angles must add to $180^{\circ}$.

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