Lesson 5 ~25 min Unit 3 · Geometry +85 XP

Exterior Angles of Triangles

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. Use this rule, together with the supplementary relationship to its adjacent interior angle, to find unknown angles quickly.

Today's hook: If you walk around a triangular block, you turn three times. How much do you turn altogether by the end, and how does that connect to the exterior angles you walked past at each corner?

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Think First

warm-up

A triangle has interior angles $40^{\circ}, 60^{\circ}$ and $80^{\circ}$. Extend the side at the $80^{\circ}$ corner. What's the angle outside the triangle next to it (the exterior angle)? Could you have predicted it from just the other two angles inside the triangle?

Record your answer in your workbook.

The Big Idea

+5 XP

An exterior angle of a triangle is the angle formed when one side is extended past a vertex. The Exterior Angle Theorem says: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. The exterior angle is also supplementary to its adjacent interior angle (they sit on a straight line, so they sum to $180^{\circ}$).

At any vertex, extend one side: the angle outside the triangle is the exterior angle $e$. The two interior angles at the OTHER two vertices are the remote interior angles $a$ and $b$. The Exterior Angle Theorem says $e = a + b$. Also, the exterior angle and its adjacent interior angle $c$ form a straight line, so $e + c = 180^{\circ}$.

Exterior angle $e = a + b$ (sum of remote interior angles)

Skip a vertex

Look at the two interior angles that don't share the exterior corner, just add them.

Always supplementary

Exterior + its adjacent interior = $180^{\circ}$, because they sit on a straight line.

Faster than angle sum

If you know the two remote interiors, just add, no need to subtract from $180^{\circ}$.

What to write in your book

An exterior angle is formed by extending one side of a triangle past a vertex.
Exterior Angle Theorem: $e = a + b$ where $a$ and $b$ are the two remote (non-adjacent) interior angles.
Reason to write: "(ext. ∠ of $\triangle$)".

Quick check, the two remote interior angles of a triangle are $40^{\circ}$ and $65^{\circ}$. What is the corresponding exterior angle?

What You'll Master

objectives

Know

An exterior angle is formed by extending one side of the triangle
Exterior angle = sum of the two non-adjacent (remote) interior angles
Exterior + adjacent interior = $180^{\circ}$ (supplementary, on a straight line)
Shorthand reason: "(ext. ∠ of $\triangle$)"

Understand

Why the theorem follows from the angle sum + linear pair
Why the exterior angle must be larger than each of the two remote interior angles
How to choose between using the exterior angle rule or the $180^{\circ}$ angle sum

Can Do

Identify the exterior angle and the corresponding remote interior angles in a diagram
Apply $e = a + b$ to find an unknown
Use the supplementary relationship to switch between interior and exterior angles

Words You Need

vocabulary

Exterior angleThe angle formed outside a triangle when one of its sides is extended past a vertex.

Interior angleAn angle inside the triangle, at a vertex.

Adjacent interior angleThe interior angle at the SAME vertex as the exterior angle. They share a vertex and form a straight line.

Remote interior anglesThe two interior angles at the OTHER two vertices, the ones not touching the exterior angle.

Exterior Angle TheoremAn exterior angle of a triangle equals the sum of the two remote interior angles.

SupplementaryTwo angles whose sum is $180^{\circ}$, in this case, the exterior and its adjacent interior.

Spot the Trap

heads-up

Wrong: "The exterior angle equals the interior angle at the same vertex." No, those two are SUPPLEMENTARY (add to $180^{\circ}$), not equal.

Right: The exterior angle equals the sum of the TWO REMOTE interior angles, not the one next to it.

Wrong: "Just add all three interior angles to get the exterior." All three interiors sum to $180^{\circ}$, you only want the two remote ones.

Right: Identify the exterior angle's vertex; the two interiors at the OTHER vertices are the remote ones. Add those.

Why $e = a + b$ Works

+5 XP

The theorem follows from two facts you already know: the angle sum of a triangle ($180^{\circ}$) and the straight-line supplementary pair ($180^{\circ}$).

Inside the triangle, the three interior angles satisfy $a + b + c = 180^{\circ}$, so $a + b = 180 - c$. On the straight line at the extended side, the exterior $e$ and the adjacent interior $c$ also satisfy $e + c = 180^{\circ}$, so $e = 180 - c$. Both expressions equal $180 - c$, therefore $e = a + b$. Two of the same $180 - c$, set equal, gives the theorem.

$a + b = 180 - c = e \Rightarrow e = a + b$

Same total, two ways

Both "two remote angles" and "exterior angle" complete $c$ to $180^{\circ}$, so they're equal.

Shortcut

Saves a step: instead of "$180 - $ ext-int $= a$, then $180 - a - $ other interior", just add the two remote interiors.

Always larger

An exterior angle of a triangle is ALWAYS bigger than either remote interior angle alone.

What to write in your book

Proof uses two facts: $a + b + c = 180^{\circ}$ (angle sum) and $e + c = 180^{\circ}$ (straight line).
Both rearrange to give $180 - c$, so $e = a + b$.
The exterior angle is always greater than either of the two remote interior angles alone.

True or false?

The exterior angle of a triangle is equal to its adjacent interior angle.

All Three Exterior Angles Sum to $360^{\circ}$

+5 XP

If you walk all the way around the outside of a triangle, you make ONE complete turn ($360^{\circ}$). At each vertex you turn by exactly the exterior angle. So the three exterior angles of any triangle add to $360^{\circ}$. This is a useful sanity check.

Let the exterior angles be $e_1, e_2, e_3$. By the theorem each one equals the sum of two remote interiors: $e_1 = b + c$, $e_2 = a + c$, $e_3 = a + b$. Adding:
$e_1 + e_2 + e_3 = 2a + 2b + 2c = 2(a + b + c) = 2 \times 180^{\circ} = 360^{\circ}$.

$e_1 + e_2 + e_3 = 360^{\circ}$ (sum of exterior angles of $\triangle$)

One full turn walking around

You face the same direction at the end, that's $360^{\circ}$ of total turning.

True for ALL polygons

The sum of exterior angles of ANY convex polygon is $360^{\circ}$, not just triangles.

Sanity check

After finding all three exterior angles, check they add to $360^{\circ}$.

What to write in your book

The three exterior angles of any triangle sum to $360^{\circ}$ (one full turn walking around).
Quick check: after finding all three exterior angles, confirm they total $360^{\circ}$.
This rule actually holds for any convex polygon, not just triangles.

Fill in the blank.

The three exterior angles of any triangle sum to °.

Watch Me Solve It · 3 examples

Watch Me Solve It · Direct application

+15 XP per step

PROBLEM

In a triangle, the two interior angles remote from a particular vertex are $45^{\circ}$ and $70^{\circ}$. Find the exterior angle at that vertex.

1
State the theorem
Exterior angle $=$ sum of remote interior angles
2
Add the two remote interiors
$45 + 70 = 115^{\circ}$
3
Conclude with a reason
Exterior $= 115^{\circ}$ (ext. ∠ of $\triangle$)
Check: the adjacent interior would be $180 - 115 = 65^{\circ}$, and $45 + 70 + 65 = 180^{\circ}$ ✓

AnswerExterior angle $= 115^{\circ}$.

Watch Me Solve It · Finding a remote interior

+15 XP per step

PROBLEM

An exterior angle of a triangle is $130^{\circ}$. One of the two remote interior angles is $55^{\circ}$. Find the other remote interior angle.

1
Set up the equation
$55 + x = 130$ (ext. ∠ of $\triangle$)
2
Solve for $x$
$x = 130 - 55 = 75^{\circ}$
3
Sanity check
Adjacent interior $= 180 - 130 = 50^{\circ}$. Sum: $55 + 75 + 50 = 180^{\circ}$ ✓

AnswerOther remote interior angle $= 75^{\circ}$.

Watch Me Solve It · Algebraic exterior angle

+15 XP per step

PROBLEM

A triangle has remote interior angles $x^{\circ}$ and $(x + 20)^{\circ}$, and the corresponding exterior angle measures $100^{\circ}$. Find $x$, then state the two remote interior angles.

1
Apply the theorem
$x + (x + 20) = 100$ (ext. ∠ of $\triangle$)
2
Simplify and solve
$2x + 20 = 100 \Rightarrow 2x = 80 \Rightarrow x = 40$
3
State the angles
Remote interiors: $40^{\circ}$ and $60^{\circ}$. Adjacent interior $= 180 - 100 = 80^{\circ}$.
Check: $40 + 60 + 80 = 180^{\circ}$ ✓

Answer$x = 40$; remote angles are $40^{\circ}$ and $60^{\circ}$.

Common Pitfalls

heads-up

Adding the adjacent interior angle

Students sometimes include the interior angle at the SAME vertex as the exterior angle. That's the supplement, not part of the sum.

Fix: Only the TWO interior angles at the OTHER two vertices count, the REMOTE ones.

Confusing exterior with reflex

The exterior angle is the small angle outside the triangle next to the vertex, not the reflex angle that goes the long way around.

Fix: Exterior + adjacent interior $= 180^{\circ}$. If your answer is >$180^{\circ}$, you've found the reflex by mistake.

Forgetting to state the reason

An answer without a reason loses marks. Always write "(ext. ∠ of $\triangle$)".

Fix: Tag the line with the shorthand: e.g. $x = 45 + 70 = 115^{\circ}$ (ext. ∠ of $\triangle$).

Copy Into Your Books

The Theorem

$e = a + b$
(remote interior angles)
Reason: (ext. ∠ of $\triangle$)

Supplementary pair

$e + c = 180^{\circ}$
Adjacent interior $c$ on straight line

Sum of exteriors

$e_1 + e_2 + e_3 = 360^{\circ}$
(true for any polygon)

Strategy

Spot the extended side
Locate the exterior vertex
Add the two REMOTE interiors

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Exterior Angles

4 problems

Four quick drills on the exterior angle theorem. Solve, then reveal.

1 The two remote interiors are $40^{\circ}$ and $55^{\circ}$. Find the exterior angle.

Exterior = sum of remote interiors.$40 + 55 = 95^{\circ}$
2 An exterior angle is $140^{\circ}$. Find the adjacent interior angle.

Supplementary on a straight line.$180 - 140 = 40^{\circ}$
3 An exterior angle is $120^{\circ}$, and one remote interior is $30^{\circ}$. Find the other remote interior.

$30 + x = 120$.$x = 90^{\circ}$
4 Two exterior angles of a triangle are $110^{\circ}$ and $130^{\circ}$. Find the third.

Sum of exteriors = $360^{\circ}$.$360 - 110 - 130 = 120^{\circ}$

Complete in your workbook.

Quick Check · 5 questions

Two remote interior angles are $50^{\circ}$ and $60^{\circ}$. Find the exterior angle.

+10 XP

An exterior angle is $135^{\circ}$. The adjacent interior angle is:

+10 XP

An exterior angle is $125^{\circ}$, and one remote interior is $40^{\circ}$. Find the other remote interior $x$.

+10 XP

Two of the three exterior angles of a triangle are $100^{\circ}$ and $130^{\circ}$. Find the third.

+10 XP

Which statement is the Exterior Angle Theorem?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 3 MARKS

Q6. Find the exterior angle in each case (the two remote interior angles are given). Show working with a reason.
(a) remote interiors $25^{\circ}, 80^{\circ}$
(b) remote interiors $60^{\circ}, 70^{\circ}$
(c) remote interiors $45^{\circ}, 45^{\circ}$

Answer in your workbook.

Apply Medium 3 MARKS

Q7. An exterior angle of a triangle is $145^{\circ}$.
(a) Find the adjacent interior angle.
(b) If one of the remote interiors is $90^{\circ}$, find the other remote interior.
(c) Verify the three interior angles sum to $180^{\circ}$.

Answer in your workbook.

Reason Hard 3 MARKS

Q8. The remote interior angles of a triangle are $(2x + 15)^{\circ}$ and $(3x - 5)^{\circ}$, and the corresponding exterior angle is $130^{\circ}$. Find $x$, then state the size of all three interior angles of the triangle.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B$110^{\circ}$. $50 + 60$.

2. D$45^{\circ}$. $180 - 135$.

3. C$85^{\circ}$. $125 - 40$.

4. A$130^{\circ}$. $360 - 100 - 130$.

5. B Exterior = sum of the two remote interiors.

Show Your Working Model Answers

Q6 (3 marks): (a) Ext. $= 25 + 80 = 105^{\circ}$ (ext. ∠ of $\triangle$) [1]. (b) Ext. $= 60 + 70 = 130^{\circ}$ (ext. ∠ of $\triangle$) [1]. (c) Ext. $= 45 + 45 = 90^{\circ}$ (ext. ∠ of $\triangle$) [1].

Q7 (3 marks): (a) Adjacent interior $= 180 - 145 = 35^{\circ}$ (∠s on str. line) [1]. (b) Other remote interior $= 145 - 90 = 55^{\circ}$ (ext. ∠ of $\triangle$) [1]. (c) Three interior angles: $35, 90, 55$, sum $= 180^{\circ}$ ✓ [1].

Q8 (3 marks): $(2x + 15) + (3x - 5) = 130 \Rightarrow 5x + 10 = 130 \Rightarrow 5x = 120 \Rightarrow x = 24$ [1]. Remote interiors: $2(24) + 15 = 63^{\circ}$ and $3(24) - 5 = 67^{\circ}$ [1]. Third interior $= 180 - 63 - 67 = 50^{\circ}$, or use $180 - 130 = 50^{\circ}$. Three angles: $63^{\circ}, 67^{\circ}, 50^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

Two Triangles, One Exterior Angle

A line segment $AC$ has a point $B$ between $A$ and $C$. From $B$, draw two more segments to a point $D$ above the line, making triangles $ABD$ and $BCD$ that share side $BD$. The angle $\angle ABD$ is $50^{\circ}$ and $\angle BCD$ is $35^{\circ}$. Use the exterior angle theorem twice (once in each triangle) to find $\angle ADC$, the full angle at $D$. Justify each step.

Reveal solution

In triangle $BCD$, the angle $\angle DBA$ is an exterior angle (at vertex $B$), so $\angle DBA = \angle BCD + \angle BDC$. We're told $\angle DBA = 50^{\circ}$ and $\angle BCD = 35^{\circ}$, giving $\angle BDC = 50 - 35 = 15^{\circ}$. The remaining angle of triangle $ABD$ at $D$, namely $\angle ADB$, satisfies $\angle ADB + \angle DBA + \angle DAB = 180^{\circ}$, but the question only asks for $\angle ADC = \angle ADB + \angle BDC$. Using the exterior angle theorem in triangle $ABD$ at vertex $B$ (where $\angle DBC$ is exterior and equals $180 - 50 = 130^{\circ}$): $\angle DBC = \angle DAB + \angle ADB$. The simplest answer comes by viewing $\angle ADC$ as the exterior angle of triangle $BCD$ at $D$, summed to the interior $\angle BDC$, or by combining: $\angle ADC = 50 + 35 = 85^{\circ}$ if $A, B, C$ are collinear with $D$ above the line and angles measured on the same side.

Quick Review

Exterior angle

Formed by extending one side of the triangle.

Theorem

$e = a + b$ (sum of remote interior angles).

Adjacent pair

$e + c = 180^{\circ}$ on a straight line.

Sum of three exteriors

$e_1 + e_2 + e_3 = 360^{\circ}$.

Reason shorthand

(ext. ∠ of $\triangle$).

Strategy

Spot the extension; add the two REMOTE interiors.

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