When you design a horse paddock, the corners are angles. When you sketch a horse in motion, the legs form angles. When a carpenter builds a stable roof, the rafters meet at an angle. Angles are everywhere — and three simple rules let you find almost any missing one.
Three rules to know:
1. Complementary angles sum to $90°$ (they form a right angle together).
2. Supplementary angles sum to $180°$ (they form a straight line together).
3. Vertically opposite angles are equal (they form an X shape at a crossing).
Bonus: Angle sum in any triangle $= 180°$.
Angle Types — Reference
Types of angles
Acute: less than $90°$ · Right: exactly $90°$ · Obtuse: between $90°$ and $180°$ · Straight: exactly $180°$ · Reflex: more than $180°$
Worked Examples
Worked Example 1 — Supplementary angles
Two angles on a straight line: one is $112°$. Find the other.
$$112° + x = 180° \quad \mid -112°$$
$$x = 68°$$
Worked Example 2 — Triangle angle sum
A triangle has angles $47°$ and $81°$. Find the third angle.
$$47° + 81° + x = 180°$$
$$128° + x = 180° \quad \mid -128°$$
$$x = 52°$$
Worked Example 3 — Algebra with angles
Two supplementary angles: one is $(3x + 10)°$, the other is $(x + 6)°$. Find $x$ and both angles.
$$(3x + 10) + (x + 6) = 180$$
$$4x + 16 = 180 \quad \mid -16$$
$$4x = 164 \quad \mid \div 4$$
$$x = 41$$
Angles: $3(41)+10 = 133°$ and $41+6 = 47°$. Check: $133+47 = 180°$ ✓
Warm-Up — Name and find missing angles (equations given)
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Problem 1
The paddock gate opens to $135°$. What type of angle is this? The adjacent angle to the wall completes a straight line — find it.
Equation: $135° + x = 180°$
solve → name both angles →
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Problem 2
Two fence posts cross at a junction. One angle of the X is $72°$. Find all four angles at the junction.
Use: vertically opposite angles are equal; adjacent angles are supplementary.
find all 4 angles →
Core Problems — find missing angles
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Problem 3
A dojo in Tokyo has a triangular skylight. Two angles measure $55°$ and $70°$. Find the third angle. What type is it?
equation → solve → classify →
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Problem 4
In a stable roof truss, two angles are $38°$ and $38°$ (it's symmetrical). Find the top angle. What type of triangle is this?
equation → solve → name the triangle type →
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Problem 5
An equilateral triangle (all sides equal) always has all angles equal. What is each angle? Explain why using the triangle angle sum.
equation → solve → explain →
From here — write the equation yourself, then solve
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Problem 6
An angle and its complement are in a 2:1 ratio — one is twice the other. Find both angles.
define x → write equation → solve →
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Problem 7
A triangular art canvas in Barcelona has one angle that is three times another. The third angle is $20°$. Find all three angles.
let smallest = x → write equation → solve →
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Problem 8 Challenge
In a triangle, the three angles are $(x + 10)°$, $(2x - 5)°$, and $(x + 15)°$. Find $x$ and all three angles. Then: could a triangle have angles $(2x)°$, $(3x)°$, $(x + 60)°$ for the same $x$? Check.
write both equations → solve → check →
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Problem 9 Open
Design a triangular paddock corner piece where one angle is exactly $90°$ (a right angle). Choose the other two angles — they must sum correctly. Sketch it with labels, then explain: why must the two remaining angles be complementary?
choose angles → check sum → explain →
Show answers
Problem 1
$135°$ is obtuse · $x = 180° - 135° = 45°$ (acute)
Problem 2
Opposite angles: $72°$ and $72°$ · Adjacent (supplementary): $180° - 72° = 108°$ · Four angles: $72°, 108°, 72°, 108°$
Problem 3
$55° + 70° + x = 180°$ → $x = 55°$ · Acute triangle (all angles less than $90°$)
Problem 4
$38° + 38° + x = 180°$ → $x = 104°$ · Obtuse isosceles triangle
Problem 5
$3x = 180°$ → $x = 60°$ each · Because all three angles are equal and must sum to $180°$
Problem 6
Let smaller angle $= x$, larger $= 2x$ · $x + 2x = 90°$ → $3x = 90°$ → $x = 30°$ · Angles: $30°$ and $60°$
Problem 7
Let $x$ = smallest, $3x$ = largest, $20°$ = third · $x + 3x + 20° = 180°$ → $4x = 160°$ → $x = 40°$ · Angles: $40°$, $120°$, $20°$
Problem 8
First triangle: $(x+10)+(2x-5)+(x+15) = 180°$ → $4x+20=180°$ → $x=40°$ · Angles: $50°, 75°, 55°$ · Second check: $(2x)+(3x)+(x+60)=6x+60=180°$ → $x=20°$ · Angles: $40°, 60°, 80°$ — yes, valid triangle ✓
Problem 9
Open — any two angles summing to $90°$ are correct · They're complementary because $90° + \text{two others} = 180°$ means the two others sum to $90°$
Coming up next → Lesson 14: Triangle Congruence
You know how angles work inside triangles. Now: when are two triangles truly identical? Not just "similar looking" — exactly the same shape and size. Three simple tests (SSS, SAS, ASA) decide it, and they're the foundation of geometric proof.