A bridge truss has two angles of 42° and 76°. What is the third angle? And why must the three always add to exactly 180°?
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The three interior angles of any triangle always add to exactly 180°. An exterior angle of a triangle equals the sum of the two remote interior angles.
| Term | Meaning |
|---|---|
| Interior angle | An angle inside the triangle, at a vertex |
| Exterior angle | Angle formed by extending a side of the triangle; supplementary to the adjacent interior angle |
| Remote interior angles | The two interior angles that are not adjacent to a given exterior angle |
| Equilateral triangle | All three sides equal; all three angles equal 60° |
| Isosceles triangle | Two equal sides; two equal base angles |
| Scalene triangle | All sides and angles different |
| Right-angled triangle | Contains exactly one 90° angle; the other two angles add to 90° |
Theorem
The sum of the interior angles of a triangle is 180°.
$$\alpha + \beta + \gamma = 180°$$
This is true for every triangle, no matter how flat, how tall, or how skewed.
To find a missing angle:
Theorem
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
$$e = \alpha + \gamma \quad \text{(remote interior angles)}$$
An exterior angle is formed when you extend a side of the triangle. It is always larger than either remote interior angle.
Why does this work?
The exterior angle and the adjacent interior angle form a straight line (180°). The adjacent interior angle = 180° − (sum of other two). So the exterior angle = sum of the other two.
By Angles
| Acute | All angles < 90° |
| Right | One angle = 90° |
| Obtuse | One angle > 90° |
By Sides
| Equilateral | 3 equal sides; 3 × 60° |
| Isosceles | 2 equal sides; 2 equal angles |
| Scalene | All sides & angles different |
Example 1, Find the missing angle
A triangle has angles 55°, 72°, and $x$°. Find $x$.
Step 1, Write the angle sum equation
$$55 + 72 + x = 180$$
Example 2, Algebra with angle sum
A triangle has angles $x$, $2x$, and $3x$. Find all three angles and classify the triangle.
Step 1, Set up equation
$$x + 2x + 3x = 180$$
Example 3, Exterior angle theorem
An exterior angle of a triangle is 110°. One of the remote interior angles is 45°. Find the other remote interior angle and the adjacent interior angle.
Step 1, Use exterior angle theorem
Exterior angle = sum of two remote interior angles:
$$110° = 45° + \text{other remote angle}$$
Find each missing angle. Click to reveal.
Adding all four angles (exterior + interior)
Only the three interior angles add to 180°. The exterior angle is not one of the three, it replaces the adjacent interior angle on a straight line.
Using the wrong pair for exterior angle theorem
The exterior angle equals the sum of the two remote (non-adjacent) interior angles, not all three. The adjacent interior angle is supplementary to the exterior angle (together they make 180°).
Forgetting isosceles base angles are equal
When a triangle is labelled isosceles (or has two equal sides marked), the two base angles are equal. Use this to set up equations, don't guess which angle is the vertex angle.
Not checking by substituting back
Always verify your answer: do the three angles add to 180°? A quick check prevents losing marks for arithmetic errors.
A triangle has angles 55° and 72°. The third angle is:
An exterior angle of a triangle is 120°. One remote interior angle is 50°. The other remote interior angle is:
A triangle has angles $x$, $2x$ and $3x$. What is $x$?
An isosceles triangle has a vertex angle of 100°. Each base angle measures:
A triangle has angles $(2x + 10)°$, $(x + 20)°$, and $x°$. What is the largest angle?
A triangle has angles $(3x + 5)°$, $(2x - 10)°$, and $(x + 45)°$.
In triangle $ABC$, the exterior angle at $C$ is $(5x + 8)°$. The two remote interior angles are $(3x - 4)°$ at $A$ and $(x + 22)°$ at $B$.
An isosceles triangle has two equal sides. The vertex angle (between the equal sides) is $(4x - 20)°$. Each base angle is $(x + 35)°$.
Q6
Q7
Q8
Clock angles: The hour and minute hands of a clock form a triangle with the centre of the clock face.
Part 1: At 3:00 the hands are 90° apart (3 out of 12 equal segments, each 30°: $3 \times 30° = 90°$).
Part 2: Hour hand moves $0.5°$ per minute. In 20 min: $20 \times 0.5° = 10°$. Minute hand is at 4 (120° from 12). Hour hand is at $90° + 10° = 100°$ from 12. Angle between hands: $120° - 100° = 20°$.
Part 3: The minute hand gains $5.5°$ per minute over the hour hand. Starting at 90° separation. For 90° again (closing gap to 0° then opening): the hands coincide at $90 \div 5.5 \approx 16.36$ min after 3:00. After coinciding, they reach 90° again in a further $90 \div 5.5 \approx 16.36$ min, i.e., about $32.7$ min after 3:00, so at approximately 3:33.
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