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.