Math worksheet for calculating angles in triangles and a regular pentagon.

Here is the complete solution to the worksheet “Angles in Triangles”, with explanations for each part.

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## Section A: Work out the missing angles.

Key Rule: The sum of the interior angles in any triangle is always 180°.

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1)

Given angles: 100° and 30°
Find `a°`

Solution:
a = 180° - 100° - 30° = 50°

✔ a = 50

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2)

Given angles: 50° and 45°
Find `b°`

Solution:
b = 180° - 50° - 45° = 85°

✔ b = 85

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3)

This is a right-angled triangle (marked with square symbol → 90°).
Given angle: 60°
Find `c°`

Solution:
c = 180° - 90° - 60° = 30°

✔ c = 30

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4)

Triangle with two equal sides (marked with tick marks) → Isosceles triangle
Top angle = 34°
Base angles are equal → both are `d°`

Solution:
Sum of base angles = 180° - 34° = 146°
Each base angle = 146° ÷ 2 = 73°

✔ d = 73

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5)

Another isosceles triangle — two sides marked equal, so base angles are equal.
One base angle = 26° → other base angle = 26°
Find `e°` (the top angle)

Solution:
e = 180° - 26° - 26° = 128°

✔ e = 128

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6)

All three sides marked equal → Equilateral triangle
All angles are equal.

Solution:
f = 180° ÷ 3 = 60°

✔ f = 60

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## Section B: More complex problems

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1)

Exterior angle = 116°
Interior angle adjacent to it = 180° - 116° = 64°
Other interior angle = 41°
Find `a°`

Solution:
a = 180° - 64° - 41° = 75°

✔ a = 75

*(Alternatively: exterior angle = sum of opposite interior angles → 116° = 41° + a → a = 75°)*

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2)

Right-angled triangle (90°)
One angle = 49°
Find `b°` — this is an exterior angle at the vertex with 49°.

First, find the third interior angle:
Third angle = 180° - 90° - 49° = 41°

Now, `b°` is the exterior angle at the 49° vertex → it’s supplementary to 49°?
Wait — looking at diagram: `b°` is the exterior angle formed by extending one side from the 49° vertex.

Actually, in a triangle, the exterior angle equals the sum of the two opposite interior angles.

So: b = 90° + 41° = 131°

✔ b = 131

*(Check: 49° + 131° = 180° — yes, they are on a straight line. So correct.)*

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3)

Two lines intersecting above the triangle — top angle = 93°
This is vertically opposite to the top angle of the triangle → so top angle of triangle = 93°
Another angle = 40°
Find `c°`

Solution:
c = 180° - 93° - 40° = 47°

✔ c = 47

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4)

The triangle is split into two smaller triangles by a line from the top vertex to the base.

Left small triangle has angles: 62°, d°, and 10° (at top)
Right small triangle has angles: 38°, 10°, and the remaining part of the top angle.

Wait — actually, the 10° is part of the top angle of the big triangle. The big triangle has:

- Bottom left: 62°
- Bottom right: 38°
- Top angle: d° + 10°? No — look again.

Actually, the 10° is an angle *inside* the big triangle, splitting the top angle. But we can treat the whole big triangle as having angles:

Bottom left: 62°
Bottom right: 38°
Top angle: let's call it T

Then T = 180° - 62° - 38° = 80°

But the top angle is split into two parts: one is 10°, the other is `d°`

So: d + 10° = 80° → d = 70°

✔ d = 70

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5)

Isosceles triangle — two sides equal → base angles equal.
One base angle = 56° → other base angle = 56°
Top angle = e°

Solution:
e = 180° - 56° - 56° = 68°

✔ e = 68

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6)

Two triangles sharing a vertex. We’re given exterior angles.

Top triangle: exterior angle = 147° → interior angle = 180° - 147° = 33°
Bottom triangle: exterior angle = 138° → interior angle = 180° - 138° = 42°

Also, top triangle has another angle = 53°
Bottom triangle has angle = f°

We need to find `f`.

Notice: the two triangles share a common vertex angle (vertically opposite angles).

In top triangle: angles are 53°, 33°, and the shared angle → let’s call it x.

x = 180° - 53° - 33° = 94°

This same angle x is vertically opposite to the top angle of the bottom triangle → so bottom triangle has angles: 42°, f°, and 94°

So: f = 180° - 42° - 94° = 44°

✔ f = 44

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## Extension: Regular Pentagon

A regular pentagon has all sides and angles equal.

Sum of interior angles of a polygon = (n - 2) × 180°
For pentagon (n=5): (5-2)×180 = 3×180 = 540°

Each interior angle = 540° ÷ 5 = 108°

In the diagram, we see one interior angle of the pentagon is split into two parts: one is labeled 72°, the other is `x°`

So: 72° + x° = 108° → x = 36°

✔ x = 36

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## ✔ Final Answers:

Section A:

1) a = 50
2) b = 85
3) c = 30
4) d = 73
5) e = 128
6) f = 60

Section B:

1) a = 75
2) b = 131
3) c = 47
4) d = 70
5) e = 68
6) f = 44

Extension:

x = 36

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