Calculating Angles (B) Worksheet | 8th Grade PDF Worksheets - Free Printable

Let's solve each of the missing angles in the worksheet step by step, using geometric principles such as:

- Sum of angles in a triangle = 180°
- Angles on a straight line = 180°
- Angles around a point = 360°
- Alternate, corresponding, and co-interior angles (when lines are parallel)
- Isosceles triangles: base angles are equal
- Parallel lines: marked with arrows

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1. First Diagram: Right triangle with 48° angle

We have a right triangle with one angle = 48°, and a right angle (90°).

- a°: Angle in triangle
$ a = 180 - 90 - 48 = 42^\circ $

- b°: Exterior angle at bottom-right vertex
Since it's on a straight line with the interior angle of the triangle:
The interior angle is 48°, so:
$ b = 180 - 48 = 132^\circ $

> ✔ a = 42°, b = 132°

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2. Triangle with two equal sides and 71° base angle

This is an isosceles triangle with one base angle = 71°, so other base angle = 71°.

- d°: Vertex angle
$ d = 180 - 71 - 71 = 38^\circ $

- c°: Angle between side and horizontal line
This is an alternate angle to the 71° angle (since the line is parallel). So:
$ c = 71^\circ $

- e°: Other angle in the triangle above
This is the same as c°, because it’s alternate to the 71° angle on the other side → $ e = 71^\circ $

- f°: Base angle of triangle → already given as 71°, but we can verify:
$ f = 71^\circ $ (base angle)

> ✔ d = 38°, c = 71°, e = 71°, f = 71°

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3. Two intersecting triangles with 82° angle

Two congruent triangles intersected at center. The vertical angles are equal.

- g°: Angle opposite 82° → vertically opposite → $ g = 82^\circ $

- h°: Angle at bottom-left — this is part of a triangle.
We know that the two triangles are congruent (marked with ticks), and the top angle is 82°.
The triangle has two equal sides (ticks), so it’s isosceles.
Let’s find the base angles.

Sum of angles = 180° → base angles = $ (180 - 82)/2 = 49^\circ $

But h° is the exterior angle at the bottom-left vertex.

The interior angle is 49°, so:
$ h = 180 - 49 = 131^\circ $

> ✔ g = 82°, h = 131°

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4. Parallelogram with 63° angle

Parallelogram properties:
- Opposite angles are equal
- Adjacent angles sum to 180°

Given one angle = 63°

- i°: adjacent to 63° → $ i = 180 - 63 = 117^\circ $
- j°: opposite to 63° → $ j = 63^\circ $
- k°: opposite to i → $ k = 117^\circ $

> ✔ i = 117°, j = 63°, k = 117°

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5. Triangle with 75° angle and tick marks

This is an isosceles triangle with two equal sides (ticks) and a 75° angle at the base.

So the two base angles are equal? Wait — the 75° is one base angle, and since the two sides are equal, the two base angles are equal.

Wait — actually, the tick marks are on the two equal sides, so the base angles are equal.

But the 75° is at the base, so the other base angle is also 75°.

Then the vertex angle l° is:
$ l = 180 - 75 - 75 = 30^\circ $

But l° is shown as an exterior angle (reflex), so:
Interior angle = 30°, so exterior angle:
$ l = 360 - 30 = 330^\circ $? No — wait.

Wait: The angle labeled l° is the exterior angle at the vertex.

But in the diagram, the triangle is drawn with a reflex angle labeled l°, so it’s the exterior angle at the vertex.

So interior angle at vertex = 30°, so:
$ l = 360 - 30 = 330^\circ $? But that seems too big.

Wait — actually, look carefully: the angle l° is the angle outside the triangle at the vertex, formed by extending one side.

But if the interior angle is 30°, then the exterior angle is:
$ l = 180 - 30 = 150^\circ $

Because it's supplementary to the interior angle.

Wait — no: the triangle has two 75° angles at the base, so vertex angle = 30°.

The l° is the exterior angle at the vertex, meaning it's adjacent to the 30° angle, forming a straight line.

So:
$ l = 180 - 30 = 150^\circ $

✔ l = 150°

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6. Right triangle with 68° angle and tick marks

Right triangle, one angle = 68°, right angle = 90°.

- m°: One of the acute angles → $ m = 180 - 90 - 68 = 22^\circ $

- n°: Exterior angle at the bottom-right vertex

Interior angle there is 68°, so:
$ n = 180 - 68 = 112^\circ $

> ✔ m = 22°, n = 112°

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7. Triangle with 118° angle and tick marks

This is a triangle with two equal sides (ticks), so isosceles.

One angle is 118° — this is the vertex angle (between the two equal sides).

Then the two base angles are equal.

Let each base angle be x:
$ x + x + 118 = 180 $
$ 2x = 62 $
$ x = 31^\circ $

Now, o° is the exterior angle at the top vertex.

So:
$ o = 180 - 118 = 62^\circ $

> ✔ o = 62°

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8. Irregular quadrilateral with 94° and tick marks

This shape looks like a parallelogram-like figure with all sides equal (tick marks), and one angle = 94°.

It's a rhombus (all sides equal), and opposite angles are equal.

Also, adjacent angles sum to 180°.

So:
- Opposite angle to 94° is also 94°
- Then the other two angles: $ (360 - 2×94)/2 = (360 - 188)/2 = 91^\circ $ each?

Wait: total sum = 360°

So: 94 + 94 + x + x = 360 → 188 + 2x = 360 → 2x = 172 → x = 86°

But p° is the interior angle at the top — which is not 94°, so it must be 86°

Wait — but p° is labeled as a reflex angle (greater than 180°). So it's the exterior angle?

No — the angle marked p° is inside the shape, but it's a "C" shape, so the angle is reflex.

So the interior angle at that vertex is 86°, so the reflex angle is:
$ p = 360 - 86 = 274^\circ $

But let's double-check.

Actually, in a rhombus, opposite angles are equal, and adjacent angles sum to 180°.

Given one angle = 94°, then adjacent angle = 180 - 94 = 86°

So the four angles are: 94°, 86°, 94°, 86°

Now, p° is the reflex angle at the top vertex — so if the interior angle is 86°, then:
$ p = 360 - 86 = 274^\circ $

✔ p = 274°

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9. Triangle with 57° angle and tick marks

This is a triangle with two equal sides (ticks), so isosceles.

One angle = 57°, but is it the vertex or base?

The 57° is at the top, and the two equal sides go down from there — so it's the vertex angle.

Then the two base angles are equal.

Let each be x:
$ x + x + 57 = 180 $
$ 2x = 123 $
$ x = 61.5^\circ $

Now, q° is the exterior angle at the bottom-right vertex.

So:
$ q = 180 - 61.5 = 118.5^\circ $

But let's check: is this correct?

Wait — the triangle has a straight line extended from the bottom-right side, and q° is the exterior angle.

Yes — so:
$ q = 180 - \text{interior base angle} = 180 - 61.5 = 118.5^\circ $

✔ q = 118.5°

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10. Two triangles with 39° and 85° angles

We have two overlapping triangles with parallel lines (arrows).

We need to find r°.

Look at the lower triangle: angles are 39°, 85°, and unknown.

Sum = 180 → third angle = $ 180 - 39 - 85 = 56^\circ $

Now, this 56° angle is corresponding to r°, because the lines are parallel (arrows), and the transversal cuts them.

So r° = 56°

✔ r = 56°

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11. Angles with 122° and 147°

We have two parallel lines, and a triangle between them.

Given:
- Top angle = 122° (on the left)
- Bottom angle = 147° (on the left)
- Need s° — the angle at the right vertex

First, note that the 122° and 147° are exterior angles.

Let’s find the interior angles.

- At top-left: 122° is exterior → interior = $ 180 - 122 = 58^\circ $
- At bottom-left: 147° is exterior → interior = $ 180 - 147 = 33^\circ $

Now, in the triangle, we have:
- Left angle = 58°
- Bottom-left angle = 33°
- So third angle (at the right vertex) = $ 180 - 58 - 33 = 89^\circ $

But s° is the exterior angle at that vertex.

So:
$ s = 180 - 89 = 91^\circ $

Wait — but s° is shown as a large arc, suggesting it's the exterior angle.

Yes — so:
$ s = 180 - 89 = 91^\circ $

✔ s = 91°

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12. Complex shape with 264° and 26°

We have a zig-zag shape with angles:

- A large angle = 264°
- Another angle = 26°
- Need t°

Let’s analyze.

We have a polygon with three sides forming a "Z" shape.

The total turn around a point is 360°.

But here, we have a reflex angle of 264°, and a small angle of 26°.

But perhaps better to consider the interior angles.

Alternatively, use the fact that the sum of exterior angles of any polygon is 360°.

But here, it's not a closed polygon — but we can think in terms of turning.

Alternatively, look at the angles around the "corner".

At the top vertex: 264° is the interior angle (reflex), so the exterior angle is $ 360 - 264 = 96^\circ $

At the bottom-left: 26° is an interior angle, so its exterior is $ 180 - 26 = 154^\circ $? No — wait.

Actually, let's trace the path.

We have a path going up, then bending at 264°, then down, then bending at t°, then back.

But perhaps simpler:

The turn at the first bend is $ 360 - 264 = 96^\circ $ (left turn)

At the second bend (bottom): 26° is the interior angle, so the turn is $ 180 - 26 = 154^\circ $? No — for a polygon, the exterior angle is $ 180 - \text{interior} $

But here, the path goes from left to right, then bends down.

Wait — let's assume the shape is made of straight segments.

We can use the idea that the sum of the turns should be 360° for a closed loop.

But here, it’s open.

Alternative approach:

Look at the triangle formed.

There is a triangle with two angles: 26° and another angle.

But the 264° is the reflex angle at the top.

So the interior angle at the top is $ 360 - 264 = 96^\circ $

Now, the triangle has:
- Top angle = 96°
- Left angle = 26°
- So right angle = $ 180 - 96 - 26 = 58^\circ $

Now, t° is the exterior angle at the right vertex.

So:
$ t = 180 - 58 = 122^\circ $

But wait — the arrow shows that t° is on the outside, so yes.

✔ t = 122°

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## ✔ Final Answers:

| Letter | Value |
|--------|-------|
| a° | 42° |
| b° | 132° |
| c° | 71° |
| d° | 38° |
| e° | 71° |
| f° | 71° |
| g° | 82° |
| h° | 131° |
| i° | 117° |
| j° | 63° |
| k° | 117° |
| l° | 150° |
| m° | 22° |
| n° | 112° |
| o° | 62° |
| p° | 274° |
| q° | 118.5°|
| r° | 56° |
| s° | 91° |
| t° | 122° |

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