Students can practice calculating unknown angles in triangles using this worksheet, which includes problems on exterior angles and vertical angles.

Let’s solve each part step by step.

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Part a)

We have a triangle with one angle = 21°, and an exterior angle = 125° at the bottom right. The angles inside the triangle are labeled x, y, z — but note: x is inside the triangle at the top right, y is at the bottom left (inside), and z is outside? Wait — looking carefully:

Actually, in diagram a):

- Top angle = 21°
- Bottom right has an exterior angle of 125° → so the interior angle next to it (which is angle x) is:
→ 180° - 125° = 55°

So now we know two interior angles of the triangle: 21° and 55°.

Sum of angles in any triangle = 180°

So third angle (y) = 180° - 21° - 55° = 104°

Now, angle z is adjacent to angle y on a straight line → so they form a linear pair.

→ z + y = 180°
→ z = 180° - 104° = 76°

Wait — let me double-check the labels.

In diagram a):

- The triangle has vertices: top (21°), bottom-left (angle y?), bottom-right (angle x?)
- But there's also an exterior angle marked 125° at the bottom-right vertex — that means the interior angle at that vertex is 180 - 125 = 55° → that must be angle x.
- Then angle y is at the bottom-left vertex — which we found as 104°.
- Angle z is shown outside, adjacent to angle y — so yes, z = 180 - y = 76°.

✔ So:
x = 55°
y = 104°
z = 76°

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Part b)

Diagram b) shows intersecting lines forming triangles and vertical angles.

We see:

- One triangle has angles: 35°, 78°, and angle c? Wait — no.

Actually, look:

There’s a triangle with angles: 35°, and then another angle labeled “b” at the intersection point, and angle “c” at the top.

But also, there’s an exterior angle of 113° at the intersection point — which is vertically opposite to angle “a”, and adjacent to angle “b”.

First, angle a and the 113° angle are vertically opposite? No — actually, if two lines cross, vertically opposite angles are equal.

Looking: the 113° angle and angle “a” are on opposite sides of the intersection — so they are vertical angles → so a = 113°

Then, angle b is adjacent to 113° on a straight line → so b + 113° = 180° → b = 67°

Now, look at the triangle that includes angles: 35°, 78°, and angle c? Wait — no.

Actually, the triangle has angles: 35°, angle b (which we just found as 67°), and angle c? Let’s check.

Wait — the triangle formed has:

- Bottom angle = 35°
- Right angle at intersection = angle b = 67°
- Top angle = angle c?

But wait — angle c is labeled at the top, between the two lines — but is it part of the triangle?

Actually, yes — the triangle has three angles: 35°, b (67°), and c.

So sum = 180° → c = 180 - 35 - 67 = 78°

Wait — but 78° is already labeled at the top! That can’t be coincidence.

Actually, looking again: the 78° is labeled at the top vertex — that IS angle c.

So maybe we don’t need to calculate c — it’s given? But the problem asks for c.

Wait — perhaps the 78° is NOT angle c — let me re-express.

Actually, in diagram b):

- There is a triangle with:
- One angle = 35° (bottom)
- Another angle = ? at the intersection (that’s angle b)
- Third angle = ? at the top — labeled as “c”? Or is 78° separate?

Wait — the label “78°” is placed near the top vertex, and “c” is also there — likely, 78° is the measure of angle c.

But then why ask for c? Maybe it’s redundant? Or perhaps I misread.

Alternative approach:

The 113° angle is an exterior angle to the triangle? Let’s think.

Actually, the 113° angle is at the intersection point, and it’s an exterior angle to the triangle that contains the 35° angle.

Recall: Exterior angle = sum of two remote interior angles.

So, if 113° is the exterior angle at the intersection point, then it equals the sum of the two non-adjacent interior angles of the triangle.

Those would be: 35° and the top angle (let’s call it T).

So: 113° = 35° + T → T = 113 - 35 = 78°

Ah! So the top angle of the triangle is 78° — which matches the label.

Now, what is angle c? In the diagram, angle c is labeled at the same location as the 78° — so probably c = 78°.

But let’s confirm all requested angles:

We need: a, b, c

From earlier:

- Angle a: vertically opposite to 113° → so a = 113°
- Angle b: adjacent to 113° on straight line → b = 180 - 113 = 67°
- Angle c: this is the top angle of the triangle, which we found as 78° (from exterior angle theorem or from triangle sum)

Alternatively, using triangle sum: angles in triangle are 35°, b=67°, and c → 35+67+c=180 → c=78°

Perfect.

✔ So:
a = 113°
b = 67°
c = 78°

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Part c)

Two triangles sharing a common vertex (where angles f and n are).

Left triangle: angles 71°, g, and f
Right triangle: angles 55°, m, and 34° — wait, 34° is at the shared vertex? Actually, 34° is labeled at the shared vertex, but on the right side.

Actually, at the intersection point, we have:

- Left triangle has angle f at the center
- Right triangle has angle 34° at the center? But that can’t be — because f and 34° are adjacent? Or vertical?

Wait — looking: the two triangles share a common vertex. At that vertex, the angles are:

- For left triangle: angle f
- For right triangle: angle labeled 34° — but that might not be the full story.

Actually, the 34° is an angle in the right triangle — at the shared vertex.

And angle f is in the left triangle — at the same vertex.

But since the two triangles are connected at that point, and assuming the lines are straight, then angle f and the 34° angle are vertical angles? Or adjacent?

Wait — typically in such diagrams, when two triangles share a vertex like this, the angles at that vertex are either vertical or supplementary.

But here, the 34° is inside the right triangle, and f is inside the left triangle — and they are on opposite sides of the intersection — so they are vertical angles → so f = 34°

Is that correct? Let me visualize.

If two lines cross, they form two pairs of vertical angles.

Here, the two triangles are formed by two crossing lines — so yes, the angle in the left triangle at the center (f) and the angle in the right triangle at the center (34°) are vertical angles → so f = 34°

Great.

Now, right triangle: angles are 55°, 34°, and m.

Sum = 180° → m = 180 - 55 - 34 = 91°

Angle p is adjacent to the 55° angle on a straight line → so p + 55° = 180° → p = 125°

Now, left triangle: angles are 71°, g, and f (which is 34°)

So: 71 + g + 34 = 180 → g = 180 - 71 - 34 = 75°

Angle n: where is n? In the diagram, n is labeled at the shared vertex, but on the other side — probably adjacent to f.

Since f and n are on a straight line (because the line goes through), then f + n = 180°

f = 34° → n = 180 - 34 = 146°

Wait — but let’s confirm.

At the intersection point, four angles are formed:

- f (in left triangle) = 34°
- The angle opposite to f (in right triangle) = 34° (given)
- The other two angles are adjacent to these — so each should be 180 - 34 = 146°

Angle n is labeled in the left triangle? No — looking back: in diagram c), n is labeled at the shared vertex, but between the two triangles — actually, it might be the angle adjacent to f.

In standard labeling, if f is one angle at the vertex, and n is the next one along the line, then yes, f and n are supplementary.

Moreover, in the answer list, we have to find n — and it’s separate from f.

So yes: n = 180° - f = 180 - 34 = 146°

Alternatively, since the right triangle has 34° at that vertex, and n is adjacent to it on the straight line, same thing.

✔ So:
f = 34°
g = 75°
m = 91°
n = 146°
p = 125°

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Final verification:

Part a:
- x = 180 - 125 = 55° (interior adjacent to exterior)
- Triangle angles: 21 + 55 + y = 180 → y = 104°
- z = 180 - y = 76° (linear pair)

Part b:
- a = 113° (vertical to 113°)
- b = 180 - 113 = 67° (linear pair)
- c = 180 - 35 - 67 = 78° (triangle sum) OR via exterior angle: 113 = 35 + c → c=78°

Part c:
- f = 34° (vertical to given 34°)
- m = 180 - 55 - 34 = 91°
- p = 180 - 55 = 125° (linear pair)
- g = 180 - 71 - 34 = 75°
- n = 180 - f = 146° (linear pair)

All checks out.

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Final Answer:
a) x = 55°, y = 104°, z = 76°
b) a = 113°, b = 67°, c = 78°
c) f = 34°, g = 75°, m = 91°, n = 146°, p = 125°