Practice worksheet for finding missing angles in triangles using geometric principles.

Let’s solve each triangle step by step. We’ll use two key rules:

1. Angles on a straight line add up to 180°.
2. The three angles inside any triangle always add up to 180°.

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Part a)

We have a triangle with one angle = 21°, and an exterior angle next to angle x = 125°.

- Since the exterior angle (125°) and angle x are on a straight line:
→ x = 180° - 125° = 55°

Now we know two angles in the triangle: 21° and x = 55°.

- So, y = 180° - 21° - 55° = 104°

Angle z is on a straight line with angle y:
→ z = 180° - 104° = 76°

✔ Check: 21 + 55 + 104 = 180 ✔️
And 104 + 76 = 180 ✔️

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Part b)

We have intersecting lines forming vertical angles and triangles.

First, look at the top right: angle labeled 113° and angle a are on a straight line.

→ a = 180° - 113° = 67°

Angle b is vertically opposite to 113°? Wait — actually, looking at the diagram: angle b and 113° are adjacent on a straight line? No — let’s think again.

Actually, angle b and 113° are on a straight line? Let me re-express.

Looking carefully: The line that has 113° also has angle a next to it → so yes, a = 180 - 113 = 67°.

Now, angle b is vertically opposite to the angle that’s part of the triangle with 35° and c.

Wait — better approach: In the small triangle on the left, we have angles: 35°, c, and the angle at the intersection point (which is same as angle b?).

Actually, angle b and the 113° angle are on a straight line? No — they’re adjacent but not necessarily supplementary unless specified.

Wait — let’s label properly.

From the diagram:

- At the top right intersection: one angle is 113°, and angle a is next to it on the same straight line → so a = 180 - 113 = 67°

- Angle b is vertically opposite to the angle that is inside the triangle with 35° and c? Actually, no — angle b is adjacent to 113°? Let me think differently.

Actually, angle b and 113° form a linear pair? Looking at the drawing: if 113° and angle a are on one side, then angle b is across from them? Hmm.

Better: Use vertical angles.

At the intersection point where 113° is marked, the angle directly opposite to it is angle b? Or is it?

Actually, standard rule: when two lines cross, vertical angles are equal.

So if 113° is one angle, the angle opposite to it (vertically) should be equal. But in the diagram, angle b is drawn adjacent to 113°? I need to reinterpret.

Wait — perhaps angle b is the vertical angle to the 113°? That would make sense.

But let’s check the triangle: there’s a triangle with angles 35°, c, and another angle which is vertically opposite to 113°? Or maybe not.

Alternative plan:

Look at the triangle that contains 35°, c, and the angle at the bottom right vertex.

That bottom right vertex angle is actually angle b? Or is it?

Actually, let’s assume:

In the lower-left triangle: angles are 35°, c, and let’s call the third angle “d”.

Then d + 35 + c = 180.

Also, angle d and angle b are on a straight line? Or vertical?

This is getting messy. Let me try a different way.

Notice: angle b and 113° are on a straight line? If you follow the line, yes — because 113° and angle a are on one side, and angle b is on the other side of the same line? Not sure.

Wait — here's a better idea:

Use the fact that around a point, angles add to 360°, but that might complicate.

Let’s go back to basics.

In part b):

There is a triangle with angles: 35°, c, and let’s say angle at the intersection = let’s call it “e”.

Then e + 35 + c = 180.

Also, angle e and 113° are on a straight line? Because they are adjacent along the transversal? Yes! Look: the line that goes through 113° continues and forms angle e with the other line.

Actually, angle e and 113° are supplementary because they form a linear pair.

So: e = 180 - 113 = 67°

Then in the triangle: 35 + c + 67 = 180 → c = 180 - 35 - 67 = 78°

Wait — but in the diagram, c is already labeled as 78°? Oh! Look: in the diagram, near the top left, it says "c" and "78" — so probably c = 78° is given? Or is 78° the value of c?

Looking back at the original image description: in part b), it shows “c” and “78” together — likely meaning angle c = 78°.

Similarly, “b” and “113” — wait no, 113 is separate.

Actually, rereading: in part b), the diagram has:

- A triangle with angles: 35°, c, and another angle.

- Outside, near the top, it says “c” and “78” — probably indicating that angle c is 78°? Or is 78° the measure of angle c?

I think the “78” written next to “c” means that angle c = 78°. Similarly, “113” is given, and “a”, “b” are unknowns.

Assume that in the diagram, angle c is labeled as 78° — so we can take c = 78°.

Then in the triangle: angles are 35°, c=78°, and the third angle (let’s call it d).

So d = 180 - 35 - 78 = 67°

Now, angle d and angle b are vertically opposite? Or on a straight line?

Actually, angle d and angle b are on a straight line? No — they are adjacent.

Wait — angle d is inside the triangle, and angle b is outside, sharing the same vertex.

Actually, angle b and angle d are on a straight line? Let’s see: if you extend the side, then yes — angle b and angle d form a linear pair.

So b = 180 - d = 180 - 67 = 113°

But 113° is already given elsewhere — and angle a is next to it.

If b = 113°, then since b and a are on a straight line? No — in the diagram, a and 113 are on the same line, so a = 180 - 113 = 67°

But if b = 113°, that matches the given 113°? Confusion.

Perhaps the 113° is angle b? Let me reinterpret the diagram based on common problems.

Standard problem: two lines intersect, forming vertical angles. One angle is 113°, so its vertical angle is also 113°, and the adjacent angles are 67° each.

In this case, likely:

- The 113° is one angle at the intersection.

- Then angle a is adjacent to it on a straight line → a = 180 - 113 = 67°

- Angle b is vertically opposite to 113° → so b = 113°

- Now, in the triangle below, we have angles: 35°, c, and the angle at the top which is vertically opposite to a? Or what?

The triangle has vertices: bottom left (35°), top left (c), and bottom right (which is angle b? Or part of it).

Actually, the triangle includes the angle that is vertically opposite to a.

Since a = 67°, and it's at the intersection, the angle inside the triangle at that vertex is also 67° (because it's the same angle or vertical?).

Let’s define:

At the intersection point of the two lines, four angles are formed:

- Top-right: 113°

- Bottom-left: also 113° (vertical angle)

- Top-left: 67°

- Bottom-right: 67°

Now, the triangle in question has:

- Bottom-left vertex: 35°

- Top-left vertex: angle c

- Bottom-right vertex: which is the 67° angle (bottom-right of the intersection)

So in the triangle: angles are 35°, c, and 67°

Sum: 35 + c + 67 = 180 → c = 180 - 102 = 78°

Which matches the "78" written next to c in the diagram.

So:

- a = 67° (adjacent to 113° on straight line)

- b = 113° (vertical angle to the given 113°)

- c = 78° (calculated from triangle)

Perfect.

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Part c)

Two triangles sharing a common vertex, with vertical angles.

Given:

Left triangle: angles g, 71°, and f

Right triangle: angles m, 55°, and n

At the intersection, angle f and angle n are vertically opposite? Or are they adjacent?

Actually, angle f and angle n are on a straight line? No — they are vertical angles.

Look: the two lines cross, so angle f and angle n are vertically opposite → so f = n

Also, there's a 34° angle shown — which is probably the angle between the two triangles, i.e., the vertical angle to both f and n? No.

Actually, the 34° is likely the angle at the intersection for one of the triangles.

Re-examining: in the diagram, it shows "n" and "34°" together — probably meaning angle n = 34°? Or is 34° a separate angle?

Looking at the text: "f n 34°" — likely, angle n is 34°, and f is adjacent or something.

Actually, standard setup: when two lines intersect, they form two pairs of vertical angles.

Here, the angle labeled 34° is probably one of the angles at the intersection, and f and n are the other angles.

But in the diagram, it seems that angle n is the same as the 34°? Or is 34° the measure of angle n?

I think "n" and "34°" are meant to indicate that angle n = 34°.

Similarly, in the right triangle, angles are m, 55°, and n=34°.

So in right triangle: m + 55 + 34 = 180 → m = 180 - 89 = 91°

Now, angle f is vertically opposite to angle n? Or adjacent?

Since the lines cross, angle f and angle n are on opposite sides — actually, angle f and the 34° angle are vertical angles? Let's see.

If angle n = 34°, and it's at the intersection, then the angle vertically opposite to it is also 34°, but that might be angle f? Or not.

In the left triangle, angle f is at the intersection, and in the right triangle, angle n is at the same intersection.

Typically, in such diagrams, angle f and angle n are vertical angles, so f = n.

But if n = 34°, then f = 34°.

Then in left triangle: angles g, 71°, and f=34°

So g = 180 - 71 - 34 = 75°

Now, what about p? Angle p is shown outside the right triangle, adjacent to the 55° angle.

So p and 55° are on a straight line → p = 180 - 55 = 125°

Confirm:

- f = 34° (vertical to n, or given as same)

- g = 75°

- m = 91°

- n = 34°

- p = 125°

Check triangles:

Left: 75 + 71 + 34 = 180 ✔️

Right: 91 + 55 + 34 = 180 ✔️

Straight line: 55 + 125 = 180 ✔️

Perfect.

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Final Answers:

a)
x = 55°
y = 104°
z = 76°

b)
a = 67°
b = 113°
c = 78°

c)
f = 34°
g = 75°
m = 91°
n = 34°
p = 125°

Note: In part c, f and n are both 34° because they are vertical angles or as per diagram labeling.

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