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

Let’s solve each problem step by step. We’ll use the key rule: The angles inside any triangle always add up to 180°.

Also, remember:
- A straight line = 180°
- Vertically opposite angles are equal
- In an isosceles triangle (two sides marked same), two angles are equal
- In an equilateral triangle (all sides marked same), all angles are 60°
- Exterior angle of a triangle = sum of the two opposite interior angles

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Section A

1) Triangle with angles: 100°, 30°, and a°
→ 100 + 30 + a = 180
→ 130 + a = 180
→ a = 50

2) Triangle with angles: 50°, 45°, b°
→ 50 + 45 + b = 180
→ 95 + b = 180
→ b = 85

3) Right triangle (one angle is 90°), given 60°, find c°
→ 90 + 60 + c = 180
→ 150 + c = 180
→ c = 30

4) Isosceles triangle (two sides marked same → two base angles equal). Top angle = 34°, so bottom two angles are equal: d° and d°
→ 34 + d + d = 180
→ 34 + 2d = 180
→ 2d = 146
→ d = 73

5) Isosceles triangle (two sides marked same → two angles equal). Given one angle = 26°, and another angle e° — but note: the two equal angles are NOT the 26° one, because it’s at the bottom left, and the marks are on the other two sides. So the two equal angles are the top and right ones? Wait — let’s look again.

Actually, in diagram 5: the two sides that are marked equal are the bottom side and the right side. That means the angles opposite them are equal. The angle opposite the bottom side is the top angle. The angle opposite the right side is the bottom-left angle (which is 26°). So if those two sides are equal, then the angles opposite them are equal → so top angle = 26°. Then the third angle (e°) is at the bottom-right.

Wait — actually, standard convention: when two sides have tick marks, the angles *opposite* those sides are equal.

In diagram 5: the two sides with ticks are the bottom side and the right side. So:

- Angle opposite bottom side = top angle
- Angle opposite right side = bottom-left angle = 26°

So top angle = 26° too.

Then e° is the remaining angle: bottom-right.

So: 26 + 26 + e = 180
→ 52 + e = 180
→ e = 128

But wait — looking at the diagram, angle e° is drawn as the large angle at the bottom-right. Yes, that makes sense for an obtuse triangle.

Alternatively, maybe I misread which sides are equal. Let me double-check.

Actually, in many diagrams like this, if two sides are marked equal, the base angles are equal. But here, the 26° is at the bottom-left, and the two equal sides are the bottom and the right — so yes, angles opposite them should be equal: top angle and bottom-left angle.

So yes: top = 26°, bottom-left = 26°, so bottom-right (e°) = 180 - 52 = 128.

✔ e = 128

6) Equilateral triangle (all three sides marked same → all angles equal)
→ Each angle = 180 ÷ 3 = 60°
→ f = 60

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Section B

1) Triangle with interior angles: 41°, a°, and exterior angle 116° adjacent to the third angle.

Exterior angle = sum of two opposite interior angles.

So: 116 = 41 + a
→ a = 116 - 41 = 75

(Alternatively: the interior angle next to 116° is 180 - 116 = 64°. Then 41 + 64 + a = 180 → 105 + a = 180 → a = 75. Same answer.)

✔ a = 75

2) Right triangle (one angle 90°), one angle 49°, and exterior angle b° adjacent to the 90° angle.

First, find the third interior angle: 90 + 49 + ? = 180 → ? = 41°

Now, b° is the exterior angle adjacent to the 90° angle → so b = 180 - 90 = 90? No — wait, the diagram shows b° outside, adjacent to the 90° angle? Actually, looking at the diagram description: it's a right triangle, with the right angle at the bottom, 49° at the right, and b° is the exterior angle at the top-left vertex.

Actually, better approach: exterior angle = sum of two opposite interior angles.

b° is exterior at the top-left vertex. The two opposite interior angles are the 90° and the 49°.

So b = 90 + 49 = 139

✔ b = 139

3) Two lines crossing, forming vertical angles. One angle is 93°, so vertically opposite is also 93°. Below that is a triangle with angles: c°, 40°, and the 93° angle (since it’s part of the triangle? Wait — no.

Looking at diagram 3: two lines cross, making four angles. One is labeled 93°. Below that intersection is a triangle sharing the vertex. The triangle has angles: c°, 40°, and the angle that is vertically opposite to 93°? Or adjacent?

Actually, the 93° angle is above the triangle. The angle inside the triangle at that vertex is vertically opposite to 93°? No — vertically opposite would be directly across. If 93° is above, then the angle below (inside the triangle) is also 93° only if they are vertical — but in the diagram, likely the 93° and the triangle’s top angle are adjacent on a straight line? Wait, no.

Standard interpretation: when two lines intersect, vertical angles are equal. The triangle uses one of the vertical angles.

Assume: the 93° angle and the triangle’s top angle are vertical angles → so triangle’s top angle = 93°.

Then triangle has angles: 93°, 40°, c°

→ 93 + 40 + c = 180
→ 133 + c = 180
→ c = 47

✔ c = 47

4) Big triangle split into two smaller triangles. Left small triangle: angles 62°, d°, and shared angle. Right small triangle: angles 38°, 10°, and shared angle.

Note: the shared angle is common to both small triangles. Also, the big triangle’s top angle is d° + 10°.

We can find the shared angle from the right small triangle: 38 + 10 + shared = 180 → shared = 132? That can’t be — too big.

Wait — no: in the right small triangle, angles are 38°, 10°, and the angle at the top of that small triangle (which is part of the big triangle’s top).

Actually, let’s label:

Big triangle: bottom-left = 62°, bottom-right = 38°, top = d° + 10°

Sum: 62 + 38 + (d + 10) = 180
→ 100 + d + 10 = 180
→ d + 110 = 180
→ d = 70

Check with small triangles:

Left small triangle: angles 62°, d=70°, and the shared angle at the top-middle.

62 + 70 + shared = 180 → shared = 48°

Right small triangle: angles 38°, 10°, and shared angle = 48°? 38+10+48=96 ≠ 180 — contradiction.

Ah, I see — the 10° is not an angle of the right small triangle; it’s the difference between the big top angle and the left small triangle’s top angle.

Better approach:

Let the shared angle (at the point where the line splits the big triangle) be x.

In left small triangle: 62 + d + x = 180 → equation 1

In right small triangle: 38 + 10 + x = 180? No — the 10° is adjacent to d, so the right small triangle’s top angle is 10°, and its other angles are 38° and x.

Yes! So right small triangle: angles 38°, 10°, and x → 38 + 10 + x = 180 → x = 132

Then left small triangle: 62 + d + 132 = 180 → 194 + d = 180 → d = -14? Impossible.

Mistake: the 10° is not an angle of the right small triangle; it’s the angle between the two segments at the top. So the big triangle’s top angle is d° + 10°, and the line splits it into d° and 10°.

The shared angle x is at the base, between the two small triangles.

So for left small triangle: angles 62°, d°, x → 62 + d + x = 180

For right small triangle: angles 38°, 10°, x → 38 + 10 + x = 180 → x = 132

Then 62 + d + 132 = 180 → d = 180 - 194 = -14 — still wrong.

This suggests my assumption is incorrect. Perhaps the 10° is not an angle of the right small triangle, but rather the angle between the two parts of the top.

Alternative: the entire big triangle has angles: 62°, 38°, and (d + 10)°

Sum: 62 + 38 + d + 10 = 180 → 110 + d = 180 → d = 70

Now, check the internal line: it creates two triangles.

Left triangle: angles 62°, d=70°, and the angle at the split point — call it y.

62 + 70 + y = 180 → y = 48°

Right triangle: angles 38°, 10°, and the same y? But 38 + 10 + 48 = 96 ≠ 180 — so no.

Unless the 10° is not an angle of the right triangle, but the angle between the two segments at the top.

Perhaps the right small triangle has angles: 38°, and the angle at the split point (y), and the top angle is 10° — but then 38 + y + 10 = 180 → y = 132, as before.

Then left triangle: 62 + d + 132 = 180 → d = -14 — impossible.

I think there's a misinterpretation of the diagram.

Let me try a different approach.

In the big triangle, the sum of angles is 180°.

Bottom-left: 62°

Bottom-right: 38°

Top: let's call it T = d + 10

So 62 + 38 + T = 180 → T = 80°

So d + 10 = 80 → d = 70

Now, the line from top to base divides the big triangle into two smaller triangles.

The left small triangle has angles: 62°, d=70°, and the angle at the division point on the base — call it P.

62 + 70 + P = 180 → P = 48°

The right small triangle has angles: 38°, 10°, and the same P? But 38 + 10 + 48 = 96, not 180.

Unless the 10° is not an angle of the right small triangle, but the angle between the two segments at the top.

Perhaps the 10° is the angle between the two lines at the top, so the right small triangle's top angle is 10°, and its other angles are 38° and Q, where Q is the angle at the division point.

But then 38 + 10 + Q = 180 → Q = 132

Then for the left small triangle: 62 + d + Q = 180 → 62 + d + 132 = 180 → d = -14 — still bad.

I recall that in such problems, the 10° might be the angle between the two segments, but not part of the triangle's angles directly.

Another idea: perhaps the 10° is the difference, and we need to use the fact that the two small triangles share the side, and their angles at the base add up.

Let's denote the angle at the division point on the base as S for the left triangle and T for the right triangle. Since they are on a straight line, S + T = 180°.

For left small triangle: 62 + d + S = 180 → S = 118 - d

For right small triangle: 38 + 10 + T = 180 → T = 132

Then S + T = 180 → (118 - d) + 132 = 180 → 250 - d = 180 → d = 70

Yes! And S = 118 - 70 = 48, T = 132, and 48 + 132 = 180 — perfect.

So d = 70

✔ d = 70

5) Isosceles triangle with two sides marked equal. Given one angle 56°, and e° is the apex angle or base?

Diagram: likely the two equal sides are the legs, so base angles are equal. But here, 56° is at the left, and e° is at the top, and there's a line from top to base, creating two triangles.

Actually, the whole figure is a triangle with a line from the apex to the base, and the two sides are marked equal, so it's isosceles with AB = AC, say.

Given angle at B is 56°, and since AB = AC, then angle at C is also 56°.

Then apex angle A = 180 - 56 - 56 = 68°

But e° is shown as the angle at the apex, but divided? No, in the diagram, e° is the angle at the apex of the whole triangle, and there's a line from apex to base, but e° is labeled as the whole apex angle.

Looking back: "5) [diagram] with 56° at left, e° at top, and a line from top to base, and the two sides are marked equal."

If the two sides are equal, and 56° is a base angle, then the other base angle is also 56°, so apex e° = 180 - 56 - 56 = 68°

But why is there a line from apex to base? Perhaps it's to confuse, or perhaps e° is not the whole apex angle.

In the diagram description, it says "e°" is at the top, and there's a line down, but e° might be the angle between the left side and the line, or something.

Re-examining: in section B, problem 5: "triangle with 56° at bottom-left, e° at top, and a line from top to base, and the two sides are marked equal."

Typically, if two sides are marked equal, and it's the whole triangle, then it's isosceles with those two sides equal.

Assume the two equal sides are the left and right sides, so base angles are equal.

Given bottom-left angle = 56°, so bottom-right angle = 56°, then top angle e° = 180 - 56 - 56 = 68°

The line from top to base might be irrelevant for finding e°, or perhaps it's to indicate that e° is the whole angle.

Perhaps e° is the angle in the left small triangle.

Another possibility: the line from apex to base creates two triangles, and e° is the angle at the apex for the left small triangle.

But the problem asks for e°, and in the diagram, it's likely the whole apex angle.

To confirm, if the whole triangle is isosceles with base angles 56°, then apex is 68°.

And the line down might be a red herring or for another purpose, but since no other information, probably e° = 68°

But let's see the answer format; perhaps it's correct.

I recall that in some diagrams, if there's a line, it might be that the 56° is not a base angle of the whole triangle.

Perhaps the two equal sides are the left side and the line down, but that doesn't make sense.

Another thought: in diagram 5, the two sides marked equal are the left side and the line from apex to base? Unlikely.

Standard interpretation: the whole triangle has two sides equal, so isosceles, and 56° is a base angle, so e° = 68°.

I'll go with that.

✔ e = 68

6) Two triangles sharing a vertex, like an hourglass.

Top triangle: angles 53°, and 147° is an exterior angle? 147° is shown as an exterior angle at the top-right of the top triangle.

Similarly, bottom triangle: f° at bottom-left, 138° exterior at bottom-right.

For the top triangle: exterior angle 147° = sum of two opposite interior angles.

One interior angle is 53°, let the other be g°.

So 147 = 53 + g → g = 94°

Then the third angle of the top triangle is at the shared vertex, say h°.

53 + 94 + h = 180 → h = 33°

Now, vertically opposite angle in the bottom triangle is also 33°.

For the bottom triangle: exterior angle 138° = sum of two opposite interior angles.

One is f°, the other is the 33° angle (vertically opposite).

So 138 = f + 33 → f = 105

Check: bottom triangle angles: f=105°, 33°, and the third angle i°.

105 + 33 + i = 180 → i = 42°

Exterior angle at bottom-right is 138°, which should be adjacent to i°, so 180 - i = 180 - 42 = 138 — yes, matches.

So f = 105

✔ f = 105

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Extension: Regular pentagon

Regular pentagon: all sides and angles equal.

Each interior angle of a regular pentagon = ((5-2)*180)/5 = 540/5 = 108°

The diagram shows a regular pentagon with a diagonal drawn, and an exterior angle of 72° at the bottom-left.

72° is given as an exterior angle, which makes sense because for a regular pentagon, exterior angle = 360/5 = 72°.

Inside, there's a triangle formed by two sides and a diagonal.

The angle x° is inside the pentagon, at the bottom-right vertex, between the side and the diagonal.

At each vertex of the pentagon, the interior angle is 108°.

When you draw a diagonal from one vertex to another, it splits the interior angle.

In a regular pentagon, drawing a diagonal from a vertex creates an isosceles triangle.

Specifically, consider the triangle formed by three consecutive vertices: A-B-C, with diagonal A-C.

Angle at B is 108°.

Sides AB = BC (regular pentagon), so triangle ABC is isosceles with AB = BC.

Thus, angles at A and C in triangle ABC are equal.

Sum of angles in triangle ABC: angle at B is 108°, so angles at A and C are (180 - 108)/2 = 36° each.

But in the diagram, x° is likely the angle between the side and the diagonal, which would be part of the interior angle.

At vertex C, the interior angle is 108°, composed of the angle from the diagonal and the side.

In triangle ABC, angle at C is 36°, which is the angle between side BC and diagonal AC.

Then the remaining part of the interior angle at C is between diagonal AC and side CD, which would be 108° - 36° = 72°.

But in the diagram, x° is shown at the bottom-right, and there's a 72° exterior angle at bottom-left.

Perhaps x° is the angle in the triangle formed.

Looking at the diagram description: "regular pentagon" with a diagonal, and x° is inside, near the bottom-right, and 72° is exterior at bottom-left.

In a regular pentagon, when you draw a diagonal, it forms a triangle with two sides.

The triangle that includes x° might be the one with vertices at the bottom-left, bottom-right, and the top or something.

Standard result: in a regular pentagon, the diagonal creates angles of 36°, 72°, etc.

Specifically, the triangle formed by two diagonals and a side is golden triangle, but here it's simpler.

Consider the pentagon ABCDE, with A at bottom-left, B at bottom-right, C at top-right, D at top, E at top-left.

Draw diagonal from A to C.

Then at vertex B, interior angle 108°.

Triangle ABC: AB = BC (sides of pentagon), so isosceles.

Angle at B is 108°, so angles at A and C in triangle ABC are (180-108)/2 = 36° each.

Now, at vertex A, the interior angle of pentagon is 108°, which is composed of angle from triangle ABC (36°) and the angle from triangle AED or something.

Actually, at vertex A, the interior angle is between sides EA and AB.

Diagonal AC is drawn, so it splits the 108° angle into two parts: angle between EA and AC, and angle between AC and AB.

In triangle ABC, angle at A is 36°, which is between AB and AC.

So the other part, between EA and AC, is 108° - 36° = 72°.

Similarly at other vertices.

In the diagram, x° is likely the angle at B between AB and the diagonal, but in our case, at B, the diagonal is not drawn from B; in this setup, diagonal is from A to C, so at B, no diagonal emanates.

Perhaps the diagonal is from bottom-left to bottom-right? But that would be a side.

Another common diagram: draw a diagonal from one vertex to a non-adjacent vertex, forming a triangle with two sides.

For example, from vertex A to vertex C, as above.

Then the triangle is ABC, with angles 36° at A, 108° at B, 36° at C.

But x° is shown at the bottom-right, which might be vertex B or C.