Angles in a Triangle Worksheets - Math Monks - Free Printable

Let's solve each triangle step by step to find the missing angles. We'll use the triangle angle sum property, which states that the sum of the interior angles in any triangle is always:

> 180°

We’ll also use properties like:
- Right angles = 90°
- Isosceles triangles: two equal sides → two equal angles
- Exterior angles: an exterior angle equals the sum of the two opposite interior angles

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Problem 1

Triangle ABC, with:
- ∠A = 90° (right angle)
- ∠B = 49°

Find ∠ACB.

Solution:
Sum of angles = 180°
So:
∠ACB = 180° – 90° – 49° = 41°

✔ Answer: ∠ACB = 41°

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Problem 2

Triangle ABC:
- ∠C = 39°
- ∠B = 58°

Find ∠BAC.

Solution:
∠BAC = 180° – 39° – 58° = 83°

✔ Answer: ∠BAC = 83°

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Problem 3

Triangle ABC:
- ∠A = 38°
- ∠C = 90° (right angle)

Find ∠ABC.

Solution:
∠ABC = 180° – 38° – 90° = 52°

✔ Answer: ∠ABC = 52°

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Problem 4

Triangle ABC:
- ∠B = 77°
- Sides AC and BC have tick marks → AC = BC → Isosceles triangle
- So, angles opposite equal sides are equal → ∠A = ∠B? Wait — let’s check:

Wait: Side AC and side BC have tick marks → so AC = BC → then angles opposite them are equal.

Side AC is opposite ∠B
Side BC is opposite ∠A

So ∠A = ∠B?

But ∠B is given as 77° → so ∠A = 77°?

Then ∠C = 180° – 77° – 77° = 26°

But wait: Let’s double-check the diagram.

In triangle ABC:
- Tick marks on AC and BC → so AC = BC → base is AB
- So angles at A and B are equal?

No! If AC = BC, then angles opposite them are equal:
- AC is opposite ∠B
- BC is opposite ∠A
→ So ∠A = ∠B

Yes → ∠A = ∠B = 77°

Then ∠ACB = 180° – 77° – 77° = 26°

✔ Answer: ∠ACB = 26°

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Problem 5

Triangle ABC:
- ∠A = 15°
- ∠C = 35°

Find ∠ABC.

Solution:
∠ABC = 180° – 15° – 35° = 130°

✔ Answer: ∠ABC = 130°

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Problem 6

Triangle ABC:
- ∠A = 120°
- Sides AB and AC have tick marks → AB = AC → isosceles triangle
- So angles at B and C are equal

Let ∠B = ∠C = x

Then:
x + x + 120° = 180°
2x = 60°
x = 30°

So ∠ACB = 30°

✔ Answer: ∠ACB = 30°

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Problem 7

Triangle ABC:
- ∠A = 100°
- Side AB and side AC have tick marks → AB = AC → isosceles triangle
- So angles at B and C are equal

Let ∠B = ∠C = x

Then:
x + x + 100° = 180°
2x = 80°
x = 40°

So ∠ACB = 40°

✔ Answer: ∠ACB = 40°

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Problem 8

Triangle ABC:
- ∠B = 30°
- ∠C = 35°
- Point D and E form a straight line through A, creating an exterior angle at A
- We are to find ∠DAE

Note: Line DE passes through point A, and forms a straight line. So ∠DAE is the exterior angle at A.

First, find ∠BAC (interior angle at A):

∠BAC = 180° – 30° – 35° = 115°

Now, since DE is a straight line passing through A, and ∠DAE is the angle between DA and AE, it's the supplementary angle to ∠BAC.

So:
∠DAE = 180° – ∠BAC = 180° – 115° = 65°

✔ Answer: ∠DAE = 65°

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✔ Final Answers:

1. ∠ACB = 41°
2. ∠BAC = 83°
3. ∠ABC = 52°
4. ∠ACB = 26°
5. ∠ABC = 130°
6. ∠ACB = 30°
7. ∠ACB = 40°
8. ∠DAE = 65°

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