Angles in a Triangle Worksheets - Math Monks - Free Printable

Here are the step-by-step solutions to find the missing angles for each triangle.

Key Rules to Remember:
1. Triangle Sum: The three angles inside a triangle always add up to 180°.
2. Straight Line: Angles on a straight line add up to 180°.
3. Right Angle: A square symbol means the angle is 90°.

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1.
* Add the known angles: $70^\circ + 45^\circ = 115^\circ$.
* Subtract from 180°: $180^\circ - 115^\circ = 65^\circ$.
* $x = 65$

2.
* Add the known angles: $82^\circ + 35^\circ = 117^\circ$.
* Subtract from 180°: $180^\circ - 117^\circ = 63^\circ$.
* $x = 63$

3.
* Add the known angles: $58^\circ + 86^\circ = 144^\circ$.
* Subtract from 180°: $180^\circ - 144^\circ = 36^\circ$.
* $x = 36$

4.
* First, find the third angle *inside* the triangle. Add the two known inside angles: $61^\circ + 52^\circ = 113^\circ$.
* The angle next to $x$ inside the triangle is: $180^\circ - 113^\circ = 67^\circ$.
* Angles on a straight line add to 180°. So, $x + 67^\circ = 180^\circ$.
* $x = 180^\circ - 67^\circ = 113^\circ$.
* *(Shortcut: The exterior angle $x$ is equal to the sum of the two opposite interior angles: $61 + 52 = 113$)*.
* $x = 113$

5.
* Find $y$: The angle $y$ and the $120^\circ$ angle are on a straight line.
* $y = 180^\circ - 120^\circ = 60^\circ$.
* Find $x$: Now look at the triangle. The angles are $x$, $50^\circ$, and $y$ ($60^\circ$).
* Add knowns: $50^\circ + 60^\circ = 110^\circ$.
* Subtract from 180°: $180^\circ - 110^\circ = 70^\circ$.
* $x = 70, y = 60$

6.
* Add the known angles: $77^\circ + 64^\circ = 141^\circ$.
* Subtract from 180°: $180^\circ - 141^\circ = 39^\circ$.
* $x = 39$

7.
* Add the known angles: $13^\circ + 29^\circ = 42^\circ$.
* Subtract from 180°: $180^\circ - 42^\circ = 138^\circ$.
* $x = 138$

8.
* Find $q$: Look inside the triangle first. Add the bottom angles: $64^\circ + 38^\circ = 102^\circ$.
* $q = 180^\circ - 102^\circ = 78^\circ$.
* Find $x$: Angle $x$ and angle $q$ are on a straight line.
* $x = 180^\circ - 78^\circ = 102^\circ$.
* $x = 102, q = 78$

9.
* Find $x$: Angle $x$ and the $69^\circ$ angle are on a straight line.
* $x = 180^\circ - 69^\circ = 111^\circ$.
* Find $y$: Look inside the triangle. The angles are $81^\circ$, $69^\circ$, and $y$.
* Add knowns: $81^\circ + 69^\circ = 150^\circ$.
* Subtract from 180°: $180^\circ - 150^\circ = 30^\circ$.
* $x = 111, y = 30$

10.
* This is a right-angled triangle (square symbol = 90°).
* Add the known angles: $90^\circ + 45^\circ = 135^\circ$.
* Subtract from 180°: $180^\circ - 135^\circ = 45^\circ$.
* $x = 45$

11.
* Add the known angles: $35^\circ + 15^\circ = 50^\circ$.
* Subtract from 180°: $180^\circ - 50^\circ = 130^\circ$.
* $x = 130$

12.
* This is a right-angled triangle (square symbol = 90°).
* Add the known angles: $45^\circ + 90^\circ = 135^\circ$.
* Subtract from 180°: $180^\circ - 135^\circ = 45^\circ$.
* $x = 45$

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Final Answer:
1. x = 65
2. x = 63
3. x = 36
4. x = 113
5. x = 70, y = 60
6. x = 39
7. x = 138
8. x = 102, q = 78
9. x = 111, y = 30
10. x = 45
11. x = 130
12. x = 45