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 any triangle always add up to $180^\circ$.
2. Straight Line: Angles on a straight line add up to $180^\circ$.
3. Exterior Angle: An exterior angle (outside the triangle) is equal to the sum of the two opposite interior angles. Alternatively, it forms a straight line ($180^\circ$) with the adjacent interior angle.

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1.
* Given angles: $70^\circ$ and $45^\circ$.
* Calculation: $180 - 70 - 45 = 65$.
* $x = 65^\circ$

2.
* Given angles: $82^\circ$ and $35^\circ$.
* Calculation: $180 - 82 - 35 = 63$.
* $x = 63^\circ$

3.
* Given angles: $58^\circ$ and $86^\circ$.
* Calculation: $180 - 58 - 86 = 36$.
* $x = 36^\circ$

4.
* First, find the third interior angle: $180 - 61 - 52 = 67^\circ$.
* Angle $x$ and the $67^\circ$ angle form a straight line ($180^\circ$).
* Calculation: $180 - 67 = 113$.
* $x = 113^\circ$

5.
* Angle $y$ and the $120^\circ$ angle form a straight line.
* $y = 180 - 120 = 60^\circ$.
* Now find $x$ using the triangle sum ($180^\circ$) with angles $50^\circ$ and $y$ ($60^\circ$).
* $x = 180 - 50 - 60 = 70$.
* $x = 70^\circ, y = 60^\circ$

6.
* Given angles: $77^\circ$ and $64^\circ$.
* Calculation: $180 - 77 - 64 = 39$.
* $x = 39^\circ$

7.
* Given angles: $13^\circ$ and $29^\circ$.
* Calculation: $180 - 13 - 29 = 138$.
* $x = 138^\circ$

8.
* First, find angle $q$ inside the triangle: $180 - 64 - 38 = 78^\circ$.
* Angle $x$ and angle $q$ form a straight line.
* Calculation: $180 - 78 = 102$.
* $x = 102^\circ$ (Note: $q = 78^\circ$)

9.
* Angle $x$ and the $69^\circ$ angle form a straight line.
* $x = 180 - 69 = 111^\circ$.
* Now find $y$ using the triangle sum with angles $81^\circ$ and $69^\circ$.
* $y = 180 - 81 - 69 = 30$.
* $x = 111^\circ, y = 30^\circ$

10.
* This is a right-angled triangle ($90^\circ$).
* Given angles: $90^\circ$ and $45^\circ$.
* Calculation: $180 - 90 - 45 = 45$.
* $x = 45^\circ$

11.
* Given angles: $35^\circ$ and $15^\circ$.
* Calculation: $180 - 35 - 15 = 130$.
* $x = 130^\circ$

12.
* This is a right-angled triangle ($90^\circ$).
* Given angles: $90^\circ$ and $45^\circ$.
* Calculation: $180 - 90 - 45 = 45$.
* $x = 45^\circ$

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