Finding missing angles. Interactive worksheet | TopWorksheets - Free Printable

To solve the missing angles in the worksheet, we will use basic geometric principles such as:

1. Sum of angles in a triangle: The sum of the interior angles of a triangle is always \(180^\circ\).
2. Right angle: A right angle measures \(90^\circ\).
3. Straight line: Angles on a straight line sum to \(180^\circ\).
4. Circle: The total angle around a point is \(360^\circ\).

Let's solve each problem step by step.

---

Problem 1


- The diagram shows a right triangle with one angle given as \(20^\circ\).
- The sum of angles in a triangle is \(180^\circ\).
- One angle is \(90^\circ\) (right angle), and another is \(20^\circ\).
- Therefore, the missing angle \(x\) is:
\[
x = 180^\circ - 90^\circ - 20^\circ = 70^\circ
\]

Answer: \(x = 70^\circ\)

---

Problem 2


- The diagram shows an angle marked as \(x\) and another angle given as \(140^\circ\).
- These two angles are supplementary (they form a straight line).
- Therefore, the missing angle \(x\) is:
\[
x = 180^\circ - 140^\circ = 40^\circ
\]

Answer: \(x = 40^\circ\)

---

Problem 3


- The diagram shows a triangle with angles \(30^\circ\), \(30^\circ\), and \(x\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore, the missing angle \(x\) is:
\[
x = 180^\circ - 30^\circ - 30^\circ = 120^\circ
\]

Answer: \(x = 120^\circ\)

---

Problem 4


- The diagram shows a triangle with angles \(15^\circ\), \(10^\circ\), and \(x\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore, the missing angle \(x\) is:
\[
x = 180^\circ - 15^\circ - 10^\circ = 155^\circ
\]

Answer: \(x = 155^\circ\)

---

Problem 5


- The diagram shows a triangle with angles \(2x\), \(60^\circ\), and \(x\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore:
\[
2x + x + 60^\circ = 180^\circ
\]
\[
3x + 60^\circ = 180^\circ
\]
\[
3x = 120^\circ
\]
\[
x = 40^\circ
\]

Answer: \(x = 40^\circ\)

---

Problem 6


- The diagram shows a semicircle with angles \(x\), \(70^\circ\), and \(90^\circ\).
- The total angle around a point on a semicircle is \(180^\circ\).
- Therefore:
\[
x + 70^\circ + 90^\circ = 180^\circ
\]
\[
x + 160^\circ = 180^\circ
\]
\[
x = 20^\circ
\]

Answer: \(x = 20^\circ\)

---

Problem 7


- The diagram shows a triangle with angles \(40^\circ\), \(50^\circ\), and \(x\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore:
\[
x + 40^\circ + 50^\circ = 180^\circ
\]
\[
x + 90^\circ = 180^\circ
\]
\[
x = 90^\circ
\]

Answer: \(x = 90^\circ\)

---

Problem 8


- The diagram shows two angles on a straight line, both marked as \(x\).
- The sum of angles on a straight line is \(180^\circ\).
- Therefore:
\[
x + x = 180^\circ
\]
\[
2x = 180^\circ
\]
\[
x = 90^\circ
\]

Answer: \(x = 90^\circ\)

---

Problem 9


- The diagram shows a triangle with angles \(x\), \(45^\circ\), and \(55^\circ\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore:
\[
x + 45^\circ + 55^\circ = 180^\circ
\]
\[
x + 100^\circ = 180^\circ
\]
\[
x = 80^\circ
\]

Answer: \(x = 80^\circ\)

---

Problem 10


- The diagram shows a circle with a central angle of \(90^\circ\) and an inscribed angle \(x\).
- The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
- Therefore:
\[
x = \frac{90^\circ}{2} = 45^\circ
\]

Answer: \(x = 45^\circ\)

---

Problem 11


- The diagram shows a triangle with angles \(92^\circ\), \(60^\circ\), and \(x\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore:
\[
x + 92^\circ + 60^\circ = 180^\circ
\]
\[
x + 152^\circ = 180^\circ
\]
\[
x = 28^\circ
\]

Answer: \(x = 28^\circ\)

---

Problem 12


- The diagram shows a triangle with angles \(x\), \(46^\circ\), and \(32^\circ\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore:
\[
x + 46^\circ + 32^\circ = 180^\circ
\]
\[
x + 78^\circ = 180^\circ
\]
\[
x = 102^\circ
\]

Answer: \(x = 102^\circ\)

---

Final Answers

\[
\boxed{
\begin{array}{ll}
1) & 70^\circ \\
2) & 40^\circ \\
3) & 120^\circ \\
4) & 155^\circ \\
5) & 40^\circ \\
6) & 20^\circ \\
7) & 90^\circ \\
8) & 90^\circ \\
9) & 80^\circ \\
10) & 45^\circ \\
11) & 28^\circ \\
12) & 102^\circ \\
\end{array}
}
\]