Angles in a circle | Lucas Lobo - Free Printable

To solve the missing angles in the given problems, we will use properties of circles and triangles, such as the fact that the sum of angles in a triangle is \(180^\circ\), and the relationship between central angles and inscribed angles.

Problem 1:

#### Given:
- Angles in the quadrilateral inside the circle: \(67^\circ\) and \(81^\circ\).
- We need to find \(k\), \(l\), and \(m\).

#### Solution:
1. Finding \(k\):
- The quadrilateral is cyclic (all vertices lie on the circle).
- The opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
- Therefore, \(k + 81^\circ = 180^\circ\).
\[
k = 180^\circ - 81^\circ = 99^\circ
\]

2. Finding \(l\):
- In the triangle formed by the sides of the quadrilateral, the sum of the angles is \(180^\circ\).
- The triangle has angles \(67^\circ\), \(81^\circ\), and \(l\).
\[
67^\circ + 81^\circ + l = 180^\circ
\]
\[
148^\circ + l = 180^\circ
\]
\[
l = 180^\circ - 148^\circ = 32^\circ
\]

3. Finding \(m\):
- The angle \(m\) is an exterior angle to the triangle with angles \(67^\circ\) and \(32^\circ\).
- The exterior angle is equal to the sum of the two non-adjacent interior angles.
\[
m = 67^\circ + 32^\circ = 99^\circ
\]

#### Final Answers for Problem 1:
\[
k = 99^\circ, \quad l = 32^\circ, \quad m = 99^\circ
\]

Problem 2:

#### Given:
- One angle in the triangle is \(43^\circ\).
- We need to find \(h\), \(i\).

#### Solution:
1. Finding \(h\):
- The angle \(h\) is the central angle subtended by the same arc as the inscribed angle \(43^\circ\).
- The central angle is twice the inscribed angle.
\[
h = 2 \times 43^\circ = 86^\circ
\]

2. Finding \(i\):
- The angle \(i\) is the remaining angle in the triangle.
- The sum of the angles in a triangle is \(180^\circ\).
\[
43^\circ + 43^\circ + i = 180^\circ
\]
\[
86^\circ + i = 180^\circ
\]
\[
i = 180^\circ - 86^\circ = 94^\circ
\]

#### Final Answers for Problem 2:
\[
h = 86^\circ, \quad i = 94^\circ
\]

Problem 3:

#### Given:
- A cyclic quadrilateral with one angle \(110^\circ\).
- We need to find \(x\) and \(y\).

#### Solution:
1. Finding \(x\):
- The angle \(x\) is an inscribed angle subtending the same arc as the central angle \(110^\circ\).
- The inscribed angle is half the central angle.
\[
x = \frac{110^\circ}{2} = 55^\circ
\]

2. Finding \(y\):
- The angle \(y\) is an exterior angle to the triangle with angles \(55^\circ\) and \(55^\circ\).
- The exterior angle is equal to the sum of the two non-adjacent interior angles.
\[
y = 55^\circ + 55^\circ = 110^\circ
\]

#### Final Answers for Problem 3:
\[
x = 55^\circ, \quad y = 110^\circ
\]

Problem 4:

#### Given:
- A triangle with one angle \(54^\circ\) and a tangent line.
- We need to find \(x\), \(y\), and \(z\).

#### Solution:
1. Finding \(x\):
- The angle \(x\) is the angle between the tangent and the chord.
- The angle between the tangent and the chord is equal to the inscribed angle subtending the same arc.
- The inscribed angle is \(54^\circ\).
\[
x = 54^\circ
\]

2. Finding \(y\):
- The angle \(y\) is the angle at the center subtending the same arc as the inscribed angle \(54^\circ\).
- The central angle is twice the inscribed angle.
\[
y = 2 \times 54^\circ = 108^\circ
\]

3. Finding \(z\):
- The angle \(z\) is the remaining angle in the triangle.
- The sum of the angles in a triangle is \(180^\circ\).
\[
z + 54^\circ + 54^\circ = 180^\circ
\]
\[
z + 108^\circ = 180^\circ
\]
\[
z = 180^\circ - 108^\circ = 72^\circ
\]

#### Final Answers for Problem 4:
\[
x = 54^\circ, \quad y = 108^\circ, \quad z = 72^\circ
\]

Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Problem 1: } k = 99^\circ, l = 32^\circ, m = 99^\circ \\
&\text{Problem 2: } h = 86^\circ, i = 94^\circ \\
&\text{Problem 3: } x = 55^\circ, y = 110^\circ \\
&\text{Problem 4: } x = 54^\circ, y = 108^\circ, z = 72^\circ
\end{aligned}
}
\]