Circle Theorems worksheet with geometry problems and diagrams.
A worksheet titled "Circle Theorems (C)" with nine geometry problems involving circles, angles, and tangents, labeled Section A, featuring diagrams and blank spaces for answers.
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Step-by-step solution for: Circle Theorems (C) Worksheet | PDF Printable Geometry Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Circle Theorems (C) Worksheet | PDF Printable Geometry Worksheet
To solve the problems in this worksheet, we will use various circle theorems. Here are the key theorems we might need:
1. Angle at the Center Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the circumference.
2. Angles in the Same Segment: Angles subtended by the same arc in the same segment are equal.
3. Cyclic Quadrilateral: Opposite angles in a cyclic quadrilateral sum to 180°.
4. Tangent-Secant Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
5. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is 180°.
Let's solve each problem step by step.
---
- Given: A right triangle inscribed in a semicircle with one angle \(29^\circ\).
- To Find: Angle \(a\).
Solution:
- Since the triangle is inscribed in a semicircle, the angle subtended by the diameter is \(90^\circ\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore, \(a = 180^\circ - 90^\circ - 29^\circ = 61^\circ\).
Answer: \(a = 61^\circ\)
---
- Given: A triangle with one angle \(52^\circ\) and another angle \(b\) subtended by the same arc.
- To Find: Angle \(b\).
Solution:
- According to the "Angles in the Same Segment" theorem, angles subtended by the same arc are equal.
- Therefore, \(b = 52^\circ\).
Answer: \(b = 52^\circ\)
---
- Given: A circle with angles \(62^\circ\) and \(85^\circ\) subtended by arcs.
- To Find: Angles \(c\) and \(d\).
Solution:
- For angle \(c\):
- The angle subtended by the arc at the center is twice the angle at the circumference.
- Therefore, \(c = \frac{1}{2} \times (180^\circ - 62^\circ) = \frac{1}{2} \times 118^\circ = 59^\circ\).
- For angle \(d\):
- Similarly, \(d = \frac{1}{2} \times (180^\circ - 85^\circ) = \frac{1}{2} \times 95^\circ = 47.5^\circ\).
Answer: \(c = 59^\circ\), \(d = 47.5^\circ\)
---
- Given: A triangle with one angle \(138^\circ\) subtended by an arc.
- To Find: Angle \(e\).
Solution:
- The angle subtended by the arc at the center is \(138^\circ\).
- The angle at the circumference is half of this:
\[
e = \frac{1}{2} \times 138^\circ = 69^\circ
\]
Answer: \(e = 69^\circ\)
---
- Given: A cyclic quadrilateral with angles \(34^\circ\), \(107^\circ\), and \(f\).
- To Find: Angle \(f\).
Solution:
- In a cyclic quadrilateral, opposite angles sum to \(180^\circ\).
- Therefore, \(f = 180^\circ - 107^\circ = 73^\circ\).
Answer: \(f = 73^\circ\)
---
- Given: A triangle with one angle \(123^\circ\) subtended by an arc.
- To Find: Angle \(g\).
Solution:
- The angle subtended by the arc at the center is \(123^\circ\).
- The angle at the circumference is half of this:
\[
g = \frac{1}{2} \times 123^\circ = 61.5^\circ
\]
Answer: \(g = 61.5^\circ\)
---
- Given: A triangle with one angle \(106^\circ\) subtended by an arc.
- To Find: Angle \(h\).
Solution:
- The angle subtended by the arc at the center is \(106^\circ\).
- The angle at the circumference is half of this:
\[
h = \frac{1}{2} \times 106^\circ = 53^\circ
\]
Answer: \(h = 53^\circ\)
---
- Given: A cyclic quadrilateral with one angle \(72^\circ\) and another angle \(i\).
- To Find: Angle \(i\).
Solution:
- In a cyclic quadrilateral, opposite angles sum to \(180^\circ\).
- Therefore, \(i = 180^\circ - 72^\circ = 108^\circ\).
Answer: \(i = 108^\circ\)
---
- Given: A triangle with angles \(34^\circ\) and \(41^\circ\) subtended by arcs.
- To Find: Angles \(j\) and \(k\).
Solution:
- For angle \(j\):
- The angle subtended by the arc at the center is twice the angle at the circumference.
- Therefore, \(j = 2 \times 34^\circ = 68^\circ\).
- For angle \(k\):
- Similarly, \(k = 2 \times 41^\circ = 82^\circ\).
Answer: \(j = 68^\circ\), \(k = 82^\circ\)
---
\[
\boxed{
\begin{array}{ll}
a = 61^\circ & b = 52^\circ \\
c = 59^\circ & d = 47.5^\circ \\
e = 69^\circ & f = 73^\circ \\
g = 61.5^\circ & h = 53^\circ \\
i = 108^\circ & j = 68^\circ \\
k = 82^\circ &
\end{array}
}
\]
1. Angle at the Center Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the circumference.
2. Angles in the Same Segment: Angles subtended by the same arc in the same segment are equal.
3. Cyclic Quadrilateral: Opposite angles in a cyclic quadrilateral sum to 180°.
4. Tangent-Secant Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
5. Sum of Angles in a Triangle: The sum of the interior angles in a triangle is 180°.
Let's solve each problem step by step.
---
Problem 1
- Given: A right triangle inscribed in a semicircle with one angle \(29^\circ\).
- To Find: Angle \(a\).
Solution:
- Since the triangle is inscribed in a semicircle, the angle subtended by the diameter is \(90^\circ\).
- The sum of angles in a triangle is \(180^\circ\).
- Therefore, \(a = 180^\circ - 90^\circ - 29^\circ = 61^\circ\).
Answer: \(a = 61^\circ\)
---
Problem 2
- Given: A triangle with one angle \(52^\circ\) and another angle \(b\) subtended by the same arc.
- To Find: Angle \(b\).
Solution:
- According to the "Angles in the Same Segment" theorem, angles subtended by the same arc are equal.
- Therefore, \(b = 52^\circ\).
Answer: \(b = 52^\circ\)
---
Problem 3
- Given: A circle with angles \(62^\circ\) and \(85^\circ\) subtended by arcs.
- To Find: Angles \(c\) and \(d\).
Solution:
- For angle \(c\):
- The angle subtended by the arc at the center is twice the angle at the circumference.
- Therefore, \(c = \frac{1}{2} \times (180^\circ - 62^\circ) = \frac{1}{2} \times 118^\circ = 59^\circ\).
- For angle \(d\):
- Similarly, \(d = \frac{1}{2} \times (180^\circ - 85^\circ) = \frac{1}{2} \times 95^\circ = 47.5^\circ\).
Answer: \(c = 59^\circ\), \(d = 47.5^\circ\)
---
Problem 4
- Given: A triangle with one angle \(138^\circ\) subtended by an arc.
- To Find: Angle \(e\).
Solution:
- The angle subtended by the arc at the center is \(138^\circ\).
- The angle at the circumference is half of this:
\[
e = \frac{1}{2} \times 138^\circ = 69^\circ
\]
Answer: \(e = 69^\circ\)
---
Problem 5
- Given: A cyclic quadrilateral with angles \(34^\circ\), \(107^\circ\), and \(f\).
- To Find: Angle \(f\).
Solution:
- In a cyclic quadrilateral, opposite angles sum to \(180^\circ\).
- Therefore, \(f = 180^\circ - 107^\circ = 73^\circ\).
Answer: \(f = 73^\circ\)
---
Problem 6
- Given: A triangle with one angle \(123^\circ\) subtended by an arc.
- To Find: Angle \(g\).
Solution:
- The angle subtended by the arc at the center is \(123^\circ\).
- The angle at the circumference is half of this:
\[
g = \frac{1}{2} \times 123^\circ = 61.5^\circ
\]
Answer: \(g = 61.5^\circ\)
---
Problem 7
- Given: A triangle with one angle \(106^\circ\) subtended by an arc.
- To Find: Angle \(h\).
Solution:
- The angle subtended by the arc at the center is \(106^\circ\).
- The angle at the circumference is half of this:
\[
h = \frac{1}{2} \times 106^\circ = 53^\circ
\]
Answer: \(h = 53^\circ\)
---
Problem 8
- Given: A cyclic quadrilateral with one angle \(72^\circ\) and another angle \(i\).
- To Find: Angle \(i\).
Solution:
- In a cyclic quadrilateral, opposite angles sum to \(180^\circ\).
- Therefore, \(i = 180^\circ - 72^\circ = 108^\circ\).
Answer: \(i = 108^\circ\)
---
Problem 9
- Given: A triangle with angles \(34^\circ\) and \(41^\circ\) subtended by arcs.
- To Find: Angles \(j\) and \(k\).
Solution:
- For angle \(j\):
- The angle subtended by the arc at the center is twice the angle at the circumference.
- Therefore, \(j = 2 \times 34^\circ = 68^\circ\).
- For angle \(k\):
- Similarly, \(k = 2 \times 41^\circ = 82^\circ\).
Answer: \(j = 68^\circ\), \(k = 82^\circ\)
---
Final Answers
\[
\boxed{
\begin{array}{ll}
a = 61^\circ & b = 52^\circ \\
c = 59^\circ & d = 47.5^\circ \\
e = 69^\circ & f = 73^\circ \\
g = 61.5^\circ & h = 53^\circ \\
i = 108^\circ & j = 68^\circ \\
k = 82^\circ &
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of circle geometry worksheet.
