https://file.aiquickdraw.com/mathbot/file/1726477630461f8j5f621.jpg

Let's work through each diagram in the image and find the angles marked with letters.

### Key Concepts:
- Angles subtended by the same arc in a circle are equal.
- The angle at the center of a circle is twice the angle at the circumference subtended by the same arc (central angle theorem).

#### Diagram (a):
- Angle at the center, marked as angle \( A \), subtended by the same arc as the 20° angle at the circumference (let's call it angle \( B \)).
- According to the central angle theorem:
  \[
  A = 2 \times B = 2 \times 20^\circ = 40^\circ
  \]
  So, angle \( a = 40^\circ \).

#### Diagram (b):
- The central angle is given as 130° (marked as angle \( A \)).
- The angle at the circumference subtended by the same arc is \( b \).
  \[
  b = \frac{A}{2} = \frac{130^\circ}{2} = 65^\circ
  \]
  So, angle \( b = 65^\circ \).

#### Diagram (c):
- The angle at the center is divided into two parts, marked as \( a \) and \( b \). The angle at the circumference is 30°.
- From the central angle theorem:
  \[
  a + b = 2 \times 30^\circ = 60^\circ
  \]
  Therefore, the sum of angles \( a \) and \( b \) is 60°. Without additional information, we cannot split \( a \) and \( b \), but their sum is 60°.

#### Diagram (d):
- Angle at the center (marked \( a \)) is subtended by the same arc as the 80° angle at the circumference (marked \( c \)).
  \[
  a = 2 \times c = 2 \times 80^\circ = 160^\circ
  \]
- The angle \( b \) is at the circumference on the other side, subtended by the same arc as \( c \), so:
  \[
  b = c = 80^\circ
  \]

### Summary of Answers:
- (a) \( a = 40^\circ \)
- (b) \( b = 65^\circ \)
- (c) \( a + b = 60^\circ \)
- (d) \( a = 160^\circ \), \( b = 80^\circ \)

Would you like further clarification on any of the steps?

Here are 5 related questions:
1. How is the central angle theorem applied in circle geometry?
2. Can you explain the properties of cyclic quadrilaterals in a circle?
3. How do angles subtended by the same arc behave?
4. What is the relationship between a diameter and an angle in a semicircle?
5. How would you determine an unknown angle if multiple arcs subtend the same angle?

**Tip:** In circle geometry, always check for symmetry and angles subtended by the same arc to find unknown angles!

Find the angles marked with letters in each of the following diagrams if O is the centre of the circle.

Learn how to find angles in circle geometry using key concepts like the central angle theorem and angles subtended by the same arc. This problem is suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation