https://file.aiquickdraw.com/mathbot/file/1727804287068gk8c06oz.jpg

In the given problem, we are asked to calculate the size of angle \( \theta \), with \( CDE \) as a tangent to the circle at point \( D \).

### Key Information:
- \( CDE \) is a tangent, and we are to use properties related to circles and tangents.
- \( \angle BDA = 56^\circ \) (this is an exterior angle formed between the tangent and a chord).
- \( \angle DBA = 37^\circ \) is given.
- \( \angle DAF = 29^\circ \), part of the cyclic quadrilateral inside the circle.

### Step-by-Step Solution:
1. **Using the Tangent-Secant Theorem:**
   - The angle between the tangent and a chord (in this case, \( \angle BDA \)) is equal to the angle in the alternate segment of the circle. Therefore, \( \angle BDA = \angle B \).
   - Since \( \angle BDA = 56^\circ \), we can conclude that \( \angle B = 56^\circ \).

2. **Angles in a Triangle:**
   - Now, consider triangle \( ABD \) inside the circle. The angles in this triangle must sum to \( 180^\circ \).
   \[
   \angle A + \angle B + \angle DBA = 180^\circ
   \]
   Substituting the known values:
   \[
   \theta + 56^\circ + 37^\circ = 180^\circ
   \]
   Simplifying this equation:
   \[
   \theta + 93^\circ = 180^\circ
   \]
   \[
   \theta = 180^\circ - 93^\circ = 87^\circ
   \]

### Conclusion:
The size of angle \( \theta \) is \( 87^\circ \).

Let me know if you'd like any further details or explanations! 

Here are some follow-up questions to reinforce the concept:

1. How does the tangent-secant theorem apply to different types of circles?
2. Why do the angles in a triangle always sum to 180 degrees?
3. What are some other properties of cyclic quadrilaterals?
4. How would the solution change if the tangent was extended beyond point \( E \)?
5. Can you apply this method to solve for angles in an ellipse?

**Tip:** Always remember that the angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. This property is key to solving many circle geometry problems!

CDE is a tangent to the circle below. Calculate the size of angle θ. Fully justify your answer.

Learn how to calculate angle θ in a circle with a tangent using the Tangent-Secant Theorem. This problem also involves angles in a triangle and cyclic quadrilaterals, making it ideal for Grades 9-11 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