Math Problem Statement
Solution
The problem involves a cyclic quadrilateral, where the points , , , and lie on the circumference of a circle. In this case, you are asked to find the size of angle , which is angle , given that angle is .
Key Concept:
For any cyclic quadrilateral (a four-sided figure with all its vertices on a circle), the opposite angles are supplementary. This means:
Solution:
Given that , we can use the supplementary angles property of a cyclic quadrilateral:
Solving for :
Therefore, the size of angle .
Justification:
The property of opposite angles of a cyclic quadrilateral being supplementary allows us to calculate . Since is , must be to satisfy the relationship .
Would you like further details or clarification?
Here are five related questions to deepen understanding:
- What are the properties of angles in a cyclic quadrilateral?
- How do we know a quadrilateral is cyclic?
- If one angle of a cyclic quadrilateral is obtuse, what can be said about the opposite angle?
- What happens if all the angles in a quadrilateral on a circle are equal?
- How does the inscribed angle theorem relate to this problem?
Tip: Always check if a quadrilateral is cyclic when working with problems on circles, as opposite angles can provide quick solutions.
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Math Problem Analysis
Mathematical Concepts
Cyclic Quadrilateral
Circle Theorems
Geometry
Formulas
Opposite angles in a cyclic quadrilateral sum to 180°
Theorems
Cyclic Quadrilateral Theorem
Supplementary Angles Theorem
Suitable Grade Level
Grades 9-11