Math Problem Statement

What is the size of angle x? Write a justification for your answer.

Solution

The problem involves a cyclic quadrilateral, where the points PP, QQ, RR, and OO lie on the circumference of a circle. In this case, you are asked to find the size of angle xx, which is angle RQO\angle RQO, given that angle OPR\angle OPR is 146146^\circ.

Key Concept:

For any cyclic quadrilateral (a four-sided figure with all its vertices on a circle), the opposite angles are supplementary. This means:

OPR+RQO=180\angle OPR + \angle RQO = 180^\circ

Solution:

Given that OPR=146\angle OPR = 146^\circ, we can use the supplementary angles property of a cyclic quadrilateral:

RQO+146=180\angle RQO + 146^\circ = 180^\circ

Solving for RQO\angle RQO:

RQO=180146=34\angle RQO = 180^\circ - 146^\circ = 34^\circ

Therefore, the size of angle x=34x = 34^\circ.

Justification:

The property of opposite angles of a cyclic quadrilateral being supplementary allows us to calculate xx. Since OPR\angle OPR is 146146^\circ, RQO\angle RQO must be 3434^\circ to satisfy the relationship OPR+RQO=180\angle OPR + \angle RQO = 180^\circ.

Would you like further details or clarification?

Here are five related questions to deepen understanding:

  1. What are the properties of angles in a cyclic quadrilateral?
  2. How do we know a quadrilateral is cyclic?
  3. If one angle of a cyclic quadrilateral is obtuse, what can be said about the opposite angle?
  4. What happens if all the angles in a quadrilateral on a circle are equal?
  5. 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