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Inscribing a Quadrilateral

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Inscribing a Quadrilateral

Study Guide

Key Definition

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary: $\angle Q + \angle S = 180^\circ$ and $\angle R + \angle P = 180^\circ$.

Important Notes

A cyclic quadrilateral is a quadrilateral inscribed in a circle.
Opposite angles of an inscribed quadrilateral are supplementary.
$\angle Q + \angle S = 180^\circ$
$\angle R + \angle P = 180^\circ$
This property is crucial for solving related geometry problems.

Mathematical Notation

$\angle$ denotes an angle

$+$ represents addition

$180^\circ$ means the sum forms a straight line

Remember to use proper notation when solving problems

Why It Works

The inscribed angle theorem proves that the opposite angles of a quadrilateral inscribed in a circle are supplementary: $\angle Q + \angle S = 180^\circ$.

Remember

If a quadrilateral is cyclic, its opposite angles are always supplementary.

Quick Reference

Supplementary Angles:$\angle Q + \angle S = 180^\circ$

Understanding Inscribing a Quadrilateral

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Video explanation of this concept

Beginner

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Beginner Explanation

A quadrilateral can be inscribed in a circle if opposite angles add up to $180^\circ$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is true for a quadrilateral inscribed in a circle?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a circular park with a walking path forming a quadrilateral inside. If $\angle Q = 65^\circ$ and $\angle R = 85^\circ$, calculate $\angle S$ and $\angle P$ so that $\angle Q + \angle S = 180^\circ$ and $\angle R + \angle P = 180^\circ$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that a quadrilateral with opposite angles adding up to $180^\circ$ can be inscribed in a circle.

Challenge Quiz

Single Choice Quiz

Advanced

Given a cyclic quadrilateral, if $\angle A = 70^\circ$ and $\angle C = 110^\circ$, what is $\angle B + \angle D$?

Recap

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Review key concepts and takeaways