Master

Angle of Intersecting Chords Theorem

Master angle of intersecting chords theorem with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Angle of Intersecting Chords Theorem

Study Guide

Key Definition

The Angle of Intersecting Chords Theorem states that: If two chords intersect inside a circle, then the measure of the angle formed is half the sum of the measure of the arcs intercepted by the angle and its vertical angle. This can be represented as $\angle 1 = \frac{1}{2} (\widehat{PQ} + \widehat{RS})$.

Important Notes

A chord is a line segment whose endpoints lie on the circle.
An arc is any connected portion of the circle’s circumference between two points on the circle.
Vertical angles are pairs of opposite angles formed by intersecting lines.

Mathematical Notation

$\angle$ represents an angle.

$\widehat{AB}$ represents the arc from A to B.

Remember to use proper notation when solving problems.

Why It Works

Proof sketch: Extend one chord to form an inscribed angle and apply the inscribed angle theorem (an inscribed angle measures half its intercepted arc). Then use vertical angle relationships to derive that the angle between two intersecting chords equals half the sum of the intercepted arcs.

Remember

The measure of the angle formed by intersecting chords inside a circle is half the sum of the measures of the intercepted arcs.

Quick Reference

Angle of Intersecting Chords:$\angle 1 = \frac{1}{2} (\widehat{PQ} + \widehat{RS})$

Understanding Angle of Intersecting Chords Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When two chords intersect inside a circle, the angle formed is half the sum of the arcs intercepted by the angle.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If two chords intersect inside a circle, and the arcs intercepted by the angle and its vertical angle are 100° and 120°, what is the measure of the angle?

Real-World Problem

Question Exercise

Intermediate

Pizza Challenge

A pizza is cut into slices, forming chord lines. If two of these lines intersect, and the arcs intercepted are 90° and 180°, what is the measure of the angle at the intersection inside the pizza?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Consider a circle in which two chords intersect. If one of the intercepted arcs measures 120°, and the angle at the intersection is 70°, what is the measure of the other intercepted arc?

Challenge Quiz

Single Choice Quiz

Advanced

The measure of an angle formed by intersecting chords inside a circle is 80°. If one of the intercepted arcs is 120°, what is the measure of the other intercepted arc?

Recap

Watch & Learn

Review key concepts and takeaways