---
title: "Circle Theorems: Cyclic Quadrilaterals and the Same Segment"
description: "Use the equal angles subtended by the same arc, and the opposite angles of a cyclic quadrilateral adding to 180 degrees."
canonical: https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1
source: https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1.md
retrieved: 2026-09-12
---

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# Circle Theorems: Cyclic Quadrilaterals and the Same Segment

Use the equal angles subtended by the same arc, and the opposite angles of a cyclic quadrilateral adding to 180 degrees.

Subject: Mathematics · Area: Geometry · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1

## Ready when they can

- Find equal angles standing on the same arc in a busy diagram
- Use opposite angles of a cyclic quadrilateral to find a missing angle
- Give the theorem name with each step rather than the answer alone

## Lesson: Cyclic quadrilaterals and shared arcs

Angles standing on the same arc are equal, but only on the same side of the chord. Points C and D on the same arc see chord AB under the same angle: if angle ACB is 42 degrees, angle ADB is 42 too.

**Example.** A cyclic quadrilateral has all four corners on the circle. Opposite angles total 180 degrees. With angle A at 70 degrees, angle C is 110, since 180 minus 70 is 110. With angle Q at 115 degrees, angle S is 65.

**Tip.** Give the theorem name with each step, not just the answer. Say Same Segment Theorem for a shared arc step, and Cyclic Quadrilateral Theorem for an opposite angle step.

**Example.** Both rules can combine in one question. In cyclic WXYZ with angle W 80 degrees, angle Y is 100 by the Cyclic Quadrilateral Theorem. Angle WXZ matches angle WMZ at 35 degrees by the Same Segment Theorem, since X and M share one arc.

**Recap.** Match shared arcs for equal angles, total opposite angles to 180, and name the theorem each time.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Circle Theorems: Angles at the Centre and in a Semicircle](https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR)

## Opens up

- [Tangents, Radii and the Alternate Segment Theorem](https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C)