Angles in Same Segment Class 9 Maths | I'm Up and Down, and Round and Round

Understand it

When two angles are formed by joining the ends of the same chord to two points on the same part of the circle, both angles are equal. This is a common theorem used in proofs and angle finding.

Example

If points P and Q lie on the same segment of chord AB, then angle APB = angle AQB.

Formula and rule

If P and Q are on the same segment standing on chord AB, then angle APB = angle AQB.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1In a circle, angle ACB = 48 degrees and D lies on the same segment of chord AB as C.
2Therefore angle ADB = 48 degrees.

Method you can reuse

1Identify the chord common to both angles.
2Check whether both angle vertices are in the same segment.
3If yes, equate the two angles.

Sketch it in your answer

Draw chord AB and two points P and Q on the same arc, then join PA, PB, QA and QB to show equal angles.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If angle APB and angle AQB stand on chord AB in the same segment and angle APB = 63 degrees, what is angle AQB?

Think first, then show answer

Angle AQB is 63 degrees because angles in the same segment of a circle are equal.

Common wrong answer

Writing the angles as supplementary is wrong when the points are in the same segment; supplementary opposite angles belong to cyclic quadrilaterals.

Don't mix this up

Students apply the theorem even when the two points are on opposite segments. Same chord alone is not enough; the angles must be in the same segment.

Exam tip

Before using the theorem, circle the common chord and check that both angle vertices lie on the same side of that chord.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions often ask for a missing angle after showing two angles standing on the same chord in the same segment.

1
Enough for 1 mark
Angles in the same segment of a circle are equal. These angles stand on the same chord and have their vertices on the same arc segment.
2
Add for 2 marks
Example: If points P and Q lie on the same segment of chord AB, then angle APB = angle AQB.
3
Add for 3 marks
This is a common theorem used in proofs and angle finding.

Use the concept graph around this idea instead of revising it in isolation.

Angle Subtended by Arc/Chord at Centre vs at Circumference

The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. In a cyclic quadrilateral, each pair of opposite angles is supplementary.

Concyclic Points

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

Definition of a Circle

A circle is the set of all points in a plane that are at the same distance from a fixed point called the centre. This fixed distance is called the radius.

Previous: Angle Subtended by Arc/Chord at Centre vs at Circumference Next: Cyclic Quadrilateral Back to full mind map

Angles in Same Segment

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

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Angles in Same Segment

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page