MathematicsChapter 10: Tangents and Angles of a Circle
Angles in A Circle
Central angle is twice the inscribed angle that stands on the same arc.
Learning objectives
- Explain angles in a circle using the correct definitions, conditions, and notation.
- Apply central = 2 x inscribed (same arc), semicircle 90 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Central angle is twice the inscribed angle that stands on the same arc. Angles in the same segment are equal. Angle in a semicircle is 90 deg (stands on a diameter). Opposite angles of a cyclic quadrilateral sum to 180 deg (if that theorem is in your book list).
Mathematical form
central = 2 x 35
central = 70 deg
angle = 2 x inscribed angle (same arc) central
Where / symbols
- ANGLES IN A CIRCLE
- Central angle is twice the inscribed angle that stands on the same arc. Angles in the same segment are equal. Angle in a semicircle is 90 deg (stands on a diameter). Opposite angles of a cyclic quadrilateral sum to 180 deg (if that theorem is in your book list).
- CENTRAL VS INSCRIBED
- Inscribed 35 deg on an arc -> central 70 deg on the same arc. Central 80 deg -> inscribed 40 deg. 'Same arc' matters; the other arc is different.
- SAME SEGMENT AND SEMICIRCLE
- Angles in the same segment (same side of a chord) are equal. Diameter as one side: the rim angle is 90. Use Pythagoras on that triangle.
Relationship
- central = 2 x inscribed (same arc)
Overview
Same arc, different seats:
Central vs Inscribed
Inscribed 35 deg on an arc -> central 70 deg on the same arc. Central 80 deg -> inscribed 40 deg. 'Same arc' matters; the other arc is different.
Examples 2 x 35 = 70 · half of 80 = 40 · Same chord/arc
Same Segment and Semicircle
Angles in the same segment (same side of a chord) are equal. Diameter as one side: the rim angle is 90. Use Pythagoras on that triangle.
Examples Angle in semicircle 90 · Hyp = diameter · Same segment: equal inscribed
WHAT — ANGLES IN A CIRCLE
Central angle is twice the inscribed angle that stands on the same arc. Angles in the same segment are equal. Angle in a semicircle is 90 deg (stands on a diameter). Opposite angles of a cyclic quadrilateral sum to 180 deg (if that theorem is in your book list). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Angles In A Circle matters because tangent and circle-angle theorems explain the geometry at a point of contact. They support proofs, numerical angle questions, and practical circle constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through CENTRAL VS INSCRIBED, then SAME SEGMENT AND SEMICIRCLE. Worked: inscribed angle on an arc is 35 deg. Find the central angle on the same arc. central angle = 2 x inscribed angle (same arc) central = 2 x 35 central = 70 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- inscribed angle on an arc is 35 deg. Find the central angle on the same arc.
- central angle = 2 x inscribed angle (same arc)
- central = 2 x 35
- central = 70 deg
In simple words
The inscribed angle must sit on the remaining circumference of that arc, not on the centre and not in the other segment unless the theorem says so.
Key points
- central = 2 x inscribed (same arc)
- semicircle 90
- twice at the centre
- same segment equal
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angles in A Circle
Sticky Notes · Quick Revision
Core Idea
- Central angle is twice the inscribed angle that stands on the same arc. Angles in the same segment are equal. Angle in a semicircle is 90 deg (stands on a diameter). Opposite angles of a cyclic quadrilateral sum to 180 deg (if that theorem is in your book list).
CENTRAL VS INSCRIBED
- Inscribed 35 deg on an arc -> central 70 deg on the same arc. Central 80 deg -> inscribed 40 deg. 'Same arc' matters; the other arc is different.
SAME SEGMENT AND SEMICIRCLE
- Angles in the same segment (same side of a chord) are equal. Diameter as one side: the rim angle is 90. Use Pythagoras on that triangle.
Worked Method
- Worked: inscribed angle on an arc is 35 deg. Find the central angle on the same arc.
- central angle = 2 x inscribed angle (same arc)
- central = 2 x 35
- central = 70 deg
Common Misconception
- The inscribed angle must sit on the remaining circumference of that arc, not on the centre and not in the other segment unless the theorem says so.
FBISE Exam Check
- The inscribed angle must sit on the remaining circumference of that arc, not on the centre and not in the other segment unless the theorem says so.
60-second revision
- Central angle is twice the inscribed angle that stands on the same arc.
- CENTRAL VS INSCRIBED: Inscribed 35 deg on an arc -> central 70 deg on the same arc.
- SAME SEGMENT AND SEMICIRCLE: Angles in the same segment (same side of a chord) are equal.
- Worked-method reminder: central = 70 deg
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