🔸 Concept 12: Angle in a Semicircle

If chord \( AB \) is a diameter, then arc \( AB = 180^\circ \) and any angle on this arc (like \( \angle ACB \)) is a right angle: \( \angle ACB = 90^\circ \).

🔸 Concept 13: Inscribed vs Circumscribed Figures

If a polygon lies inside a circle with all vertices touching the circle, it is inscribed, and the circle is called circumscribed.

If a circle lies inside a polygon and touches all sides, it is inscribed within the polygon.

🧩 Question 5

ABCDE is inscribed in a circle and \( AE \) is diameter. Find \( \angle B + \angle D \)?

✅ Solution to Question 5 Show/Hide Solution

Since \( AE \) is the diameter, \( \angle B = 90^\circ \). Also, \( \angle D = 180^\circ - \angle B + 90^\circ = 180 - B + 90 = 270 - B \).
Hence, \( \angle B + \angle D = B + (270 - B) = 270^\circ \).

🔸 Concept 14: Important Angle Sums

• \( \angle ACE = 90^\circ \)
• \( \angle ABC + \angle CDE = 270^\circ \)
• \( \angle AHG + \angle GFE = 270^\circ \)
• \( \angle ABC + \angle IDE + \angle AHE = 360^\circ \)
• Total: \( 270 + 270 = 540^\circ \)

🔸 Concept 15: Two Chord Angle Formulae

When chords intersect inside a circle: \[ \theta = \frac{m(BC) + m(AD)}{2} \]
When chords intersect outside: \[ \theta = \frac{m(AD) - m(BC)}{2} \]