Angles in inscribed quadrilaterals

Key Notes :

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🔵 What is an Inscribed Quadrilateral?

• An inscribed quadrilateral is a four-sided shape where all vertices (corners) lie on a single circle — this circle is called the circumscribed circle.
• Each side acts like a chord of the circle.

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📌Key Theorems and Properties

✅ Property 1: Opposite Angles are Supplementary

• In an inscribed quadrilateral, the opposite angles add up to 180° (they are supplementary).

∠A + ∠C = 180^∘ and ∠B + ∠D = 180^∘

✅ Property 2: Exterior Angle Rule

• An exterior angle of an inscribed quadrilateral equals the interior opposite angle.

∠E = ∠C

(where E is an exterior angle at vertex B opposite angle C)

✅ Property 3: Cyclic Quadrilateral Condition

• A quadrilateral can only be inscribed in a circle if its opposite angles are supplementary.
• If you check this property and it holds true, you confirm the quadrilateral is cyclic.

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🛠️Example Problem

✅ Example 1:

If an inscribed quadrilateral has angles ∠A = 80∘ and ∠C = 100∘, find ∠B and ∠D.

👉 Solution:

• Opposite angles are supplementary:

∠A + ∠C = 180^∘  ⟹  80^∘+100^∘ = 180^∘✅ Correct

For the other pair:

∠B + ∠D = 180^∘

If ∠B = 110∘, then:

∠D = 180^∘ − 110^∘ = 70^∘

👉 Final Answer: ∠B = 110^∘ , D = 70^∘

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🔥Common Mistakes to Avoid

⚠️ 1. Confusing adjacent angles with opposite angles — Only opposite angles are supplementary, not adjacent ones.

⚠️ 2. Forgetting the exterior angle rule — Exterior angles are equal to the interior opposite angle, not adjacent angles.

⚠️ 3. Assuming all quadrilaterals are cyclic — A quadrilateral is only cyclic if the opposite angles are supplementary.

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🧠Real-Life Applications

• Architecture: Circular windows with decorative inscribed designs.
• Engineering: Gear and wheel designs in machines.
• Astronomy: Calculating angles in planetary orbits.

Learn with an example

let’s practice!