ABCD is a cyclic quadrilateral- Tangents at A and C intersect at P- If -ABC - 100- then what is the measure of -APC-

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Properties of Cyclic Quadrilateral

ABCD is a cyc...

Question

ABCD is a cyclic quadrilateral. Tangents at A and C intersect at P. If $∠$ ABC = $100^{\circ}$ then what is the measure of $∠$ APC?

20^{\circ}

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30^{\circ}

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35^{\circ}

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40^{\circ}

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Solution

The correct option is A $20^{\circ}$
We know that sum of opposite angles of a cyclic quadrilateral is $180^{\circ}$
$∴ ∠ B + ∠ D = 180^{\circ}$
$\Rightarrow 100 + ∠ D = 180^{\circ}$
$\Rightarrow ∠ D = 80^{\circ}$
$∴ ∠ A C P = ∠ P A C = 80^{\circ}$
In $Δ$ PAC,
$∠ P + ∠ P A C + ∠ P C A . = 180^{\circ}$
$\Rightarrow ∠ P + 80^{\circ} + 80^{\circ} = 180^{\circ}$
$\Rightarrow ∠ P = 180^{\circ} - 160^{\circ} = 20^{\circ}$

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ABCD is a cyclic quadrilateral. Tangents at A and C intersect at P. If ∠ABC = 100∘ then what is the measure of ∠APC?

The correct option is A 20∘We know that sum of opposite angles of a cyclic quadrilateral is 180∘∴∠B+∠D=180∘⇒100+∠D=180∘⇒∠D=80∘∴∠ACP=∠PAC=80∘In ΔPAC,∠P+∠PAC+∠PCA.=180∘⇒∠P+80∘+80∘=180∘⇒∠P=180∘−160∘=20∘

ABCD is a cyclic quadrilateral. Tangents at A and C intersect at P. If $∠$ ABC = $100^{\circ}$ then what is the measure of $∠$ APC?