Question

ABCD is a cyclic quadilateral
Find the measure of angle ABF.
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Answer 1

Answer ⤵️⤵️

Given: ABCD is a cyclic quadrilateral. Side AB is extended to point F, and AD is extended to point E. The measure of angle FBC is 130∘130∘ , and the measure of angle ECD is x.

To determine: The value of x.

Since ABF are collinear, we have angles CBA and CBF form a linear pair

Hence, we have

∠CBF+∠CBA=180∘∠CBF+∠CBA=180∘

Substituting the value of ∠CBF∠CBF, we get

130∘+∠CBA=180∘130∘+∠CBA=180∘

Subtracting 130 on both sides, we get

∠CBA=50∘∠CBA=50∘

Now, we know that the sum of measures of opposite angles of a cyclic quadrilateral is 180∘180∘. Since angles CDA and CBA are opposite angles of the cyclic quadrilateral ABCD, we have

∠CDA+∠CBA=180∘∠CDA+∠CBA=180∘

Substituting the value of ∠CBA,∠CBA, we get

∠CDA+50∘=180∘∠CDA+50∘=180∘

Subtracting 50 from both sides of the equation, we get

∠CDA=130∘∠CDA=130∘

Now, since A,D and E are collinear, we have the angels EDC and CDA form a linear pair

Hence, we have

∠EDC+∠CDA=180∘∠EDC+∠CDA=180∘

Substituting the value of ∠EDC∠EDC and ∠CDA∠CDA, we get

x+130∘=180∘x+130∘=180∘

Subtracting 130 on both sides, we get

x=50∘x=50∘

Hence the value of x is 50∘50∘

Hence option [a] is correct.

Note: Alternative solution:

We know that the exterior angle of the cyclic quadrilateral is equal to the interior opposite angle.

Since ∠EDC∠EDC is an exterior angle of the cyclic quadrilateral ABCD and ∠CBA∠CBA is the corresponding interior opposite angle, we have

x=∠CBA=50∘x=∠CBA=50∘

Hence, the value of x is 50∘

Answer 2

Answer:

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