Question

In Fig 10.39, A,B,C,D are four points on a circle . AC and BD intersect at a point E such that Angle(BEC = 130)and ANGLE(ECD = 20 ) . Find Angle(BAC). Can I know is that is It is having Minor arc or Major arc​

Answer 1

Given:

A circle.

AC and BD intersect each other at point E.

∠BEC  = 130°

∠ECD = 20°

To Find:

∠BAC

Does ∠BAC subtend a major or minor arc?

Solution:

In line BD:

➝ ∠BEC + ∠DEC = 180° [Linear Pair]

➝ 130° + ∠DEC = 180°

➝ ∠DEC = 180° - 130°

➝ ∠DEC = 50°

In ΔEDC:

➝ ∠DEC + ∠ECD + ∠CDE = 180° [ASP of a triangle]

➝ 50° + 20° + ∠CDE = 180°

➝ 70° + ∠CDE = 180°

➝ ∠CDE = 180° - 70°

➝ ∠CDE = 110°

We know that;

Angles in the same segment of a circle subtended on the circumference are equal.

Here, both ∠BAC and ∠BDC lie in the same segment, therefore they are equal.

➝ ∠BAC = ∠BDC

[∠BDC is the same as ∠EDC as E and B lie on the same line]

➝ ∠BAC = ∠EDC

➝ ∠BAC = 110°

∠BAC subtends a major arc BQC.