Question

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In △ABC m∠CAB = 40° and m∠ABC = 60°. Circle, going through points A and C, intersects sides AB and BC in points D and E respectively. Find all angles in quadrilateral ACED.

Answer 1

m∠A=40°, m∠C=80°, m∠E=140°, m∠D=100°.

Explanation:

Quadrilateral ACED is inscribed into the circle (see attached diagram).

Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary (add up to 180°).

Since angle CAB has the measure of 40°, then opposite quadrilateral's angle CED has the measure of

180°-40°=140°.

Since angle ABC has the measure of 60°, then the third triangle's angle BCA has the measure

180°-40°-60°=80°.

Since angle BCA has the measure of 80°, then opposite quadrilateral's angle ADE has the measure of

180°-80°=100°.

So, in quadrilateral ACED,

m∠A=40°, m∠C=80°, m∠E=140°, m∠D=100°.