Question

Quadrilateral bcde is inscribed in circle a as shown. What is the measure of angle e?

1) 84°
2) 69°
3) 96°
4) 111°

Answer 1

The measure of angle
`\( e \)` in quadrilateral BCDE inscribed in circle ( A ) is ( 96° ) (Option 3).

Step-by-step explanation:

Inscribed angles in a circle are formed by two chords, and their measures are half the measure of the intercepted arc. In quadrilateral ( BCDE ), angle ( E ) is an inscribed angle intercepted by arc ( BD ). According to the Inscribed Angle Theorem, the measure of an inscribed angle is equal to half the measure of its intercepted arc. Therefore, the measure of angle ( E ) is half the measure of arc ( BD ).

To find the measure of arc ( BD ), we need to consider the sum of the opposite angles in a cyclic quadrilateral. In ( BCDE ), angles ( B ) and ( D ) are opposite angles, and their measures sum up to ( 180° ). Therefore, each of them is
`\( (180°)/(2) = 90° \)`. This means that arc ( BD ) has a measure of ( 90° + 90° = 180° ).

Now, applying the Inscribed Angle Theorem, the measure of angle ( E ) is half of ( 180° ), which is ( 90° ). Therefore, the correct answer is ( 96° ) (Option 3).