Question

Chords AB and CD intersect each other at point E. Find m∠BEC, if the measures of arc AD is 54°, and BC is 70°.

Answer 1

62°

Explanation:

consider the solution with short explanation (find it in the attachment).

Answer 2

`{\angle}BEC=62^(\circ)`

Explanation:

Given:It is given that Chords AB and CD intersect each other at point E.

To find: The measure of ∠BEC.

Construction: Join AC.

Solution:

It is given that Chords AB and CD intersect each other at point E.

Now, we know that the inscribed angle is the half of the intercepted arc, thus

`{\angle}ACD={(1)/(2)}(AD)`

`{\angle}ACD={(1)/(2)}(54^(\circ))`

`{\angle}ACD=27^(\circ)`

And,
`{\angle}CAB={(1)/(2)}(CB)`

`{\angle}CAB={(1)/(2)}(70^(\circ))`

`{\angle}CAB=35^(\circ)`

Now, in ΔAEC, we have

`{\angle}CAE+{\angle}ACE+{\angle}AEC=180^(\circ)` (Angle sum property of triangles)

`35+27+{\angle}AEC=180`

`{\angle}AEC=118^(\circ)`

Also, using the straight line property, we have

`{\angle}AEC+{\angle}CEB=180^(\circ)`

`118+{\angle}CEB=180`

`{\angle}CEB=62^(\circ)`

Therefore, the measure of
`{\angle}BEC` is
`62^(\circ)`.