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In the figure shown, chords AB and CD intersect at E. The measure of arc AC is 170. The measures of BD is (x+10)°, and the measure of

What is the degree measure of BEC?

In The Figure Shown Chords AB And CD Intersect At E The Measure Of Arc AC Is 170 The Measures Of BD Is X10 And The Measure Of What Is The Degree Measure Of BEC class=

Sagot :

Given:

In a circle, chords AB and CD intersect at E.

The measure of arc AC is 170°.

The measures of BD is (x+10)°.

To find:

The measure of angle BEC.

Solution:

We know that, if two chord intersect each other inside the circle, then the measure of angle at the intersection is half of the sum of subtended arcs.

Using the above theorem, we get

[tex]m\angle DEB=\dfrac{1}{2}(arc(AC)+acr(BD))[/tex]

[tex]2x=\dfrac{1}{2}(170+x+10)[/tex]

[tex]4x=x+180[/tex]

[tex]4x-x=180[/tex]

[tex]3x=180[/tex]

Divide both sides by 3.

[tex]x=60[/tex]

The measure of angle DEB is:

[tex]m\angle DEB=(2x)^\circ[/tex]

[tex]m\angle DEB=(2\times 60)^\circ[/tex]

[tex]m\angle DEB=120^\circ[/tex]

Now,

[tex]m\angle DEB+m\angle BEC=180^\circ[/tex]            (Linear pair)

[tex]120^\circ+m\angle BEC=180^\circ[/tex]

[tex]m\angle BEC=180^\circ-120^\circ[/tex]

[tex]m\angle BEC=60^\circ[/tex]

Therefore, the measure of angle BEC is 60 degrees.

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