Respuesta :

Given:

m(ar AB) = 116°

To find:

1. The measure of arc BEA.

2. The measure of ∠CAB

3. The measure of ∠BAD

Solution:

1. The measure of arc of full circle is 360°.

m(ar AB) + m(ar BEA) = 360°

116° + m(ar BEA) = 360°

Subtract 116° from both sides.

m(ar BEA) = 244°

2. Using the tangent-chord angle theorem:

[tex]$\Rightarrow m\angle CAB = \frac{1}{2} m(ar \ AB)[/tex]

[tex]$\Rightarrow m\angle CAB = \frac{1}{2} \times 116^\circ[/tex]

⇒ m∠CAB = 58°

3. Using the tangent-chord angle theorem:

[tex]$\Rightarrow m\angle BAD = \frac{1}{2} m(ar \ BEA)[/tex]

[tex]$\Rightarrow m\angle BAD = \frac{1}{2} \times 244^\circ[/tex]

⇒ m∠BAD = 122°

Using the fact that it is supplementary to ∠CAB.

⇒ m∠CAB + m∠BAD = 180°

⇒ 58° + m∠BAD = 180°

Subtract 58° from both sides.

⇒ m∠BAD = 122°

In circle o where chord AB and tangent CAD are drawn the measure of BEA is 244 degrees and the measure of ∠CAB is 58 degrees.

What is inscribed angle theorem?

Inscribed angle theorem is the theorem, which state that the angle inscribed in a circle will be half of the angle which delimits the same arc on the circle.

In circle o, chord AB and tangent CAD are drawn. It is known that mAB=116.

  • (a) The measure of BEA

It is known that the measure of the arc of a full circle is 360 degrees. Thus, the sum of the arc BEA and arc AB is equal to the 360 degrees. Thus,

[tex](m\text{arc}\; AB)+m(\text{arc}\; BEA)=360\\116+m(\text{arc}\; BEA)=360\\m(\text{arc}\; BEA)=360-116\\m(\text{arc}\; BEA)=244^o[/tex]

  • (b) The measure of ∠CAB

According to the tangent-chord angle theorem, the measure of the angle CAB is half of the measure of the arc BEA.

[tex]m\angle CAB=\dfrac{1}{2}m\text{(arc}BEA)\\m\angle CAB=\dfrac{1}{2}\times 116\\m\angle CAB=58^o[/tex]

Thus, in circle o where chord AB and tangent CAD are drawn, the measure of BEA is 244 degrees and the measure of ∠CAB is 58 degrees.

Learn more about the inscribed angle theorem here;

https://brainly.com/question/3538263

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