Question

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in figure ray bc is a tangent at point b and ray ba is secant .
angle ABC interupts arc aXb
if m arc aXb = 130° then find measure of angle ABC

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Answer 1

Figure:-

Given:-

• BC is a tangent at point B and ray BA is secant. <ABC inetercepts arc AXB.
• Measure of (arcAXB) = 130°.

To find:-

• Find the measure of <ABC.?

Solution:-

• BC is a tangent to circle at B.
• Measure of (arc A × B) = 130°
• ∠BOA = 130° [by arc theorem]

.:. ∆AOB,

OA = OB (Radius)

.:. ∠OBA = ∠OAB => x°

In ∆AOB,

=> ∠O + ∠A + ∠B = 180°

=> 130° + 2x = 180°

=> 2x = 180° - 130°

=> 2x = 50°

=> x = 50°/2

=> x = 25°

By tangent theorem, Radius will be perpendicular to tangent at point of contact, ∠OBC = 90°.

=> ∠OBA + ∠ABC = 90°

=> x + ∅ = 90°

=> 25° + ∅ = 90°

=> ∅ = 90° - 25°

=> ∅ = 65°

Therefore,

.:. ∠ABC = 65°

Hence, the required value of ∠ABC = 65°.

Additional Information:-

• Straight line = Angle which measures 180°.
• Supplementary Angle = The two angle are supplementary, if there sum are 90°.
• Complementary Angle = The two angle are complementary, if these sum are 90°.

Some Important:-

• A transversal intersects two parallel lines.
• The corresponding angles are equal.
• The vertically opposite angles are equal.
• The alternate interior angles are equal.
• The alternate exterior angles are equal.
• The pair of interior angles on the same side of the transversal is supplementary.

Answer 2

Answer:

$\tt \blue{Angle \: ABC = 65°} is \: the \: answer$

REFER THE ATTACHMENT

Jo.in the pointt A and B by centre O.

So, the angle AOB = 130°

Let the angle be = ϴ

OA = OB

Since , OB are the radius .

Therefore , angle OBA = angle OAB = x° _____[ eq (1) ]

[ equal sides have equal angle opp. to it ]

Now

In the figure

BC is the tangent to the circle.

And

measure of a(AXB) = 130°

Therefore , angle BOA = 130° [By arc theorem ]

Since,

In triangle AOB

OA = OB [ Radius]

angle OBA = angle OAB = x° [ From eq 1 ]

In triangle AOB,

angle O + angle A + angle B = 180°

=> 130° + 2x = 180°

=> 2x = 50°

=> x = 25°

Now,

By tangent theorem

Radius will be perpendicular to tangent at pointt of contact , angle OBC = 90°

angle OBA + angle ABC = 90°

=> x° + ϴ = 90°

=> 25° + ϴ = 90°

=> ϴ = 65°

Therefore , angle ABC = 65°

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