In the shown figure, AB=OB and CT is the tangent to the circle at O. If ∠COA =125^∘, then ∠OAB i

step 1 In this question figure is given so in this question given AB equal to OB and CT is the tangent

In the shown figure, AB=OB and CT is the tangent to the...

In the shown figure, $\mathrm{AB}=\mathrm{OB}$ and $\mathrm{CT}$ is the tangent to the circle at $\mathrm{O}$. If $\angle \mathrm{COA}$ $=125^{\circ}$, then $\angle \mathrm{OAB}$ is
(1) $55^{\circ}$
(2) $271 / 2^{\circ}$
(3) $821 / 2^{\circ}$
(4) $45^{\circ}$

Key Concepts

Isosceles Triangle Properties

In an isosceles triangle, two sides are equal in length, which directly implies that the angles opposite these equal sides are also congruent. This property is fundamental when analyzing relationships within triangles and solving for unknown angles, as it provides a basis for setting up equations based on the equality of base angles.

Circle Tangency and Radius Perpendicularity

A tangent to a circle at a given point is perpendicular to the radius drawn to that point of tangency. This principle is crucial in problems involving circles because it allows the formation of right angles which can simplify computations and provide additional relationships between angles and segments in geometric configurations.

Central Angles

A central angle is an angle whose vertex is at the center of a circle, and its measure is equal to the measure of the intercepted arc. Understanding central angles is key in circle geometry as they provide direct insight into the relationships between angles formed at the center and the corresponding segments or chords, which can be used to resolve unknown measures in the circle.

Angle Sum in Triangles

The sum of the interior angles of any triangle is always 180°. This fundamental fact of Euclidean geometry allows one to establish relationships among the angles of a triangle, especially when some of the angles are already known or are expressed in terms of each other as in the case of isosceles triangles.

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