Math Problem Statement

what is angle oab

Solution

To find the angle OAB\angle OAB in the given diagram, let's use the information we've already determined.

Key Information:

  • OBC=90\angle OBC = 90^\circ because OBOB is the radius and ABAB is the tangent.
  • OCB=64\angle OCB = 64^\circ (as given in the problem).
  • ABC=154\angle ABC = 154^\circ (calculated previously).

To find OAB\angle OAB:

OAB\angle OAB is an external angle to triangle OABOAB, and it equals the difference between ABC\angle ABC and OBC\angle OBC. The reasoning is:

  • OAB=ABCOBC\angle OAB = \angle ABC - \angle OBC
  • OAB=15490=64\angle OAB = 154^\circ - 90^\circ = 64^\circ

So, OAB=64\angle OAB = 64^\circ.

This result is consistent with the fact that OCB\angle OCB and OAB\angle OAB are equal because they subtend the same arc BCBC of the circle (this follows from the Inscribed Angle Theorem and its corollary related to angles in the same segment).

Would you like more details on this result, or have any other questions? Here are some related questions:

  1. What is the Inscribed Angle Theorem and how is it used here?
  2. How do we know that OAB\angle OAB and OCB\angle OCB are equal?
  3. Can we determine the length of side ABAB using trigonometry?
  4. What if the circle had a different radius—how would the angles change?
  5. What properties of tangents are applied in this problem?

Tip: In circle geometry, external angles formed by tangents often give key insights into solving problems, especially when using properties like the Inscribed Angle Theorem.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Geometry
Inscribed Angle Theorem

Formulas

-

Theorems

Inscribed Angle Theorem

Suitable Grade Level

High School