__Category__: Plane Geometry"Published in Newark, California, USA"

Given: ∆ABC inscribed in circle O

Prove: m∠BAC = ½ m(arc BC)

Photo by Math Principles in Everyday Life |

__Solution__:

Consider the given figure and we will label further as follows

Photo by Math Principles in Everyday Life |

Proof:

1. Statement: From point A, draw a line that passes through the center of a circle.

Reason: Line AO will be used later to get m∠BOC.

2. Statement: Draw radius OB and let m∠BAO = x.

Reason: Radius OB will be used to create an isosceles ∆BOA and x as the base angle.

3. Statement: OB

**≅**OA and m∠BAO

**≅**m∠OBA.

Reason: OB and OA are the radius of a circle and the two sides of an isosceles triangles are congruent. The two base angles of an isosceles triangles are congruent.

4. Statement: m∠BOA = 180° - 2x

Reason: The sum of the three angles of a triangle is always equal to 180°.

5. Statement: m∠BOD = 180° - (180° - 2x) = 2x

Reason: The sum of supplementary angles is always 180°. m∠BOA and m∠BOD are supplementary angles.

6. Statement: Draw radius OC and let m∠CAO = y.

Reason: Radius OC will be used to create an isosceles ∆COA and y as the base angle.

7. Statement: OC

**≅**OA and m∠CAO

**≅**m∠OCA.

Reason: OC and OA are the radius of a circle and the two sides of an isosceles triangles are congruent. The two base angles of an isosceles triangles are congruent.

8. Statement: m∠COA = 180° - 2y

Reason: The sum of the three angles of a triangle is always equal to 180°.

9. Statement: m∠COD = 180° - (180° - 2y) = 2y

Reason: The sum of supplementary angles is always 180°. m∠COA and m∠COD are supplementary angles.

10. Statement: m∠BAC = x + y and m∠BOC = 2x + 2y.

Reason: Addition property of angles

11. Statement: m∠BAC = ½ m∠BOC

Reason: If OA, OB, and OC are equidistant from its center at O, then m∠BAC = ½ m∠BOC.

12. Statement: m∠BOC = m(arc BC)

Reasons: If m∠BOC is a central angle formed by the two radii OB and OC, then m(arc BC) is an arc formed by a central angle.

13. Statement: m∠BAC = ½ m∠BOC = ½ m(arc BC)

Reasons: Substitution proposition.