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Given: ∆ABC inscribed in circle O
Prove: m∠BAC = ½ m(arc BC)
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure and we will label further as follows
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| 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.