Question

The circumference of the triangle ABC Is 0 prove that angle OBC + angle BAC = 90

Answer 1

Answer:

Question: How do I prove that if the circumcentre of triangle ABC is O. angle OBC + angle BAC = 90 degrees?

Step-by-step explanation:

From the original △ ABC:

m∠ BAC + m∠ ABC + m∠ BCA = 180.

Since O is the center of the circumscribed circle OA = OB = OC (all radii of a circle are congruent). Therefore, there are three isosceles triangles (△ OAB, △ OBC, and △ OAC). In an isosceles triangle, the base angles are congruent (as marked and labeled in the diagram).

m∠ BAC =β−α

m∠ ABC = β+θ

m∠ BCA = θ−α

Substituting into the triangle angle sum equation:

(β−α)+(β+θ)+(θ−α)=180

Combining like terms:

2β+2θ−2α=180

Dividing by 2:

β+θ−α=90

Reordering:

θ+(β−α)=90

Substituting:

m∠ OBC + m∠ BAC = 90