Finding the Angle OAC in Triangle ABC with Given Angles at the Circumcenter

In the context of Euclidean geometry, understanding the relationships between different angles within a triangle is a fundamental aspect of solving geometric problems. In this article, we are tasked with determining the measure of angle OAC in triangle ABC, where point O is the circumcenter and given that angle BAC 85° and angle BCA 75°.

Step-by-Step Solution

The solution involves several key geometric principles:

Step 1: Find Angle ABC

First, we need to calculate the measure of angle ABC. Recall that the sum of the interior angles in any triangle is 180°.

angle ABC 180° - angle BAC - angle BCA

Substituting the given values:

angle ABC 180° - 85° - 75° 20°

Thus, angle ABC 20°.

Step 2: Determine Angle AOC

Next, we use the property that the angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the circumference. This property is directly applicable to the angle angle AOC at the circumcenter O, which is twice the angle angle ABC.

angle AOC 2 times; angle ABC 2 times; 20° 40°

Therefore, angle AOC 40°.

Step 3: Find Angle OAC

Finally, to find the measure of angle angle OAC, we rely on the fact that O is the circumcenter, implying that OA OC (radii of the circumcircle). This means that triangle AOC is isosceles, with angle OAC and angle OCA being equal. We can use the angle sum property of a triangle to find angle OAC.

angle AOC angle OAC angle OCA 180°

Substituting the known values:

40° angle OAC angle OCA 180°

Since angle OAC angle OCA in the isosceles triangle, we have:

40° 2 times; angle OAC 180°

Subtracting 40° from both sides:

2 times; angle OAC 140°

Dividing by 2:

angle OAC 70°

Therefore, the measure of angle angle OAC is 70°.

Conclusion

The detailed steps show that the angle angle OAC in triangle ABC measures 70°, given the circumcenter O and the angles at the vertices.

Relevant Keywords

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Additional Reading

For those interested in exploring related concepts in greater depth, we recommend the following resources:

Understanding Angles and Circles in Geometry Properties and Calculations of Triangles