Given Information
- AB = AC (Isosceles triangle)
- BD = DC (Isosceles triangle)
- O is the center of the circle
- Angle ACO = 35°
Findings
To find angles ABD and BDC, we can use the properties of isosceles triangles and the angles in a circle.
Step 1: Analyzing Triangle ACO
- Since AB = AC, triangle ACO is isosceles.
- In triangle ACO, angle AOC = 2 * angle ACO (Angle at the center theorem).
- Therefore, angle AOC = 2 * 35° = 70°.
Step 2: Finding Angle ABC
- In triangle ABC, the exterior angle (angle AOB) is equal to the sum of the opposite interior angles (angle ACO + angle ABC).
- Hence, angle ABC = angle AOB - angle ACO.
- But angle AOB = angle AOC + angle ABC = 70° + angle ABC.
- Thus, we have angle ABC + 35° = 70°, leading to angle ABC = 35°.
Step 3: Finding Angle ABD
- Since BD = DC, triangle BDC is also isosceles.
- Therefore, angle BDC = angle ABD.
- The sum of angles in triangle ABC gives us:
angle ABC + angle ABD + angle ACB = 180°.
- Substituting angle ABC = 35° and angle ACB = 35°, we find:
35° + angle ABD + 35° = 180°.
- This simplifies to angle ABD = 180° - 70° = 110°.
Step 4: Finding Angle BDC
- Since triangle BDC is isosceles with BD = DC, angle BDC = angle ABD = 110°.
Final Angles
- Angle ABD = 110°
- Angle BDC = 110°
This concludes the calculation of angles in the given geometric configuration.