Final answer:
The size of apex angle C in an isosceles triangle where AD bisects the base angle A can be found by using the property that the sum of the angles in a triangle is 180 degrees and setting up an equation with the bisected angles. Since AD bisects base angle A, we can solve for C after determining the measure of the bisected angles.
Step-by-step explanation:
To determine the size of the apex angle C in an isosceles triangle ABC, where segment AD bisects the base angle A, we first note that in an isosceles triangle the base angles are equal. Let's denote the base angles as x. Since AD bisects the base angle A, it means that each angle formed by a bisector AD will be x/2. Thus angle BAD and angle DAC both measure x/2.
By the properties of triangles, we know that the sum of the interior angles of a triangle is 180 degrees. Therefore, we can set up an equation: x (the base angle) + x (the base angle) + C (the apex angle) = 180 degrees. Simplifying, we get 2x + C = 180 degrees. Since x is twice the angle of x/2, we can use that to find the measure of angle C.
If we denote each of the bisected angles as y, then x=2y. The equation now becomes 4y + C = 180 degrees. We then solve for C to find the measure of the apex angle.
Remembering that the sum of the angles in any triangle is always 180 degrees is crucial as it is a fundamental theorem in geometry. Using this theorem and understanding that isosceles triangles have two equal angles allows us to solve for the unknown angle.