In triangle ABC, AC=BC. The point D is on BC such that AB=AD=DC. Compute the measure of angle C.
I'm not sure how to do this...
angle ADB is an exterior angle to triangle ACD.
So, it is equal to angles ACD+CAD.
Thus angle C is 45 degrees.
To solve this problem, we can utilize the fact that in an isosceles triangle, the base angles are congruent. Let's break down the steps:
1. Draw triangle ABC with AC = BC and point D on BC, such that AB = AD = DC.
2. Since AB = AD, triangle ABD is an isosceles triangle. Therefore, angle BAD = angle ABD.
3. Similarly, since AC = BC, triangle ACD is also an isosceles triangle. Hence, angle ADC = angle ACD.
4. In triangle ABC, the sum of all angles is always equal to 180 degrees. Therefore, we can sum up angles A, B, and C to get 180 degrees: angle A + angle B + angle C = 180 degrees.
5. We know that angle BAD = angle ABD and angle ADC = angle ACD, so we can substitute these values into the equation from step 4: angle A + angle B + angle C = 180 degrees.
6. Given that angle BAD = angle ABD and angle ADC = angle ACD, we can replace them with angle C, since AD and DC are congruent (AB = AD = DC): angle C + angle C + angle C = 180 degrees.
7. Simplify the equation from step 6: 3 * angle C = 180 degrees.
8. Divide both sides of the equation by 3 to solve for angle C: angle C = 180 degrees / 3.
9. Calculate the value: angle C = 60 degrees.
Thus, the measure of angle C in triangle ABC is 60 degrees.