Final answer:
To find the measure of ∡ADC in an isosceles triangle with angle bisectors AD and CD, use the properties of isosceles triangles and angle bisectors. Since two angles at the base are equal and the angle bisectors divide the apex and base angles into halves, ∡ADC is equal to angle C. Calculate ∡C using the sum of angles in a triangle, and thereby determine ∡ADC.
Step-by-step explanation:
To determine the measure of ∡ADC in an isosceles triangle ΔABC where line segments AD and CD are angle bisectors, we need to use the properties of isosceles triangles and angle bisectors. Since ΔABC is isosceles, two of its sides are equal in length, and the angles opposite these sides are equal. Let's assume AB = AC, meaning ∡B = ∡C. When AD and CD are angle bisectors, they split ∡A and ∡C evenly. Therefore, if ∡A is bisected into two equal parts, each will be ∡A/2. Similarly, ∡C is also split into two equal parts by CD, each being ∡C/2. Because ∡B = ∡C, it follows that ∡ADB = ∡BDC = ∡C/2. Now, ∡ADC is comprised of ∡ADB and ∡BDC, so ∡ADC = ∡ADB + ∡BDC or ∡ADC = ∡C/2 + ∡C/2, which simplifies to ∡ADC = ∡C.
Given that the sum of angles in any triangle is 180 degrees, we know that ∡A + ∡B + ∡C = 180 degrees. Because ΔABC is isosceles with AB = AC, we have 2∡B + ∡A = 180 degrees. Given ∡A's value, we can solve for ∡B, and since ∡ADC = ∡C, we can determine the measure of ∡ADC.