Final answer:
The triangle with base AC and angle bisector AD has its angles measured as follows: both angle measures at A are 35 degrees each, resulting from the angle bisector dividing angle A equally, and angle BAC is 110 degrees. This determination uses the properties of the triangle's angle sum and the angle bisector.
Step-by-step explanation:
To determine the measures of the angles in the triangle with base AC and angle bisector AD, given that m∠ADB = 110°, we must use the properties of triangles and angle bisectors. Since AD is the angle bisector of ∠A, it divides ∠A into two equal angles. Let's denote the measure of each of these angles as x. Given m∠ADB = 110°, and knowing that the sum of angles in any triangle is 180°, we have:
m∠ADB + 2x = 180°
Substituting the known value of m∠ADB, we have:
110° + 2x = 180°
Solving for x gives us:
2x = 180° - 110°
2x = 70°
x = 35°
Therefore, the measures of the angles at A are both 35°. Now, AD is both an exterior angle to triangle ADC and the sum of the interior opposite angles, which gives us:
m∠ADC + m∠CAD = 110°
Since m∠CAD = 35°, we have:
m∠ADC = 110° - 35°
m∠ADC = 75°
Lastly, to find the measure of angle BAC (or ∠C), we use the fact that triangle ABC's angle sum is 180°:
35° + 35° + m∠BAC = 180°
m∠BAC = 180° - 70°
m∠BAC = 110°
So, the measures of the angles of the triangle are 35°, 35°, and 110°.