What are the measures of angles a, b, and c? show your work and explain your answers.

Answers: ∠a = 30° ; ∠b = 60° ; ∠c = 105°.

1) The measure of Angle a is 30°. (m∠a = 30°).
Proof:  All vertical angles are congruent, and we are shown in the diagram that angle A — AND the angle labeled with the measurement of 30°— are vertical angles. 

2) The measure of Angle b is 60°. (m∠b = 60°).
Proof:  All three angles of a triangle add up to 90 degrees. In the diagram, we can examine the triangle formed by Angle A, Angle B, and a 90° angle. This is a right triangle, and the angle with 90∠ degrees is indicated as such (with the "square" symbol).  So we know that one angle is 90°.  We also know that m∠a = 30°.  If there are three angles in a triangle, and all three angles must add up to 180°, and we know the measurements of two of the three angles, we can solve for the unknown measurement of the remaining angle, which in this case is: m∠b.  
 90° + 30° + m∠b = 180° ;
180° - (90° + 30°) = m∠b ; 
180° - (120°) = m∠b = 60°

Now we need to solve for the measure of Angle c (m∠c).

All angles on a straight line (or straight "line segment") are called "supplementary angles" and  must add up to 180°.  As shown, Angle c is on a "straight line".  The measurement of the remaining angle represented ("supplementary angle" to Angle c is 75° (shown on diagram).  As such, the measure of "Angle C" (m∠c) = m∠c = 180° - 75° = 105°.