Answers: ∠a = 30° ; ∠b = 60° ; ∠c = 105<span>°. </span>_____________________________________________ 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<span>°). 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</span>° 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<span>° ; </span>180° - (<span>90° + 30°) = m∠b ; </span>180° - (120°) = m∠b = 60<span>° </span>___________________________ Now we need to solve for the measure of Angle c (<span>m∠c). ___________________________________________ All angles on a straight line (or straight "line segment") are called "supplementary angles" and must add up to 180</span>°. 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°.
