Explanation:
We know that the exterior angle at vertex C is equal to the sum of the interior angles at vertices A and B that are not adjacent to C. Thus, we have:
∠ACB = ∠A + ∠B
Since ∠AQC is the angle bisector of ∠BAC, we have:
∠AQD = ∠CQB
Since BQ is perpendicular to AC, we have:
∠BQC = 90° - ∠QCB
Now, let x be the measure of angle ABC. Then we have:
∠A = 180° - ∠ACB - x
∠B = 180° - ∠ACB - (90° - ∠QCB)
Substituting the given values and simplifying, we get:
∠A = 68° - x
∠B = 22° + ∠QCB
Now, we use the fact that the angles in a triangle add up to 180°:
∠A + ∠B + ∠C = 180°
Substituting the above expressions and simplifying, we get:
68° - x + 22° + ∠QCB + 112° = 180°
Simplifying further, we get:
∠QCB + x = 22°
But we also know that ∠QCB + x is equal to ∠ABC. Thus, we have:
∠ABC = 22°
Therefore, the measure of angle ABC is 22 degrees.