Question Video: Proving a Geometric Property Using Vertical Opposition of Angles Mathematics • First Year of Preparatory School

Video Transcript

In the given figure, find the measure of angle 𝑃𝐶𝑄.

Observe that angle 𝐶𝐴𝐵 is vertically opposite angle 𝑀𝐴𝐿. This means that the two angles are congruent. Similarly, angle 𝐴𝐵𝐶 is vertically opposite and, therefore, congruent to angle 𝑂𝐵𝑁. Since 𝐴𝐵𝐶 is a triangle, we know that its internal angles sum to 180 degrees.

The internal angles of triangle 𝐴𝐵𝐶 are angles 𝐶𝐴𝐵, 𝐴𝐵𝐶, and 𝐵𝐶𝐴. Therefore, angle 𝐵𝐶𝐴 has measure 180 minus 80 minus 30 equals 70 degrees. Finally, since angle 𝑃𝐶𝑄 is vertically opposite angle 𝐵𝐶𝐴, the two angles are congruent. Therefore, angle 𝑃𝐶𝑄 has measure 70 degrees.