Question

: In triangle PQR, RX || PQ and QY || PR, RX and QY intersect at O. Prove that angle QOR = 180° - angle P.

A) Angle P + angle QOR = 180°
B) Angle QOR = angle P
C) Angle QOR = 180° - angle P
D) Angle QOR = 2 * angle P

Answer 1

To prove that angle QOR equals 180° minus angle P in triangle PQR, we use the alternate interior angles theorem and the properties of a triangle to show that angle QOR is the supplement of the sum of angles RQO and ROQ, which are congruent to angles P and Q respectively.

Step-by-step explanation:

To prove that angle QOR is equal to 180° minus angle P in triangle PQR, where RX || PQ and QY || PR intersecting at O, we can use the following steps:

1. Since RX is parallel to PQ and QY is parallel to PR, we can deduce by the alternate interior angles theorem that angle RQO is congruent to angle P and angle ROQ is equal to angle Q.
2. By the properties of a triangle, the sum of angles P, Q, and R in triangle PQR is 180°.
3. Because angle RQO and angle ROQ are congruent to angle P and angle Q respectively, we can say that the sum of angles RQO, ROQ, and R is also 180°.
4. Subtracting angles RQO and ROQ from 180° gives us the measure of angle QOR, which can be represented as 180° - (angle RQO + angle ROQ).
5. Since angle RQO is equal to angle P, and angle ROQ is equal to angle Q, substituting these values gives us angle QOR as 180° - (angle P + angle Q).
6. As the angles P and Q already make up 180° along with angle R, this simplifies further to angle QOR = 180° - angle P.