# Angular Reflection

Weekly Problem 28 - 2013

Two lines meet at a point. Another line through this point is reflected in both of these lines. What is the angle between the image lines?

Two lines meet at a point. Another line through this point is reflected in both of these lines. What is the angle between the image lines?

Image

In the diagram, $\angle MON = 130^{\circ}$. The reflection of $OP$ in $OM$ is $OQ$ and the reflection of $OP$ in $ON$ is $OR$.

What is the size of $\angle QOR$?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

Image

As $OQ$ is the reflection of $OP$ in $OM$, $\angle QOM = \angle POM$.

Similarly, $\angle RON = \angle PON$.

Hence reflex $\angle QOR = 2 \times \angle MON = 260^{\circ}$

Therefore $\angle QOR =360^{\circ} - 260^{\circ} = 100^{\circ}$.