If lines $k$ and $m$ intersect at point $P$,
then a reflection in line $k$ followed by
a reflection in line $m$ is the same as a
rotation about point $P.$
The angle of rotation is $2x^(∘)$ where $x^(∘)$ is
the measure of the acute or right angle
formed by lines $k$ and $m.$
Let's visualize this.
So how do we go about proving this? Well let's start by considering the definition of reflections. Line k and line m are perpendicular bisectors of AA' and A'A'' respectively. Using the definition of a perpendicular bisector, we know that
AK≅ KA' and A'M≅ MA''
∠ AKP and ∠ A'KP are right angles
∠ A'MP and ∠ A''MP are right angles
Let's add these pieces of information to our diagram. We will also add a couple of segments, AP, A'P, and A''P which creates four separate triangles which are colored below.
Notice that △ AKP and △ A'KP share a common side KP. This is also true for △ A'MP and △ A''MP, which share the common side MP. Using the Reflexive Property of Congruence, we know that these sides are congruent to themselves.
Now we have enough information to claim, by the SAS Congruence Theorem, that
△ AKP≅ A'KP and △ A'MP≅ A''MP.
Since the triangles are congruent, we can identify a couple of congruent corresponding angles: ∠ APK ≅ ∠ A'PK and ∠ A'PM ≅ ∠ A''PM. Let's mark them below.
Using the Angle Addition Postulate, we can write a couple of equations regarding the angles we have marked in the diagram above:
&m∠ MPK=m∠ A'PK+m∠ A'PM
&m∠ APA''=m∠ APK+m∠ A'PK+m∠ A'PM+m∠ A''PM
Notice that we can, by using the Substitution Property of Equality, replace a couple of angles in the second equation.
