# I need help finding the answer to this question. My initial thought is that x=120 degrees. Is that right?  If a reflection in line k followed by a reflection in line m maps point A to A", which...

I need help finding the answer to this question. My initial thought is that x=120 degrees. Is that right?

If a reflection in line k followed by a reflection in line m maps point A to A", which results in the same image as a 120degree rotation about point P, then what is the value of x?

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steveschoen | Certified Educator

It would actually be 60.  The angle AP makes to line k is the same as the angle A'P makes to line k.  Similarly with A'P to A"P to line m.

Let's assume that all 4 angles are congruent.  Then, each angle would have to be 30.  Then, x = 60 degrees.  The thing is, it doesn't make a difference what the angles actually are, as long as the "composition" is equivalent to 120 degrees.  For instance, if the first angles, to line k, are 10 each, 20 total, then the angles with line m would have to be 50 each, 100 total, both together for 120 degrees.  And, x would still be 60 degrees.

sbernabel | Certified Educator

(1) Given ∠ k P A + ∠k P A' + ∠m P A' + ∠m P A'' = 120∘.

(2) ∠ k P A = ∠k P A' (by reflection through the line k).

(3) ∠m P A' = ∠m P A'' (by reflection through the line m).

Now substituting (2) and (3) into (1) gives

2 ∠k P A' + 2 ∠m P A' = 2 (∠k P A' + 2 ∠m P A') = 120∘.

Since x∘ = ∠k P A' + 2 ∠m P A', then 2 x∘ = 120∘. Thus x∘ = 60∘.