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## Answered question

2022-09-06

Is it given that two lines are perpendicular if a right angle is shown?
If I have a diagram, like the following:

### Answer & Explanation

Brendan Bradley

Beginner2022-09-07Added 11 answers

Step 1
So, If the right angle (the rectangle in yellow) is given. I think is okay to say that $AD\perp BC$. Now, you say that then $AB=AC$ iff $BD=BC$ by using the perpendicular bisector theorem.
Even though this is fine, I think this is overkilling since it feels that the thing you want to prove follows trivially from this theorem.
Step 2
Another way to prove this is by using Pythagorean theorem. $B{D}^{2}+A{D}^{2}=A{B}^{2}$ and DC2+AD2=AC2. If you substract these two equalities, you get $D{C}^{2}+A{D}^{2}=A{C}^{2}$. So, $BD=DC$ iff $AB=BC$.

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