Prove: Perpendicular lines form congruent, adjacent angles?

Question

Anyone know how to do proofs?

Prove: perpendicular lines form congruent, adjacent angles.

Answer 1

What your proof will consist of depends on which statements you are allowed to use as axioms. For instance, Euclid, in his Elements, _defined_ perpendicular lines as those that created congruent, adjacent angles, and thus a proof of this statement in his system would be superflorous.

Some axiomatic systems in high school give you the twin statements "the angle between perpendicular lines is 90°" and "a straight angle is 180°". In which case your proof would be:

Let a and b be perpendicular lines, and P be their point of intersection. Let A be a point other than P on a, and B and C be on b such that P is between B and C. Then APB and APC are adjacent angles (as they share a common side, and niether one is within the other) which therefore sum to angle BPC. Since BPC is a straight angle, it is 180°. Further, since a and b are perpendicular, at least one of APB and APC are 90°, so the other must be 180°-90°, which is 90° also. Since both angles are 90°, they are congruent. Q.E.D.

Answer 2

Well,

a perpindicular line is when a verticle line hits a horizontal line at 90 degrees, thats why its called perpindicular. So basically if you make a perpindicular line, there are two angles on either side of the lines which are adjacent which are congruent at 90 degrees each

Answer 3

This will be short and without any symbols. Drop an altitude from the vertex of the right angle to the hypotenuse. The two triangles formed are similar to the original triangle and their areas sum to the original. Also a corresponding side of each of the three triangles (namely, each hypotenuse) is a side of the original and areas of similar triangles are proportional to the squares of corresponding sides. Since the areas of the smaller two sum to the original we may divide by the constant of proportionality to get that the sum of the squares of the legs of the original square is equal to the square of the hypotenuse.