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ABCD is a rectangle in which diagonals AC bisects angle A as well as angle C. Show that ABCD is a squre.

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1 Answer

Hi Angel,

Here is one way of solving this.  The diagonal AC divides the rectangle into two triangles ABC and ACD.  Let's focus on triangle ABC.  We're told that the diagonal AC bisects angles A and C.  But all the interior angles of a rectrangle are 90º.  So, the angles within the triangle ABC at A and C must both be 45º.  But this means that triangle ABC must be an isosceles triangle which means that the sides opposite the 45º angles are congruent.  But these sides are just AB and BC.  Since ABCD is a rectangle, the opposite sides are equal.  So, now all 4 sides are equal and ABCD must be a square.

You might wonder how the equality of the two 45º angles makes triangle ABC is isosceles since the definition of isosceles is opposite sides are equal.  Well, there is a theorem which states that if a triangle has two angles that are congruent, then the sides opposite those angles are congruent.  And that means that the triangle is isosceles.  But if you did not want to take this for granted, you could prove it by bisecting angle B and then noticing that this divides triangle ABC into two triangles that have two pairs of equal angles (A and C and the two half angles at B) and one pair of congruent sides (the bisector itself); you could then use SAA to show that the two triangles are congruent and then use the CPCTC theorem to show that sides AB and BC are congruent.

Of course, you could lay this out in a more formal proof form; but this should give you the concepts you need to do that.

Hope this helps,

Roger