Question

 Prove that the diagonal of a square it into two congruent triangle.

Answer 1

Hint: Draw a square of the name ABCD. Then join AC, after that consider two triangles ABC, ADC and prove them they are congruent using side-side-side action of congruent.
Complete step by step answer:
Let us assume a square as ABCD. Now according to properties of squares we can say that AB=BC=CD=DA as all the sides of the square are equal. We can also write that \[\angle DAB=\angle ABC=\angle BCD=\angle CDA=90{}^\circ \] as all the interior angles of the square is 90°.
Now in square ABCD, AC is joined.

So to solve the question we have to prove that triangle ABC and ADC is congruent to each other. We can prove them by only one of the following axioms which are:

• (I) Side-Side-Side (SSS): which means that if three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.

(II) Side-Angle-Side (SAS): which means that if two sides are one included angle is congruent to corresponding parts of another triangle, the triangles are congruent.

(III) Angle-Side-Angle (ASA): This means that if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

(IV) Angle-Angle-Angle (AAA): which means that if two angles and one non included side of triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

(V) Hypotenuse-Leg (HL): This means that if the hypotenuse and leg of one right triangle are congruent to the congruent parts of another triangle, the right triangles are congruent.

Now let’s consider triangles ABC and ADC we see,

• (I) AB=AD as all the sides of the square are equal.

(II) BC=DC as all the sides of square is equal

(III) AC is the common side to both triangles ABC and ADC.

By the above three points we say that ΔABC ≅ ΔADC using Side-Side-Side axiom of congruency. Hence proved that the diagonal of square divides the square into two congruent triangles

Note: Before solving the problem students should know the properties of squares and they should also know the axioms for congruence of triangles.