How can you prove " a diagonal of a parallelogram divides the parallelogram into two congruent triangles?

Answer 1

no, that's not a theorem. you can first prove the diagonal congruent to itself by reflexive property. then you can prove opposite sides congruent by property of parallelogram. So that would prove SSS. or you can do opposite angles of a parallelogram are congruent and that would be SAS. either way, you've gotta to do it the long way because there is no such theorem that you're stating.

Answer 2

the properties of parallelograms state that the opposite sides are both congruent and parallel. You can prove that a diagonal of a parallelogram creates two congruent triangles using any triangle-congruency postulates, such as SSS, AAS, SAS, ASA, and HL, but HL won't work (explained below). S standing for sides, A for angles, H for hypotenuse, and L for leg. This means to use these postulates, the corresponding parts have to be congruent. So to use SSS, all 3 sides of 2 triangles must be congruent to prove that the two triangles are congruent. ASA, will need 1 congruent angle, 1 congruent side, and 1 congruent angle, in that order. The congruent side has to be between the congruent angles to use that theorem. Note that AAA and @SS (A is written in "at", due to the fact that it won't let it show) does not work.

Let's use SSS as the postulate. Since a property of a parallelogram states that opposite sides are both congruent and parallel, we already have 2 sides of the triangle. Now draw a diagonal. Using the Reflexive property, we can assume the diagonal is congruent to itself. So now you have 3 sets of congruent sides, enabling you to prove that the two triangles are congruent.
To use ASA or AAS, first draw a diagonal. Because opposite sides are parallel, the alternate interior angles are congruent. This means that if you draw a diagonal from the lower right to the upper left, you can see a Z. The angles of the Z (the > and the <) are congruent. Same goes for the other way. Now you have 2 congruent angles. To use ASA, use the property that opposite angles are congruent in parallelograms, and to use AAS, confirm that one of the sides to both triangles are congruent.
To use HL, first you need a parallelogram with a right angle, because hypotenuses are formed in right triangles. So the figure has to be proven to be either a rectangle or a square. But even then, you can't use HL because the hypotenuse of both triangles are the same, so use Reflexive property.
Got all that?

Answer 3

Can you use SSS (side side side)?

In a parallelogram, the opposite sides are congruent.
The diagonal would be congruent to itself.

So you could prove all three pairs of sides are congruent,
and by SSS, then the two triangles are congruent.

Answer 4

The two triangles formed have equal bases( opposite sides of a parallelogram) and equal heights ( between parallel lines). Thus the areas .5xbxh is the same for both.

Answer 5

SSS/SAS !!!@!@!