a A conditional statement has a hypothesis and a conclusion.
If two triangles are congruent,
then their areas are equal.
b The converse of p→ q is q→ p. The converse of our conditional statement of part A is as follows.
If the areas of two triangles are equal,
then two triangles are congruent.This statement is false.
We can show this by giving a counterexample.
In other words, we show that the statement is not true by showing two triangles that have the same area, but are not congruent.
On the figure above we can count squares to find the lengths of the legs of the two right triangles. Using the length of the legs, we can find the area.
Color of the Triangle
Length of the Legs
Area 1/2bh
Red
3, 4
1/2* 3* 4=6
Yellow
6, 2
1/2* 6* 2=6
The areas are equal, but the triangles are not congruent, because the sides are not congruent.
c Equilateral triangles all have the same shape.
If the area is also the same, then the size is the same.
If the shape and size of two triangles are the same, then they are congruent.
It is not possible to draw two equilateral triangles that have the same area, but not congruent.
d The area of a rectangle is the product of its width and length. If, for example, we double the width and halve the length, the area does not change.
On the figure above we can count squares to confirm that the area of both rectangles is 12 unit squares. Since the side lengths are not the same, these are not congruent rectangles.
It is possible to draw two rectangles that have the same area, but not congruent.
e Squares all have the same shape.
If the area is also the same, then the size is the same.
If the shape and size of two squares are the same, then they are congruent.
It is not possible to draw two squares that have the same area, but are not congruent.
f We can look at the pattern emerging in the previous examples and try inserting polygon in the given statement.
If a pair of polygons are congruent,
then they have the same area.
This of course is true, because congruent figures have the same size.
However, the previous examples show that the converse is only true for regular (equilateral) triangles and regular quadrilaterals (squares).
So let's modify the statement.
If a pair ofregularn-gonsare congruent,
then they have the same area.
The converse of this is the following statement.
If two regularn-gons have the same area,
then they are congruent.
To show that this is also true, we can use similar argument as in Part C and E.
For any fixed n, the regular n-gons have the same shape.
If the area is also the same, then the size is the same.
If the shape and size of two regular n-gons are the same, then they are congruent.
This shows, that for the following statement both the statement and its converse is true.
If a pair ofregularn-gonsare congruent,
then they have the same area.
