Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Squares and rectangles are special types of parallelograms with special properties. A square is a type of equiangular parallelogram and square properties include congruent diagonals and diagonals that bisect each other. A rectangle is a type of regular quadrilateral. Rectangle properties include (1) diagonals that are congruent, (2) perpendicular diagonals that bisect each other and (3) diagonals that bisect each of the angles.
It's important to know the properties of a rectangle and a square because you're going to use them in proofs, you're going to use them in true and false, fill in the blank, multiple choice, you're going to see it all over the place. So the key facts about a rectangle is that it's an equiangular parallelogram which is another way of saying if you have 360 degrees and 4 angles are congruent then each of these angles must be 90 degrees, it's also a parallelogram which means opposite sides are congruent and that you have 2 pairs of parallel side.
The diagonals of a rectangle are congruent to each other, which is a trick that people use in construction, if they're trying to build a rectangular room. And the diagonals will bisect each other. So these segments are going to be congruent. Now I didn't use a different number of marking here since the diagonals are congruent. These 4 segments will be congruent to each other. Moving on to a square the key things about a square is that you can name it a whole bunch of different ways. You could call it a regular quadrilateral, you could call it equal regular rhombus but the other key thing is that the diagonals are congruent, so basically a square is just putting together everything that you know about a parallelogram, about a rhombus and a rectangle.
Just kind of smoosh them all together and the square has all those properties. So the diagonals are congruent to each other, the diagonals bisect each other and are perpendicular, so I can mark these 4 as congruent and they bisect the angles that they intercept. So you're going to be creating 45 degree angles when you draw in your diagonals.
Now the last key thing that I want to touch on are these two statements. A square is a rhombus and rhombus is a square, one of those things that parents basically that's the only thing they remember of Geometry is being confused by these two statements. Well the first statement is true, the second statement is not. If we examine it the keys to a rhombus is that it's a parallelogram and it's a equilateral. So equilateral means all sides are congruent, so if we look at a square we have 2 pairs of parallel sides and all sides are congruent. So this one is true, a rhombus however is not a equiangular. The rhombus does not have 4 angles that are all congruent, which is one of the facts about a square so that's why this one will be false. So keep these facets in mind when you're writing proofs or answering your true and false questions.