How can you be certain that a square is a rhombus?
Hint
You don't need to write a formal proof. We won't stop you if that's what you want to do, but we're just saying it isn't necessary.
Answer
We know that a square is a type of parallelogram because opposite sides are congruent and parallel to each other. We also know that its diagonals are perpendicular bisectors of one another. Since a rhombus is a parallelogram with perpendicular diagonals, we've shown that a square is a rhombus.
Example 2
The quadrilateral ABCD is a square. If the length of AB is 2a, how long are AC and BG in terms of a?
Hint
Remember all that talk about 45-45-90 triangles? Well, it's important.
Answer
and
Example 3
The quadrilateral ABCD is a square. What is the measure of ∠AGD?
Hint
What can we say about the diagonals of a square?
Answer
90°
Example 4
The quadrilateral ABCD is a square. Show that ∠GCB measures 45°.
Hint
A square is a square, sure. But what other types of quadrilaterals can it be?
Answer
A square has all four internal angles equal to 90° because it's also a rectangle. A square's diagonals bisect the internal angles because a square is also a rhombus. This means ∠GCB, which is half the internal angle of the square, measures 45°.
Example 5
If ABCF and FCDE are both squares, show that ΔACE ~ ΔCDE.
Hint
With this question, as with photography, it's all about angles.
Answer
We'll start with the fact that all the interior angles of a square are 90°. All squares are also rhombi, which means that their diagonals are angle bisectors. That means angles like ∠AEC, ∠CED, ∠ECD, and ∠EAC all have measures of 45° and are therefore congruent. That means two angles in each of ΔACE and ΔCDE are 45° in measure. (Basically, ∠AEC ≅ ∠CED and ∠EAC ≅ ∠ECD.) According to the AA postulate for similar triangles, this means that ΔACE ≅ ΔCDE.
Example 6
JOHN is a square and HA ≅ NA. Prove that ΔJAO is isosceles.
Hint
Congruent triangles are there for you, always and forever. Even in squares.
and 
