When $ABCD$ is a rhombus then: $∢AED=∢AEB=∢DEC=∢CEB=90^o$

The diagonals of a rhombus bisect the angles of the rhombus.

Diagonals of a rhombus

When $ABCD$ is a rhombus then:

$∢A1=∢A2$ $∢B1=∢B2$ $∢C1=∢C2$ $∢D1=∢D2$

Observe: These three statements are properties of the diagonals of a rhombus that, in case you had a rhombus in front of you, you should not prove them, but simply use them. Anyway, to help you understand the logic behind them, we will demonstrate the properties below.

Let's look at the following example

Diagonals of a rhombus

Given: $ABCD$ rhombus

To prove: The diagonals of a rhombus intersect and also bisect the angles of the rhombus. Solution:

It can be argued that $AB∥DC$ and $AD∥DC$ therefore, the rhombus is also a parallelogram.

One of the properties of the parallelogram is that its diagonals intersect. Thus, we have already proven the first property. Now we can prove that all triangles created from thediagonals are congruent. All are formed by shared sides and by equal sides of the rhombus. Let's see it clearly in the illustration:

Each triangle, composed of a blue side, another green, and another pink, the triangles are congruent by SSS. Therefore, all corresponding angles are equal. The $2$ corresponding angles are also adjacent. For the angles to be corresponding and also equal they must be right angles. Consequently, the diagonals are also perpendicular – The second property.

Furthermore, according to congruence, we can argue that all angles that are bisected by the diagonals are equivalent to each other and, therefore, we determine that the diagonals of a rhombus also bisect the angles - the third property. Remember, this demonstration is only so we can understand it more deeply. You should not have to prove these three properties of the diagonals.

Now let's move on to the properties of the diagonals of a rhombus that we do have to prove in order to use them:

The diagonals of a rhombus form four congruent triangles.

The diagonals of a rhombus create equal alternate angles.

Note that we have proven these claims in the example.

Useful Information: We can deduce the area of the rhombus based on its diagonals!

Multiply the diagonals, divide by $2$ and we get the area of the rhombus. Let's see it in the formula:

Let's place in the formula when $DB=X$ $\frac{X\cdot4}{2}=40$

Multiply crosswise and it will give us: $X\cdot4=80$ $X=20$

Therefore, the diagonal $DB$ measures $20$.

Watch out, don't miss it!

You might sometime be asked if the diagonals of a rhombus are of the same length. The answer is no. The lengths of the diagonals of a rhombus are not equal.

Great! Now you know everything about the diagonals of a rhombus and you will be able to use some of its properties when it suits you.

Examples and exercises with solutions of the diagonals of a rhombus

Exercise #1

Look at the following rhombus:

Are the diagonals of the rhombus parallel?

Step-by-Step Solution

The diagonals of the rhombus intersect at their point of intersection, and therefore are not parallel

Answer

No.

Exercise #2

Look at the following rhombus:

Are the diagonals of the rhombus perpendicular to each other?

Step-by-Step Solution

The diagonals of the rhombus are indeed perpendicular to each other (property of a rhombus)

Therefore, the correct answer is answer A.

Answer

Yes.

Exercise #3

Do the diagonals of the rhombus above intersect each other?

Step-by-Step Solution

In a rhombus, all sides are equal, and therefore it is a type of parallelogram. It follows that its diagonals indeed intersect each other (this is one of the properties of a parallelogram).

Therefore, the correct answer is answer A.

Answer

Yes

Exercise #4

Given the rhombus:

How much is it worth? $∢A$?

Video Solution

Step-by-Step Solution

The rhombus is a type of parallelogram, therefore its opposite angles are equal (property of parallelograms), so:

$∢A =50\degree$Therefore, the correct answer is answer A.

Answer

50

Exercise #5

Look at the following rhombus:

Can a rhombus have diagonals that are equal?

Video Solution

Answer

Yes.

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Test your knowledge

Question 1

Look at the following rhombus:

Are the diagonals of the rhombus perpendicular to each other?