Diagonals of a Rhombus - Examples, Exercises and Solutions

First, let's mark the vertices of the rhombus with the letters ABCD, then proceed to draw the diagonals AC and BD, and finally mark their intersection point with the letter E:

Now let's examine the following properties:

a. The rhombus is a type of parallelogram, therefore its diagonals intersect each other, meaning:

$AE=EC=\frac{1}{2}AC\\ BE=ED=\frac{1}{2}BD\\$

b. A property of the rhombus is that its diagonals are perpendicular to each other, meaning:

$AC\perp BD\\ \updownarrow\\ \sphericalangle AEB=\sphericalangle BEC=\sphericalangle CED=\sphericalangle DEA=90\degree$

c. The definition of a rhombus - a quadrilateral where all sides are equal, meaning:

$AB=BC=CD=DA$

Therefore, from the three facts mentioned in: a-c and using the SAS (Side-Angle-Side) congruence theorem, we can conclude that:

d.
$$$\triangle AEB\cong\triangle CEB\cong\triangle AED\cong\triangle CED$(where we made sure to properly and accurately match the triangles according to their vertices in correspondence with the appropriate sides and angles).

Indeed, we found that the diagonals of the rhombus create (together with the rhombus's sides - which are equal to each other) four congruent triangles.

Therefore - the correct answer is answer a.

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.

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

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

Therefore, the correct answer is answer A.

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

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