Question

Proof #1 If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus.

#2 Continue the proof above to prove: In a rhombus, the diagonals bisect the angles. Write a few more sentences:

Answer 1

In a rhombus, the diagonals bisect the angles at their intersection point.

Step-by-step explanation:

To prove that in a rhombus, the diagonals bisect the angles, we can use the properties of a rhombus and the given information. Here's the proof:

Proof: Let ABCD be a rhombus. We are given that the diagonals AC and BD both bisect each other (intersect at their midpoints) and they are perpendicular (form a 90-degree angle).

Since AC and BD bisect each other, we can conclude that the line segment OB (where O is the intersection point of AC and BD) is congruent to OD (since it's a rhombus). Similarly, OA is congruent to OC.

Now, we consider triangles ABO and CDO. These triangles share side OB (or OD) and side OA (or OC) which are congruent by properties of a rhombus.

We also know that AC is perpendicular to BD, so angle ABO (or angle CDO) is congruent to angle CBO (or angle ADO) by vertical angles theorem.

Therefore, in a rhombus, the diagonals AC and BD bisect the angles at their intersection point.