Question

how to prove a quadirateral a rhomus

Answer 1

You can use the following six methods to prove that a quadrilateral is a rhombus. The last three methods in this list require that you first show (or be given) that the quadrilateral in question is a parallelogram:

If all sides of a quadrilateral are congruent, then it’s a rhombus (reverse of the definition).If the diagonals of a quadrilateral bisect all the angles, then it’s a rhombus (converse of a property).If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it’s a rhombus (converse of a property).Tip: To visualize this one, take two pens or pencils of different lengths and make them cross each other at right angles and at their midpoints. Their four ends must form a diamond shape — a rhombus.If two consecutive sides of a parallelogram are congruent, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).If either diagonal of a parallelogram bisects two angles, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).If the diagonals of a parallelogram are perpendicular, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

Here’s a rhombus proof for you. Try to come up with a game plan before reading the two-column proof.

Statement 1:

Reason for statement 1: Given.

Statement 2:

Reason for statement 2: Opposite sides of a rectangle are congruent.

Statement 3:

Reason for statement 3: Given.

Statement 4:

Reason for statement 4: Like Divisions Theorem.

Statement 5:

Reason for statement 5: All angles of a rectangle are right angles.

Statement 6:

Reason for statement 6: All right angles are congruent.

Statement 7:

Reason for statement 7: Given.

Statement 8:

Reason for statement 8: A midpoint divides a segment into two congruent segments.

Statement 9:

Reason for statement 9: SAS, or Side-Angle-Side (4, 6, 8)

Statement 10:

Reason for statement 10: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Statement 11:

Reason for statement 11: Given.

Statement 12:

Reason for statement 12: If a triangle is isosceles, then its two legs are congruent.

Statement 13:

Reason for statement 13: Transitivity (10 and 12).

Statement 14:

Reason for statement 14: If a quadrilateral has four congruent sides, then it’s a rhombus.

Answer 2

Answer:

if the diagonals of a quadilateral bisect each other , quadilateral is a parallelogramDescribe the social lifestyle of people of your geographical region of Nepal.

ABCD is an quadrilateral with AC and BD are diagonals intersecting at O.

It is given that diagonals bisect each other.

∴  OA=OC and OB=OD

In △AOD and △COB

⇒  OA=OC                                [ Given ]

⇒  ∠AOD=∠COB                  [ Vertically opposite angles ]

⇒  OD=OB                               [ Given ]

⇒  △AOD≅△COB                 [ By SAS Congruence rule ]

∴  ∠OAD=∠OCB         [ CPCT ]  ----- ( 1 )

Similarly, we can prove 

⇒  △AOB≅△COD

⇒  ∠ABO=∠CDO       [ CPCT ]     ---- ( 2 )

For lines AB and CD with transversal BD,

⇒  ∠ABO and ∠CDO are alternate angles and are equal.

∴  Lines are parallel i.e. AB∥CD

For lines AD and BC, with transversal AC,

⇒  ∠OAD and △OCB are alternate angles and are equal.

∴  Lines are parallel i.e. AD∥BC

Thus, in ABCD, both pairs of opposite sides are parallel.

∴  ABCD is a parallelogram.