## Exercise 8.1

Question 1: The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

So, 3x + 5x + 9x + 13x = 360°
Or, 30x = 360°
Or, x = 12°
Hence, 3x = 36°, 5x = 60°, 9x = 108° and 13x = 156°

Question 2: If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Answer; In the following parallelogram both diagonals are equal: So, ΔABC ≅ ΔADC ≅ ΔABD ≅ ΔBCD
Hence, ∠A=∠B=∠C=∠D=90°

As all are right angles so the parallelogram is a rectangle.

Question 3: Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus. Answer; In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle. We have to prove that AB=BC=CD=AD

In ΔAOB and ΔAOD
DO=OB (O is the midpoint)
AO=AO (common side)
∠AOB=∠AOD (right angle)
So, ΔAOB≅ ΔAOD

So, AB=AD
Similarly AB=BC=CD=AD can be proved which means that ABCD is a rhombus.

Question 4: Show that the diagonals of a square are equal and bisect each other at right angles.

Answer: In the figure given above let us assume that

∠DAB=90°
So, ∠DAO=∠BAO=45°
Hence, ∠AOD=90°

DO=AO (Sides opposite equal angles are equal)
Similarly AO=OB=OC can be proved
This gives the proof of diagonals of square being equal.

• Sum of the angles of a quadrilateral is 360°.
• A diagonal of a parallelogram divides it into two congruent triangles.
• In a parallelogram,
• opposite sides are equal
• opposite angles are equal
• diagonals bisect each other
• A quadrilateral is a parallelogram, if
• opposite sides are equal or
• opposite angles are equal or
• diagonals bisect each other or
• a pair of opposite sides is equal and parallel
• Diagonals of a rectangle bisect each other and are equal and vice-versa.
• Diagonals of a rhombus bisect each other at right angles and vice-versa.
• Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
• The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
• A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
• The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.