Quadrilaterals Test - 8

Perimeter of ABCD = AB + BC + CD + DA = 30

In parallelogram, opposite sides are equal.

AB= CD = 9.5 and BC = DA = x

So, 9.5 + x + 9.5 + x = 30

2x = 30 - 19

x = 5.5

AB = 9.5 = CD and BC = DA = 5.5

If opposite angles are bisected by diagonals in parallelogram, all four bisected angles become equal which leads to equal adjacent side. That is not true in case of parallelogram.

In rhombus ,diagonals bisect each other at right angles. Which proves that all the angles of quadrilateral formed by joining the mid points of rhombus is equal to 90. A quadrilateral with both pair of opposite angles equal is a parallelogram

A parallelogram having one right angle is a rectangle.

If DE = EF then triangle AED become congruent to triangle CEF by SSS congruence rule.

By CPCT, angle ECF = angle EAD which forms a pair of an alternate angles.

which proves that AD is parallel to CF

Let the half of the angles A ,B, C and D are denoted by a,b,c and d respectively.

a + b + angle APB = 180 and c + d + angle DRC = 180 ( angle sum property)

angle APB = angle QPS and angle DRC = angle QRS ( vertically opposite angles)

So, angle QPS + angle QRS = (180 - a - b) + ( 180 - c - d)

 = 360 - ( a + b + c + d)

 = 360 - 1/2 of ( angle A + angle B + angle C + angle D)

= 360 - 1/2 0f 360 = 180

therefore, angle QPS and angle QRS are supplementary.

angle DAC = angle ACB = 32 ( alternate angles)

angle AOB + angle COB = 180 ( linear pair)

Angle COB = 180 - 70 = 110

In triangle BOC, angle BOC + angle OCB + angle CBO = 180 ( angle sum property)

110 + 32 + angle CBO = 180

angle CBO = 180 - 142 = 38

A quadrilateral formed by joining mid points of the sides of the rectangle is a Rhombus.

In Rhombus, all sides are equal.

In triangle ABC, P and Q are mid points of the sides AB and BC respectively. By mid point theorem, PQ is parallel to AC and PQ is half of the AC.

let diagonal AC = 8 cm . So, PQ =4 cm

therefore, PQRS is a rhombus which all sides equal to 4 cm