1. Parallelogram :
A quadrilateral with opposite sides parallel is called a parallelogram.
Properties :
1) The opposite sides of a parallelogram are congruent.
2) The opposite angles of a parallelogram are congruent.
3) The diagonals of a parallelogram bisect each other.
2. Rhombus :
A quadrilateral with all four sides congruent is called a rhombus.
Properties :
1) The diagonals of rhombus bisect each other.
2) Each diagonal of a rhombus is the perpendicular.
3) The opposite angles of rhombus are congruent.
3. Rectangle :
A quadrilateral with all angle right angles is called a rectangle.
Properties :
1) The opposite sides of a rectangle are congruent.
2) The diagonals of a rectangle are congruent.
3) The diagonals of a rectangle bisect each other.
4. Square :
A quadrilateral with all sides congruent and every angle a right
angle is called a square.
Properties :
1) The diagonals of a square are congruent.
2) The diagonals of a square bisect each other.
3) Each diagonal of a square is perpendicular of the other.
5. Kite :
1) l(AB) = .......
2) l(BC) = .......
3) m<AMD = .......
4) ABCD is a kite.
a. Fill in the blanks by observing the adjoining figure.
1) m<STR = .......
2) l(PT) = .......
3) m<SPQ = .......
4) PQRS is a ............... .
b. Fill in the blanks by observing the adjoining figure.
2) l(HF) = .......
3) l(CM) = .......
4) HDEF is a..................... .
c. Fill in the blanks by observing the adjoining figure.
1) l(PN) = .......
2) l(TN) = .......
3) m<UTN = .......
4) PUNE is a ............. .
Problems on a square :
1) Find the length of diagonal QS of a square PQRS if the length of diagonal PR is 8 cm.
Solution : The diagonals of a square are congruent. So in a square PQRS,
l(QS) = l(PR)
l(QS) = 8 cm ............ (Given l(PR)= 8 cm )
2) If in the square ABCD, l(AB)= 4.5 cm, What are the lengths of the other sides of the square?
Solution : All sides of a square are congruent.So for a square ABCD
l(AB) = l(BC) = l(CD) = l(AD) = 4.5 cm.
3) The diagonasl seg DF and seg EG of the square DEFG intersect each other in point M. If
l(DM) = 7 cm. Find l(EG).
Solution :
The diagonals of a square bisect each other.
So in a square DEFG,
l(DF) = l(DM) + l(MG)
= 7 + 7 = 14 cm.
The diagonals of a square are congruent.
l(DF) = l(EG) = 14 cm.
Problems on a rectangle :
1) In a rectangle PQRS, l(PQ) = 7 cm, l(PS) = 9 cm, Find l(QR) and l(SR).
Solution :The opposite sides of a rectangle are congruent.
l(PQ) = l(SR) = 7 cm
l(PS) = l(QR) = 9 cm.
2) The diagonals AC and BD of a rectangle ABCD intersect in point K. If l(AK) = 3.5 cm. then l(KC) =? and l(AC) =?
Solution : The diagonals of a rectangle bisect each other. So the point K is mid point of seg AC.
l(AK) = l(KC) = 3.5 cm and l(AC) = l(AK) + l(KC) = 3.5 + 3.5 = 7 cm.
3) The diagonasl XZ and WY of rectangle XYZW intersect each other in point M. If
l(XZ) = 8 cm. Find l(XM) and l(YM) .
Solution :
The diagonals of a rectangle are congruent and bisect each other.
So in a rectangle XYZW,
l(XZ) = l(YW)= 8 cm and M is mid point of seg XZ and YW.
l(XM) = l(MZ)= 8/2= 4 cm
Also, l(YM) = l(MW)= 8/2= 4 cm
Also, l(YM) = l(MW)= 8/2= 4 cm
Problems on a rhombus :
1) If the length of one side of a rhombus is 7.5 cm, find the length of the remaining sides.
Solution : All sides of rhombus are congruent. So lengths of all the remaining sides is 7.5 cm.
2) The diagonals AC and BD of a rhombus ABCD intersect in point O. Find m<AOD and m<BOC.
Solution : The diagonals of rhombus are perpendicular to each other. So the diagonals AC and BD of a rhombus ABCD makes an angle right angle to each other. m<AOD = m<BOC = 90
3) If m<QPS in rhombus PQRS is 65, find m<QRS.
Solution :
The opposite angle of rhombus are congruent.
m<QPS = m<QRS = 650.
Problems on a parallelogram :
1) The diagonals LN and MT of parallelogram LMNT intersect in point O. If l(MO) = 5 cm, l(LN) = 6 cm, find l(OT) and l(NO).
Solution : The diagonals of a parallelogram bisect each other. So in a rectangle LMNT,
l(LO) = l(ON)= 1/2xl(LN)= 1/2 x 6 = 3 cm
l(MO) = l(MO)= 5 cm
2) In parallelogram PQRS, m<Q = 1300. Find the measures of the angles of PQRS.
Solution : The opposite angles of parallelogram are congruent.
m<Q=m<S= 1300,
Also m<P+ m<Q+ m<R m<S = 360
m<P +130+ m<R+ m<R + 130 = 360
m<P+ m<R = 360 -260=100
m<P+m<P=100
2m<P=100
m<P=50.
m<P=m<R=50.
Also m<P+ m<Q+ m<R m<S = 360
m<P +130+ m<R+ m<R + 130 = 360
m<P+ m<R = 360 -260=100
m<P+m<P=100
2m<P=100
m<P=50.
m<P=m<R=50.
3) The measures of the opposite angles of a parallelogram are (3x-2)0 and (50-x)0. Find the measure of each angle of the parallelogram.
Solution :
Let the parallelogram be PQRS
The opposite angles of parallelogram are congruent.
m<P=m<R
(3x-2) =(50 -x)
3x + x = 50 + 2
4x = 52
x = 13
m<P=m<R(3x-2) =(50 -x)
3x + x = 50 + 2
4x = 52
x = 13
m<P=m<R= (3x-2)= 3x 13 - 2 = 39 -2 = 37 and
m<Q=m<S=180 - 2 x 37= 180 - 74 = 146.