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Remember
1. A triangle does not have a diagonal as each vertex is adjacent to the other two.
2. If all sides of a quadrilateral are equal, its diagonals are perpendicular.
3. If all angles of a quadrilateral are equal, diagonals are equal.
4. If opposite sides of a quadrilateral are parallel and equal, diagonals bisect each other.
5. In a kite only the obtuse angles opposite to each other are equal not the acute angles.
6. In a trapezium the angels are not equal unless it is an isosceles trapezium.
7. A square is also a trapezium, rectangle and rhombus.
WE KNOW THAT
A curve which does not cut itself is called an open curve whereas a curve which cuts itself is called a closed curve. The closed curves which do not cross themselves are called simple curves. A curve (in mathematics) can be straight also.
POLYGONS
A simple closed curve made up of only the line segments is called a polygon. The simplest polygon is a triangle which is made up of 3 line segments. Let us classify the polygons according to the number of sides (or vertices) they have:
Number of sides or vertices  Name of the polygon 
3  Triangle 
4  Quadrilateral 
5  Pentagon 
6  Hexagon 
7  Heptagon 
8  Octagon 
9  Nonagon 
10  Decagon 
n  ngon 
The line connecting any two nonconsecutive vertices is called a diagonal. Convex polygons have no portion of their diagonals in their exteriors. Thus a polygon each of whose interior angles is less than 180°, is called a convex polygon, otherwise it is concave polygon.
REGULAR AND IRREGULAR POLYGONS
A regular polygon has sides of equal length and angles of equal measure, i.e. a regular polygon is both ‘equiangular’ and ‘equilateral’.
Examples: (i) An equilateral triangle has sides of equal length and angles of equal measure.
Therefore, it is a regular polygon.
(ii) A square has sides of equal length and angles of equal measure. So, it is a regular polygon.
(iii) A rectangle is equiangular but not equilateral. Therefore, it is not a regular polygon.
ANGLE SUM PROPERTY
(i) The sum of the measures of angles of a quadrilateral is 360°.
(ii) If the sides of a quadrilateral are produced in order, the sum of four exterior angles so formed is 360°.
Note: If we denote some angles by ∠1, ∠2, ∠3, etc. Then their corresponding measures are m∠1, m∠2, m∠3, etc.
Kinds of Quadrilaterals
1. Trapezium
A quadrilateral having a pair of parallel sides is called a trapezium. In the figure, ABCD is a trapezium in which AD  BC. AB and DC are nonparallel sides. In case the nonparallel sides are equal then the trapezium is called an isosceles trapezium.
In the figure, PQRS is an isosceles trapezium in which PQ  RS and nonparallel sides PS = QR.
A kite is a quadrilateral having two distinct consecutive pairs of sides of equal lengths. In the figure ABCD is a kite having AB = AC and BD = CD.
2. Parallelogram
A quadrilateral having its opposite sides parallel is called a parallelogram. In the figure, ABCD is parallelogram such that AB  CD and AD  BC.
PROPERTIES OF A PARALLELOGRAM
(i) The opposite sides of a parallelogram are parallel.
(ii) The opposite sides of a parallelogram are equal.
(iii) The opposite angles of a parallelogram are equal.
(iv) The adjacent angles of a parallelogram are supplementary.
(v) The diagonals of a parallelogram bisect each other; but they are not equal.
Note: I. The four sides and four angles of a parallelogram are called its elements.
II. Two pairs of opposite sides are AB  CD and BC  AD. ∠A and ∠C are a pair of opposite angles and ∠B and ∠D form another pair of opposite angles. ∠A and ∠D are adjacent angles. ∠B and ∠A; ∠B and ∠C; ∠C and ∠D, etc. are pairs of adjacent angles. Similarly, and are adjacent sides, other pairs of adjacent sides are and ; and ; and .
Solved Question:
Question: A rectangle is also a kind of parallelogram discuss why?
Solution: A rectangle is also a parallelogram because all its opposite sides are parallel to each other. Despite forming an angle of 90°, rectangle due to its parallel sides can be said to be a parallelogram.
Question: A parallelogram FAST is shown in the figure. Find the missing angles.Parallelogram
Solution: FAST is a parallelogram. Let us first recollect the angle related properties of a parallelogram.
Opposite angles are equal.
Adjacent angles are supplementary.
Using the first property, let us find the value of ∠FAS
∠FAS is opposite to ∠FTS, so ∠FAS shall be 80°
Now let us use the 2nd angle property, which says adjacent angles are supplementary.
According to this property, ∠FAS + ∠AST = 180°
80° + ∠AST = 180°
∠AST = 18080
= 100°
For finding the value of ∠AFT we need to use the opposite angle property.
∠AFT =∠AST
∠AST = 100°
So, ∠AFT = 100°
Question: Find the value of the following angles in the rhombus ABCD:
a. ∠XAB
b. ∠XCB and
c. ∠AXB
Solution:
a) DXB is the diagonal that bisects the ∠ABC
Since ∠ABX = 40 °, ∠ABC = 2× 40 = 80°
Now in a rhombus, the sum of adjacent angles is 180° so ∠ABC+∠BAD = 180°
80° + ∠BAD = 180°
∠BAD= 18080=100°
∠BAD = 100°
∠BAD is bisected by diagonal,AXC, so ∠ XAB = 50 °
b) ∠XAB is opposite to ∠ XCB, therefore ∠XCB=50°
c) ∠AXB is formed by the diagonal that perpendicularly bisect each other, therefor ∠AXB is 90°.
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