A presentation on Triangle and Quadrilateral.

1. Triangle and Quadrilateral
For class VIII
Prepared by
G R Ahmed
K V Khanapara Guwahati Assam.

2. Quadrilaterals
A quadrilateral is a four sided figure.
The four angles of a quadrilateral sum to 360. b 
a 
c 
d 
a  + b  + c  + d  = 360
(This is because a quadrilateral can be divided up
into two triangles.)
Note: Opposite angles in a cyclic quadrilateral sum to 180.
a  + c  = 180
b  + d  = 180

3. ..
...
.
Parallelogram
1. Opposite sides are parallel 2. Opposite sides are equal
3. Opposite angles are equal 4. Diagonals bisect each other

4. Rhombus
1. Opposite sides are parallel 2. All sides are equal 3. Opposite angles are equal
..
...
.
4. Diagonals bisect each other 5. Diagonal intersects at right
angles
6. Diagonals bisect opposite
angles
..
...
.

5. Rectangle
1. Opposite sides are parallel 2. Opposite sides are equal
3. All angles are right angles 4. Diagonals are equal and bisect each
other

6. Square
1. Opposite sides are parallel 2. All sides are equal 3. All angles are right angles
4. Diagonals are equal and
bisect each other
5. Diagonals intersect at
right angles
6. Diagonals bisect each
angle
..
....
..

7. Types of Triangles
Equilateral Triangle
.
.
.
3 equal sides
3 equal angles
Isosceles Triangle
a b
2 sides equal
Base angles are equal
a = b
(base angles are the angles
opposite equal sides)
Scalene triangle
3 unequal sides
3 unequal angles

8. Congruent triangles
Congruent means identical. Two triangles are said to be congruent if they have
equal lengths of sides, equal angles, and equal areas. If placed on top of each other
they would cover each other exactly.
The symbol for congruence is . For two triangles to be congruent (identical), the
three sides and three angles of one triangle must be equal to the three sides and three
angles of the other triangle. The following are the ‘ tests for congruency’.
a
b c
x
y z
abc xyz

9. Case 1
= Three sides of the other triangleThree sides of one triangle
SSS
Three sides

10. Case 2
Two sides and the included angle of
one triangle
Two sides and the included angle of
one triangle
=
SAS
(side, angle, side)

11. Case 3
One side and two angles of
one triangle
Corresponding side and two
angles of one triangle
=
ASA
(angle, side, angle)

12. Case 4
A right angle, the hypotenuse and
the other side of one triangle
A right angle, the hypotenuse and
the other side of one triangle
=
RHS
(Right angle, hypotenuse, side)

13. Theorem: Vertically opposite angles are equal in measure.
Given:
To prove :
Construction:
Proof: Straight angle
Straight angle
 1=2
Label angle 3
1=2
Intersecting lines L and K, with vertically
opposite angles 1 and 2.
1+3=180
2+3=180
Q.E.D.
L
K
1 2
 1+3=3+2 .....Subtract 3 from both sides
3

14. Theorem: The measure of the three angles of a triangle sum to 180.
Given:
To Prove: 1+2+3=180
Construction:
Proof: 1=4 and 2=5 Alternate angles
1+2+3=4+5+3
But 4+5+3=180 Straight angle
 1+2+3=180
The triangle abc with 1,2 and 3.
4 5
a
b c
1 2
3
Q.E.D.
Draw a line through a, Parallel to
bc. Label angles 4 and 5.

15. Theorem: An exterior angle of a triangle equals the sum of the two interior opposite
angles in measure.
Given: A triangle with interior opposite angles 1 and 2 and the exterior angle 3.
To prove: 1+ 2= 3
Construction: Label angle 4
Proof: 1+ 2+ 4=180
3+ 4=180
Three angles in a triangle
 1+ 2+ 4= 3+ 4
Straight angle
 1+ 2= 3
a
b c
3
1
2 4
Q.E.D.

16. a
b c1 2
Theorem: If to sides of a triangle are equal in measure, then the angles
opposite these sides are equal in measure.
Given: The triangle abc, with ab = ac and base angles 1 and 2.
To prove: 1 = 2
Construction: Draw ad, the bisector of bac. Label angles 3 and 4.
Proof:
ab = ac given
3 = 4 construction
ad = ad common
SAS
 1 = 2 Corresponding angles
d
3 4
abd acd
Consider abd and acd:
Q.E.D.

17. Theorem: Opposite sides and opposite angles of a parallelgram are respectively
equal in measure.
Given: Parallelogram abcd
a
b c
d
To prove:
Construction: Join a to c. Label angles 1,2,3 and 4.
Proof:
1= 2 and 3= 4 Alternate angles
ac = ac common
ASA
 ab = dcand ad = bc Corresponding sides
And abc = adc Corresponding angles
Similarly, bad = bcd
1
23
4
ab = dc , ad = bc
abc = adc, bad = bcd
Consider abc and adc :
abc adc
Q.E.D.

18. Theorem:A diagonal bisects the area of a parallelogram.
a
b c
d
Given: Parallelogram abcd with diagonal [ac].
To prove: Area of abc = area of adc.
Proof:
ab = dc
Opposite sides
ad = bc Opposite sides
ac = ac Common
SSS
Consider abc and adc:
abc adc
area abc = area adc
Q.E.D.