CONGRUNECY OF TRIANGLES
Two triangles will be congruent if at least one of the following conditions is satisfied: Three sides of one triangle are respectively equal to the three sides of the second triangle (normally referred to at as the S-S-S rule, i.e., the side-side-side congruency).
Two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the second triangle (normally referred to as the S-A-S rule, i.e., side-angle-side congruency).
Two angles and one side of a triangle are respectively equal to two angles and the corresponding side of the second triangle (normally referred to as the A-S-A rule, i.e., angle-side-angle congruency).
Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to hypotenuse and one side of the second right-angled triangle.
In two congruent triangles
- The corresponding sides (i.e., sides opposite to corresponding angles) are equal.
- The corresponding angles (angles opposite to corresponding sides) are equal.
- The areas of the two triangles will be equal.
Some more useful points about triangles
A line drawn parallel to one side of a triangle divides the other two sides in the same proportion. For example, in Fig. 1.21, PQ is drawn parallel to BC in . This will divide the other two sides AB and AC in the same ratio, i.e., AP/PB = AQ/QC.
Conversely, a line joining two points (each) dividing two sides
of a triangle in the same ratio is parallel to the third side. In
Fig. 1.22, P divides AB in the ratio m: n and Q divides AC in the
ratio m : n. Now, the line joining P and Q will be parallel to
the third side BC and the length of PQ will be equal to times the length of BC.
We can say that a line drawn through a point on a side of the triangle parallel to a second side will cut the third side in the same ratio as the first side is divided.
The line joining the midpoints of two sides of a triangle is parallel to the third side and it is half the third side.
Two triangles having the same base and lying between the same
pair of parallel lines have their areas equal (Fig.1.22 (a)). AD
is parallel to BC. Hence
QUADRILATERALS
Any four-aided closed figure is called a quadrilateral. By imposing certain conditions on the sides and/or angles of a quadrilateral, we can get the figures trapezium, parallelogram, rhombus, rectangle, and square.
The sum of four angles of a quadrilateral is equal to
The perpendiculars drawn to a diagonal (in a quadrilateral) from the opposite vertices are called "offsets". In Fig.1.23, BE and DF are the offsets drawn to the diagonal AC.
If the four vertices of a quadrilateral lie on the circumference
of a circle (i.e., if the quadrilateral can be inscribed in
circle) it is called a cyclic quadrilateral (refer to Fig. 1.24).
In a cyclic quadrilateral, sum of opposite angles . i.e in Fig 1.24
and
Also, in a cyclic quadrilateral, exterior angle is equal to the
interior opposite angle, i.e., in Fig. 1.24 is
equal to
Now, we will look at different quadrilaterals and their properties.
TRAPEZIUM
If one side of a quadrilateral is parallel to its opposite side, then it is called a trapezium. The two sides other than the parallel sides in a trapezium are called the oblique sides.
In Fig. 1.25, ABCD is a trapezium where AD is parallel to BC.
If the midpoints of the two oblique sides are joined, it is equal in length to the average of the two parallel sides, i.e., in Fig.1.25, PQ = 1/2 [AD + BC]
In general, if a line is drawn in between the two parallel sides of the trapezium such that it is parallel to the parallel sides and also divides the distance between the two parallel sides in the ratio m : n (where the portion closer to the shorter of the two parallel sides is m), the length of the line is given by:
In Fig. 1.21, RS is the line parallel to AD and BC and the ratio
of the distance of the distance DT and TE is
PARALLELOGRAM
A quadrilateral in which opposite sides are parallel is called a parallelogram.
In a parallelogram
- Opposite sides are equal
- Opposite angles are equal
- Sum of any two adjacent angels is
- Each diagonal divides the parallelogram into two congruent triangles.
- The diagonals bisect each other. Conversely, if in a quadrilateral
- the opposite bisect are equal or
- the opposite angles are equal or
- the diagonals bisect each other or
- A pair of opposite sides are parallel and equal such a quadrilateral is a parallelogram.
If two adjacent angles of a parallelogram are equal then all four angles will be equal and each in turn will be equal to other and the figure will be a rectangle.
If any two adjacent sides of a parallelogram are equal, then all four sides will be equal to each other and the figure will be a rhombus.
If any point inside a parallelogram is taken and is joined to all the four vertices the four resulting triangles will be such that the sum of the areas of opposite triangles is equal. In Fig. 1.27, P is a point inside the parallelogram ABCD and it is joined to the four vertices of the parallelogram by the lines PA, PB, PC and PD respectively. Then Area of triangle PAB + Area of triangle PCD = Area of triangle PBC + Area of triangle PAD = Half the area of parallelogram ABCD.
If there is a parallelogram and a triangle with the same base and between the same parallel lines, then the area of the triangle will be half that of the parallelogram.
If there is a parallelogram and a rectangle with the same base and between the same parallel lines, then the areas of the parallelogram and the rectangle will be the same. The figure formed by joining the midpoints of the sides of any quadrilateral taken in order, is a parallelogram
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