Congruent Figures :- Congruent means equal in all respects or figures whose shape and size are both the same.
For example:- Two circles of same radii are congruent, also two squares of the same sides are congruent.
Congruent Triangle :- Two triangle are congruent of and only if one of them can be made to superpose on the other, so as to cover it exactly.
Ø If two triangles ABC and PQR are congruent under the correspondence
Ø In two congruent triangles corresponding parts are equal and we write ‘CPCT’ for corresponding part of congruent triangle.
Criteria for congruency of triangle :-
SAS congruency Rule :- Two triangle are congruent of two sides and the included angle of one triangle are equal to the two sides and included angle of other triangle.
Note :- Remember that included angle should be equal.
· ASA congruency Rule : Two triangle are congruent if two angles and included side of one triangle are equal to two angles and included side of other triangle.
For example :-
AAS Congruency Rule :- Two triangle are congruent if any two pairs of angle and one pair of corresponding sides are equal .
For example :-
· SSS congruency Rule :- If three sides of one triangle are equal to the three sides of another triangle the two triangles are congruent.
RHS Congruency Rule :- If in two right angle triangle the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle then both triangle are congruent.
For example :-
Two triangles are said to be similar, if -
· Their corresponding angles are equal and
· Their corresponding sides are in proportion (or are in the same ratio)
Criteria for similarity of Triangles :-
1 AAA or AA criterion :- If two corresponding angles of two triangle are equal, then both triangle are similar.
Note :- If in two triangles, two angles are equal then third angle will also equal, so this rule is also called AAA.
2 S-S-S (side-side-side) criterion : If three sides of two triangles are in proportion then both triangles are similar
3. S.A.S (side-angle-side) Criterion:- If two sides of two triangles are in proportion and their included angle is equal , then both triangles are similar.
· Some important Results of similarity.
(i) In a triangle, line drawn parallel to one side divides the remaining two sided in same ratio
2 Ratio of corresponding sides of two similar triangle is equal to ratio of their corresponding medians, corresponding height and corresponding angle bisectors.
3 Ratio of areas of two similar triangles is equal to ratio of squares of corresponding sides, height, Medians, in radius and circum radius.
4 if two triangles are similar and they have equal area then both triangle will be congruent.
5 Triangles between two parallel line are similar.
· Some Important Theorems and Results related to triangle:-
(1) Mid Point Theorem :- The line segment joining the mid points of two sides of a triangle is parallel to the third side and equal to half of the third side.
(2) Converse of Midpoint theorem :- A line through the midpoint of a side of a triangle parallel to another side bisects the third side.
3 Apollonian theorem: -
4 Angle Bisector theorem : Bisector of an interior angle of triangle divides the opposite side internally in ratio of the sides containing angle.
5 Similar to (4) Bisector of an exterior angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.
6 Median divides the area of triangle in two equal parts.
7 Centroid divides the area of triangle in six equal parts.
8 Three times of the sum of the square of three sides of a triangle is equal to four times the sum of squares of its medians.
9 In a right triangle ABC, and a perpendicular is drawn point B to hypotenuse AC.
10 Area of Triangle:-
If length of medians are given :
11 In an equilateral Triangle having side ‘a’.
(i) All centres lies on one point.
12 In an isosceles Triangle :-
15. According to figure, if angle bisector of and exterior angle R meet at point T, then-
16 If are the length of perpendiculars on three sides drawn from a point inside the equilateral triangles then
17 The area of triangles having same base and between two parallel lines are equal.
18 the perimeter of a triangle is always greater than sum of all three altitudes.
19. The perimeter of a triangle is always greater than sum of all three medians.
20. On Joining the midpoint of side of a triangle four triangles are formed, then
Note : It can be proved using midpoint theorem.
21 According to figure , In a right angle triangle there are two points x and y on sides AB and BC respectively then-
22 If two medians cut each other at 90° on centroid, then –