Triangle
Definition : A triangle is a polygon with three sides.
It also has three vertices and three angles.
Types
of triangle:
1.
Equilateral Triangle
An equilateral triangle is a triangle in which all three sides are the same length.
2.
Isosceles Triangle:
An isosceles triangle is a triangle in which two sides are the same length and the third side is a different length.
3.
Scalene Triangle:
A scalene triangle is a triangle in which all three sides are different lengths.
4. Acute Triangle
An acute triangle is a triangle in which all three of the angles are acute angles.
5. Obtuse Triangle
An obtuse triangle is a triangle in which one of the angles is a obtuse angle.
6. Right Triangle
A right triangle is a triangle in which one of the angles is a right angle.
- Pythagoras theorem
In a right triangle, the square of the hypotenuse is equal to the sum of the
square of the other two sides.
Properties
- The sum of the interior angles of a triangle is 180 degrees.
- The sum of the exterior angles of a triangle is 360 degrees.
- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles of the triangle.
- Each interior opposite angle always less than that of exterior angle.
- Angles opposite to two equal sides of a triangle are equal.
- Sides opposite to two equal angles of a triangle are equal.
- In an isosceles triangle altitude bisect the base.
- If two sides of a triangle are unequal, then the longer side has greater angle opposite it.
- The greater angle of a triangle has the longer side opposite to it.
- The sum of any two side of a triangle is greater than the third side.
- In a right angle triangle, the hypotenuse is the longest side.
- The sum of the three altitudes of a triangle is less than the sum of the three sides of the triangle.
- The point of concurrency of the bisector of the angles of a triangle is called incenter.
- The point of concurrency of the perpendicular bisector of the sides of a triangle is called circumcenter.
- The point of concurrency of the three altitudes of a triangle is called orthocenter.
- Median is a line segment joining a vertex of a triangle to the midpoint of its opposite side.
- The medians of triangle passes through the same point known as centroid of triangle and it divides each median in the ratio of 2:1.
- Triangles on the same base and between the same parallels are equal in area.
- A median divides a triangle into two triangles of equal area.
- If equilateral triangles are drawn on the sides of a right angle triangle, then the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
- If a perpendicular drawn from the opposite vertex to the hypotenuse of right triangle, the triangles on the opposite sides of the perpendicular are similar to each other.
- The line segment joining the midpoint of the hypotenuse of a right triangle to the vertex of the right angle is equal to half the hypotenuse.
Theorem about Similarity of triangles
Two triangles are said to
be similar if
- If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
- If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
- If three corresponding sides of two triangles are in proportion, then the triangles are similar.
- If the corresponding sides of two triangles are proportional, then the triangles are similar.
- The ratio of area of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
- The ratio of perimeters of similar triangles is equal to the ratio of the square of the corresponding sides/altitudes/angle bisectors and medians.
- Ratio of corresponding altitudes of two similar triangles is equal to the ratio of their corresponding sides.
Congruence
Axiom
1. SAS (Side-Angle-Side) Congruence theorem:
The two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and angle of other triangle.
2. ASA (Angle-Side- Angle) Congruence theorem:
The two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding angles and side of other triangle.
3. SSS (Side-Side-Side) Congruence theorem:
The two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of other triangle.
4. AAA (Angle-Angle- Angle) Congruence theorem:
The two triangles are congruent if three angles of one triangle are equal to the corresponding three angles of other triangle.
5. RHS (Right-Hypotenuse-Side) Congruence theorem:
Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side.
Properties
- The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it.
- If a line is drawn parallel to one side of a triangle the other two sides are divided in the same ratio.
- If two lines intersect, then vertically opposite angles are equal.
- If one of the four angles formed by two intersecting lines is a right angle, then each of the four angles is a right angle.
- If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
- If a transversal intersects two parallel lines, then each pair of alternate and corresponding angles are equal.
- If a transversal intersect two lines in such a way that a pair of consecutive interior angles are supplementary, then two lines are parallel.
- If two lines are perpendicular to the same line, then they are parallel to each other.
- If two parallel lines are interested by a transversal, then bisectors of any two corresponding angles and alternate angles are parallel.
- Two line segments are congruent if and only if their lengths are equal.
- Two line angles are congruent if and only if their measures are equal.
ü
Quadrilateral
- In a regular polygon, all sides and interior angles are equal.
- The sum of the interior angles of a quadrilateral is 360 degrees.
- The exterior angle of a regular polygon is 360 degree/n.
- The interior angle of a regular polygon is (n-2/n) x 180 degree.
- The perimeter of a quadrilateral is greater than the sum of diagonals.
1. Parallelogram
A quadrilateral with each pair of opposite sides parallel.
Properties:
ØThe opposite angles are equal.
Ø
The opposite angles are equal.
Ø
The diagonals bisect each other.
2. Rhombus
A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.
Properties:
Ø
The opposite angles are equal.
Ø
The diagonals perpendicularly bisect each other.
3. Rectangle
A quadrilateral with each pair of opposite sides parallel and equal.
Properties:
Ø
All four angles are right angles.
Ø
The diagonals are of equal length.
Ø
The diagonals bisect each other.
4. Square
A quadrilateral with each pair of opposite sides parallel and all the four sides are equal.
Properties:
Ø
All four angles are right angles.
Ø
The diagonals are of equal length.
Ø
The diagonals perpendicularly bisect each other.
5. Trapezium
A quadrilateral with one pair of the opposite sides is parallel.
Property:
The diagonal of a trapezium cut each other in the same ratio.
6.
Kite
A quadrilateral with exactly two pair of equal consecutive sides.
Property:
Ø
The diagonal are perpendicular to one another.
Ø
One of the diagonal bisects the other.
Ø
In the figure < B = < D But < A not equal to < C
- The quadrilateral formed by joining the mid points of consecutive sides of a quadrilateral is a parallelogram.
- A diagonal of a parallelogram divides it into two triangles of equal area.
- Diagonals of a parallelogram divide it into four triangles of equal area.
- Any straight line parallel to the parallel sides of a trapezium cuts the other two sides proportionally.
