The figure formed by a non-self-intersecting closed broken line together with the part of the plane bounded by this line is called a polygon. The sides and vertices of this broken line are called respectively sides and vertices of the polygon, and the angles formed by each two adjacent sides (interior) angles of the polygon. The smallest number of sides in a polygon is three. Polygons are named according to the number of their sides: triangles, quadrilaterals, pentagons, hexagons, and so on.
TRIANGLE
The Triangle is a geometrical figure that is contained by three straight lines. Every triangle has three sides and three angles. It is the most basic as well as the most important shape. Triangle is a prime figure, prime figure is one which cannot be divided into any other figures more simple then itself. Triangle serves as a basis to understand all other geometrical figures and their properties. The reason being that the triangle is the only rigid figure among all polygons as there is no scope for “wiggling” unlike all other higher polygons, a triangle can be completely defined using the least number of parameters.
Classification of Triangles
Triangles are generally classified according to type of sides and angles.
- Triangle classification according to type of sides:
- Equilateral: The triangles which have all sides of equal in length.
- Isosceles: The triangles which have two of the sides of equal in length.
- Scalene: The triangles which have all sides of different lengths.
- Acute: A triangle is said to be acute when all of its angles are acute angles.
- Obtuse: A triangle is said to be obtuse when one of its angles is an obtuse angle.
- Right: A triangle is said to be right when one of its angles is a right angle.
- SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.
- SSS (Side-Side-Side): Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other.
- ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of the one are equal respectively to two angles and the included side of the other. ASA automatically implies AAS or SAA congruence. Hence the theorem can be rephrased as, If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle.
- RHS (Right-angle-Hypotenuse-Side): Two right triangles are congruent if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other.
- Two points each equidistant from the extremities of a line determine the perpendicular bisector of the line.
- The locus of a point equidistant from two given intersecting lines is a pair of lines bisecting the angles formed by those lines.
- The bisectors of the angles of a triangle are concurrent in a point equidistant from the sides of the triangle.
- The perpendiculars from the vertices of a triangle to the opposite sides are concurrent.
- The medians of a triangle are concurrent in a point two thirds of the distance from each vertex to the middle of the opposite side.
- If two triangles have two angles of the one equal to two angles of the other, the third angles are equal.
- In a triangle there can be but one right angle or one obtuse angle.
- The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
- If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater.
- If two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater.
- If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
- If two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first triangle greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second.
- In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.
- In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.
- Triangles which are on the same base and in the same parallels equal one another.
- Triangles which are on equal bases and in the same parallels equal one another.
- Equal triangles which are on the same base and on the same side are also in the same parallels.
- Equal triangles which are on equal bases and on the same side are also in the same parallels.
- Convex quadrilateral: The diagonals intersect in the interior region.
- Concave quadrilateral: One of the diagonal is in the exterior region.
- In any parallelogram, opposite sides are congruent, opposite angles are congruent, and the sum of angles adjacent to one side is two right angles. Corollary: If one of the angles of a parallelogram is right, then the other three are also right.
- Conversely, If in a convex quadrilateral, opposite sides are congruent to each other, or two opposite sides are congruent and parallel, then this quadrilateral is a parallelogram.
- If a quadrilateral is a parallelogram, then its diagonals bisect each other. Vice versa, in a quadrilateral, if the diagonals bisect each other, then this quadrilateral is a parallelogram.
- Two angles whose sides are parallel each to each are either equal or supplementary. Corollary: The opposite angles of a parallelogram are equal, and any two consecutive angles are supplementary.
- The opposite sides of a parallelogram are equal.
- Corollary:
- Segments of parallel lines cut off by parallel lines are equal.
- Two parallel lines are everywhere equally distant from each other.
- Corollary:
- The diagonals of a parallelogram bisect each other.
- Two parallelograms are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.
- Corollary: Two rectangles having equal bases and equal altitudes are congruent.
- In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.
- Parallelograms which are on the same base and in the same parallels equal one another.
- Parallelograms which are on equal bases and in the same parallels equal one another.
- If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.
- Each of the four angles of a rectangle is a right angle.
- The diagonals of a rectangle are of equal length
- Square
- The diagonals of a square are equal and perpendicular to each other.
- If the diagonals of a parallelogram are equal and intersect at right angles, then the parallelogram is a square.
- Trapezium
- A trapezoid is isosceles if and only if the base angles are congruent.
- A trapezoid is isosceles if and only if the diagonals are congruent.
- If a trapezoid is isosceles, the opposite angles are supplementary.
- The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
- If a quadrilateral is a kite, then its diagonals are perpendicular.
- If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
- The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.
- If a line is parallel to one side of a triangle and bisects another side, it bisects the third side also.
- The line which joins the mid-points of two sides of a triangle is parallel to the third side, and is equal to half the third side.
- The line joining the mid-points of the nonparallel sides of a trapezoid is parallel to the bases and is equal to half the sum of the bases.