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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 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.
- B) Triangle classification according to type of angles:
- 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.
Geometrical objects are said to be congruent when they have the same shape and size. ‘Congruent’ can be understood as the geometrical equivalent of the common term ‘equal’. two geometric figures are called congruent if they can be identified with each other by superimposing.
Congruent and non-congruent segments
Two segments are congruent if they can be laid one onto the other so that their endpoints coincide.
Congruent and non-congruent angles
In accordance with the general definition of congruent figures two angles are considered congruent if by moving one of them it is possible to identify it with the other.
Congruence of Triangles
The necessary and sufficient conditions to decide congruence in case of triangles are as follows:
- 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.
Important lines in a triangle
One of a triangle’s sides is often referred to as the base, in which case the opposite vertex is called the vertex of the triangle, and the other two sides are called lateral. Any triangle has three altitudes, three medians, and three bisectors, since each side of the triangle can take on the role of the base.
The path of a point that moves in accordance with certain given geometric conditions is called the locus of the point.
The perpendicular dropped from the vertex to the base or to its continuation is called an altitude.
A line that bisects a given line and is perpendicular to it is called the perpendicular bisector of the line. In other words, the locus of a point equidistant from the extremities of a given line is the perpendicular bisector of that line.
- 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.
A line from any vertex of a triangle to the midpoint of the opposite side is called a median of the triangle.
If two or more lines pass through the same point, they are called concurrent 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.
Angle Sum Property
The sum of the three angles of a triangle is equal to two right angles.
- 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.
Triangle Inequality Theorems
- 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.
Exterior Angle Theorems
- 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.
Triangle and Parallel lines Theorems
- 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.
In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Conversely, If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.
A quadrilateral (or quadrangle or quadragon) is a rectilinear figure bound by four straight lines called edges or sides.The word ‘quad’ means four and the word lateral means sides. Quadrilateral also has four angles defined by the points where the edges meet (called vertex).
Various Parts of Quadrilateral
Diagonals : It is formed by the segment joined by the opposite vertices.
Adjacent angles : Two angles of a quadrilateral are called adjacent angles, if they have a common side as an arm.
Opposite angles : Two angles of a quadrilateral are called opposite angles which are not adjacent angles.
On the basis of diagonals, there are two types of quadrilaterals.
- Convex quadrilateral: The diagonals intersect in the interior region.
- Concave quadrilateral: One of the diagonal is in the exterior region.
We will only be studying the various types of convex quadrilaterals in the following sections.
Classification of Quadrilaterals
There are many types of quadrilaterals. They are organized in different classes according to certain characteristics. The most common classes of quadrilaterals are: Parallelogram, Trapezium and Kite (or deltoid).
A quadrilateral whose opposite sides are pairwise parallel is called a parallelogram. Such a quadrilateral is obtained by intersecting any two parallel lines with two other parallel lines.
- 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.
2. 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.
3. 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.
4. 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.
5. The opposite sides of a parallelogram are equal.
a. Segments of parallel lines cut off by parallel lines are equal.
b. Two parallel lines are everywhere equally distant from each other.
6. The diagonals of a parallelogram bisect each other.
7. 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.
- Special Parallelogram
When all the angles contained in the parallelogram are right angles it is called a rectangle.
- Each of the four angles of a rectangle is a right angle.
- The diagonals of a rectangle are of equal length
If all the sides containing the parallelogram are equal to each other it is called a rhombus.
The diagonals of a rhombus are perpendicular to each other. Conversely, If the diagonals of a parallelogram are perpendicular, then it is a rhombus.
A square is a unique parallelogram which has all sides equal as well as all angles as right angles. It can be considered a special case of rectangle as well as rhombus.
- 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.
When none of the sides containing the quadrilateral are parallel it is called a trapezium. When two sides of the quadrilateral are mutually parallel it is called a trapezoid. If the non parallel sides are equal in that case the trapezoid is called isosceles. If it is composed of two right angles then it is referred to as right angled trapezoid. In modern usage, the term trapezium is often used to imply trapezoid.
- 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.
4. Kite (Deltoid)
A quadrilateral is a kite,if it has two pairs of equal adjacent sides and unequal opposite sides.
- 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.
Polygons in General
A portion of a plane bounded by a broken line is called a polygon. The terms sides, perimeter, angles, vertices, and diagonals are employed in the usual sense in connection with polygons in general.
Polygons classified as to Sides
A polygon is a triangle, if it has three sides; a quadrilateral, if it has four sides; a pentagon, if it has five sides; a hexagon, if it has six sides. These names are sufficient for most cases. The next few names in order are heptagon, octagon, nonagon, decagon, undecagon, dodecagon. A polygon is equilateral, if all of its sides are equal.
Polygons classified as to Angles
A polygon is equiangular, if all of its angles are equal; convex, if each of its angles is less than a straight angle; concave, if it has an angle greater than a straight angle.
A polygon that is both equiangular and equilateral is called a regular polygon.
Relation of Two Polygons
Two polygons are mutually equiangular, if the angles of the one are equal to the angles of the other respectively, taken in the same order; mutually equilateral, if the sides of the one are equal to the sides of the other respectively, taken in the same order; congruent, if mutually equiangular and mutually equilateral, since they then can be made to coincide.
- 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.
a. The sum of the angles of a quadrilateral equals four right angles; and if the angles are all equal, each is a right angle.
b. Each angle of a regular polygon of n sides is equal to (n – 2)/n right angles.
2 The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is equal to four right angles.
Generalized Mid-Point/Intercept Theorem
If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal.
- 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.
A quadrilateral is a rectilinear figure bound by four straight lines called edges or sides. Quadrilateral also has four angles defined by the points where the edges meet (called vertex).
Kinds of quadrilaterals:
Trapezium: When none of the sides containing the quadrilateral are parallel.
Trapezoid: When two sides of the quadrilateral are mutually parallel it is called a trapezoid. If the non parallel sides are equal in that case the trapezoid is called isosceles.
Parallelogram: If the pair of opposite sides are parallel, such quadrilateral is called a parallelogram. Kinds of parallelogram:
Rectangle: When all the angles contained in the parallelogram are right angles it is called a rectangle.
Rhombus: If all the sides containing the parallelogram are equal to each other it is called a rhombus.
Square: A square is a unique parallelogram which has all sides equal as well as all angles as right angles. It can be considered a special case of rectangle as well as rhombus.
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A circle is a closed curve lying in a plane such that all of its points are at equal distance from a fixed point in the plane (called the center of the circle).
Radius: A straight line from the center to the circle.
Diameter: A straight line through the center, terminating at each end of the circle.
Arc: A portion of the circle is called an arc. An arc equal to half of the circle is called a semicircle. An arc less than the semicircle is called a minor arc while an arc greater than the semicircle is called a major arc.
Chords, Secants and Tangents
Secant: A line intersecting a circle in two places is referred to as a secant.
Chord: The portion of secant that is contained within the circle is called a chord.
Tangent: If a line intersects (touches) a circle at only one single point, it is called a tangent.
An inscribed polygon is a polygon in which all vertices lie on a circle. The polygon is inscribed in the circle and the circle is circumscribed about the polygon. It is a polygon in a circle.
A circumscribed polygon is a polygon in which each side is a tangent to a circle.The circle is inscribed in the polygon and the polygon is circumscribed about the circle. It is a polygon outside the circle or in other words the circle is inside a polygon.
Mensuration is the part of geometry that is concerned with ascertaining lengths, areas, and volumes. Mensuration concerns with the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.
Perimeter of polygons
Perimeter can be defined as the total length of the boundary of a geometrical figure.
Area of polygons
Area is the measure of the surface contained inside a geometrical figure. It is generally measured against the unit square, or say, how much surface in comparison with a square of unit side length.