Geometry

Geometry

Lines And Angles

• An angle is formed at the intersection of two line segments, rays or lines. The point of intersection is called the vertex. Angles are measured in degrees.
• Angles are classified according to their degree measures.
• An acute angle measures less than 90^o
• A right angle measures 90^o
• An obtuse angle measures more than 90^o but less than 180^o
• A straight angle measures 180^o
• If two or more angles combine together to form a straight angle, the sum of their measures is 180^o.
  a^o + b^o + c^o + d^o = 180^o
• The sum of all the measures of all the angles around a point is 360^o
  a^o + b^o + c^o + d^o + e^o = 360^o
• When two lines intersect, four angles are formed, two angles in each pair of opposite angles are called vertical angles. Vertical angles, formed by the intersection of two lines, have equal measures.
  a^o = c^o, b^o = d^o
• If one of the angles formed by the intersection of two lines is a right angle, then all four angles are right angles. Such lines are called perpendicular lines
  a^o = b^o = c^o = 90^o
• In the figure below a line l divides the angle in two equal parts. This line is said to bisect the angle. The other line k bisects another line into two equal parts. This line is said to bisect a line.
  
• Two lines are said to be parallel, if they never intersect each other. However, if a third line, called a transversal, intersects a pair of parallel lines, eight angles are formed. And the relationship among theses angles is shown in the following diagram.
  
  • All four acute angles are equal a^o = c^o = e^o = g^o
  • All four obtuse angles are equal b^o = d^o = f^o = h^o
  • The sum of any pair of acute and obtuse angle is 180°, e.g. a^o + d^o =180^o,
    d^o + e^o =180^o, b^o + g^o = 180° etc.

Triangles

• In any triangle, the sum of the measures of the three angles is 180^o.
  x^o + y^o + z^o = 180^o
• In any triangle:
  • The longest side of triangle is opposite the largest angle.
  • The shortest side is opposite the smallest angle.
  • Sides with the same length are opposite the angles with the same measure.
• Triangles are classified into three different kinds with respect to the lengths of sides.
  • Scalene: in which all three sides are of different lengths.
  • Isosceles: in which two of the sides of triangle are equal in length, the third is different.
  • Equilateral: in which all three sides are equal in length.
• Triangles are also classified with respect to the angles.
  • Acute triangle: in which all three angles are acute.
  • Obtuse triangle: in which one angle is obtuse and two are acute.
  • Right triangle: This has one right and two acute angles.
• In a right triangle, the opposite to the right angle is known as hypotenuse and is the longest side. The other two sides are called legs.
• In any right triangle, the sum of the measures of the two acute angles is 90^o.
• By Pythagorean Theorem, the sum of squares of the lengths of legs of a right triangle is always equal to the square of length of hypotenuse.
  a² + b² = c²
• In any triangle, the sum of any two sides is always greater than the third one. And the difference of any two sides is always less than the third one.
  a + b > c and a - b < c
• The perimeter of a triangle is calculated by adding the lengths of all the sides of that triangle.
  perimeter = a + b + c
• The area of a triangle is calculated by the formula: area = (1/2)bh where b is the base of the triangle and h is the height of the triangle.
  • Any side of triangle can be taken as the base.
  • Height is the altitude (perpendicular) drawn to the base from its opposite vertex.
  • In a right triangle any leg could be taken as the base, the other will be the altitude.
    

Quadrilateral and other Polygons

• A polygon is a closed geometric figure, made up of line segments. The line segments are called sides and the end points of lines are called vertices (plural of vertex). Lines, inside the polygon, drawn from one vertex to the other, are called diagonals.
  
• The sum of the measures of the n angles in a polygon with n sides is
  always (n − 2)×180^o.
• In any quadrilateral, the sum of the measures of the four angles is 360^o.
• A regular polygon is a polygon in which all of the sides are of the same length. In any regular polygon, the measure of each interior angle is
  ((n - 2) x 180^o)/n and the measure of each exterior angle is 360^o/n.
• A parallelogram is a special quadrilateral, in which both pairs of opposite sides are parallel. The Following are some properties of parallelogram.
  
  • Lengths of opposite sides are equal. AB = CD and AD = BC
  • Measures of opposi te angles are equal. a^o = c^o and b^o = d^o
  • Consecutive angles add up to 180^o. a^o + b^o =180^o , b^o + c^o =180^o etc.
  • The two diagonals bisect each other. AE = EC and BE = ED
  • A diagonal divides the parallelogram into two triangles that are congruent.
• A rectangle is a parallelogram in which all four angles are right angles. It has all the properties of a parallelogram. In addition it has the following properties:
  • The measure of each angle in a rectangle is 90^o.
  • The diagonals of a rectangle are equal in length.
• A square is a rectangle that has the following additional properties:
  • A square has all its sides equal in length.
  • In a square, diagonals are perpendicular to each other.
    
• To calculate the area, the following formulas are required:
  • For a parallelogram, Area = bh , where b is the base and h is the height.
  • For a rectangle, Area = lw, where l is the length and w is the width.
  • For a square, Area = s² , where s is the side of the square.
• Perimeter for any polygon is the sum of lengths, of all its sides.

Circles

• A circle consists of all the points that are the same distance from one fixed point called the center. That distance is called the radius of a circle. The word radius is also used to represent any of the line segments joining the center and a point on the circle. The plural of radius is radii.
  
• Any triangle, such as CED in the figure, formed by connecting the end points of two radii, is an isosceles.
• A line segment, such as ED in the diagram above, both of whose end points are on a circle is called a chord.
• A chord that passes through the center of the circle is called the diameter of the circle. The length of the diameter is always double the radius of the circle. The diameter is the longest cord that can be drawn in a circle.
• The total length around a circle is known as the circumference of the circle.
• The ratio of the circumference to the diameter is always the same for any circle. This ratio is denoted by the symbol π (pronounced as pi).
• π = C/d ⇒ C=πd ⇒ C= 2πr where C is the circumference, d is the diameter and r is the radius of the circle.
• Value of π is approximately 3.14
• An arc consists of two points in a circle and all the points between them. E.g. PQ is an arc in the diagram.
  
• An angle whose vertex is at the center of the circle is called the central angle. ∠PCQ in the diagram above is a central angle.
• The degree measure of a complete circle is 360^o.
  
• The degree measure of an arc is the measure of the central angle that intercepts it. E.g. the degree measure of PQ is equal to the measure of ∠PCQ in the diagram above.
• If x is the degree measure of an arc, its length can be calculated as (x/360)C where C is the circumference.
• The area of a circle can be calculated as πr².
• The area of a sector formed by the arc and two radii can be calculated as (x/360)A, where A is the area of a circle.

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