Today we will learn the basics of Lines and Angles in Geometry.
Line Segment: The straight path between two points P and Q is called a line segment.
Ray: A ray extends indefinitely in one direction. This is exhibited by ab arrow i.e.
Line: In geometry, the word "line" refers to a straight line that extends without end in both directions.
The part of the line from P to Q is called a line segment. P and Q are the endpoints of the segment. The notation PQ is used to denote both the segment and the length of the segment. The intention of the notation can be determined from the context.
Collinear points: Three or more points are colinear if a single straight line passes through them.
Non-Collinear points: Three or more points not lying on a single line are called non-collinear points.
Intersecting lines: Two lines having a common point are called intersecting lines. The common point is the point of intersection.
Concurrent Lines: Three or more lines intersecting at the same point are said to be concurrent.
Parallel Lines: If there are two lines on the same plane and they do not intersect when produced on the either side, they are called to be parallel.
Angles: A figure consisting of two rays with the same initial points is called an angle.
Vertical Angles:If two lines intersect, the opposite angles are called vertical angles and have the same measure.
∠PRQ and ∠ SRT are vertical angles and ∠QRS and ∠PRT are vertical angles. Also, x + y = 180 since PRS is a straight line.
Reflex Angles: An angle greater than 180°, but less than 360° is called a reflex angle.
Complementary angles: Two angles whose sum is 90° are called complementary angles.
Supplementary angles: Two angles having a sum of 180° are called supplementary angles
When two parallel lines are intersected by a third line then,
1. When two lines intersect, two pairs of vertically opposite angles are equal. As per the above figure: 1 and 4, 2 and 3, 5 and 8, 6 and 7 are equal.
2. The sum of 2 adjacent angles is 180°. As per the above figure: 1°+2°=180°. Similarly for: 3 and 4 , 5 and 6 , 7 and 8
3. Pairs of the corresponding angles are always equal. As per the above figure: 1 and 5, 2 and 6, 3 and 7, 4 and 8 are equal respectively.
4. Pairs of alternate interior angle are equal. As per the figure: Angles 3° and 6° are equal and 4° and 5° are also equal as they are alternate interior angles.
Conditions for two lines to be paralel:
- They are parallel to a 3rd
- They are opposite sides of a rectangle/square/rhombus/parallelogram.
- If they are perpendicular to a 3rd
- If one of them is a side of the triangle & other joins the midpoints of the remaining two sides.
- If one of them is a side of a triangle & other divides other 2 sides proportionately
Conditions for two lines to be perpendicular:
- They are arms of a right-angle triangle.
- If the adjacent angles formed by them are equal and supplementary.
- They are adjacent sides of a rectangle or a square.
- If they are diagonals of a rhombus.
- If one of them is a tangent & other is the radius of the circle through the point of contact.
- If the sum of their squares is equal to the square of the line joining their ends.