This document discusses various geometry concepts related to angles and lines including: - Types of angles such as acute, obtuse, straight, and reflex angles - Relationships between pairs of angles such as complementary, supplementary, vertical, corresponding, alternate interior, and alternate exterior angles - Properties of lines such as intersecting lines, parallel lines, and transversals - Angle theorems about parallel lines cut by a transversal and the angle sums of triangles

1. Class IX project of MATHS

2. A line is a collection of points along a straight path. Aline has
no endpoints.
A line segment is a part of a line with two end points.
a part of line with one end point is called a ray.
All the points that lie on the same line are collinear points.
Consider the two rays AB and AC originating from the same
point A.
The union of two rays AB and AC is called an angle.
The rays that form the angle are called the arms of the angle.
The intersection point is called the vertex of the angle.
The size of an angle is measured in degrees
An angle that measures less than 90 but more than 0 is
called an acute angle.
A right angle is an angle measuring ninety degrees, formed
by the intersection of two perpendicular lines.
Angles greater than 90 degrees but less than 180
degrees are known as obtuse angles.
An angle that is equal to 180 degrees is called a straight
angle.

3. A reflex angle is greater than 180
degrees but less than 360 degrees.
Two angles are said to be adjacent if they
have a common arm and a common vertex.
Linear pair of angles: Two adjacent
angles are said to form a linear pair if
their sum is 1800.
Vertically opposite angles: When two lines
intersect four angles are formed. The angles
that are opposite to each other are
called vertically opposite angles.
Two angles are said to be complementary, if
their sum is 90 degrees.
Two angles are supplementary if their sum is
180 degrees.
They may or may not be adjacent angles.
Intersecting lines can be defined as two or
more lines that meet at one point.
Parallel lines can be defined as lines on the
same plane that never intersect.

4. Intersecting lines
• Definition of Intersecting Lines
• Lines that have one and only one point in common are
known as intersecting lines.
• More about Intersecting Lines
• A minimum of two lines are required for intersection.
• The common point where all the intersecting lines
meet is called the Point of Intersection.
• All the intersecting lines form angles at the point of
intersection.
• Related Terms for Intersecting Lines
• Common
• Line
• Point

5. • Pairs of Angles
• In geometry, certain pairs of angles can have special
relationships. Using our knowledge of acute, right,
and obtuse angles, along with properties of parallel
lines, we will begin to study the relations between
pairs of angles.
• Complementary Angles
• Two angles are complementary angles if their
degree measurements add up to 90°. That is, if we
attach both angles and fit them side by side (by
putting the vertices and one side on top of each
other), they will form a right angle. We can also say
that one of the angles is the complement of the
other.

6. Complementary angles

7. Supplementary Angles
Another special pair of angles is called
supplementary angles. One angle is
said to be the supplement of the
other if the sum of their degree
measurements is 180°. In other
words, if we put the angles side by
side, the result would be a straight
line.

9. Vertical Angles
Vertical angles are the angles
opposite of each other at the
intersection of two lines. They are
called vertical angles because they
share a common vertex. Vertical
angles always have equal
measures.

11. Alternate Interior Angles
Alternate interior angles are formed
when there exists a transversal. They
are the angles on opposite sides of
the transversal, but inside the two
lines the transversal intersects.
Alternate interior angles are
congruent to each other if (and only
if) the two lines intersected by the
transversal are parallel.

13. Alternate Exterior Angles
Similar to alternate interior angles,
alternate exterior angles are also
congruent to each other if (and only if)
the two lines intersected by the
transversal are parallel. These angles are
on opposite sides of the transversal, but
outside the two lines the transversal
intersects.

15. • Corresponding Angles
• Corresponding angles are the pairs of angles
on the same side of the transversal and on
corresponding sides of the two other lines.
These angles are equal in degree measure
when the two lines intersected by the
transversal are parallel.
• It may help to draw the letter "F" (forwards
and backwards) in order to help identify
corresponding angles. This method is
illustrated below.

17. In geometry, a transversal is a line that
passes through two lines in the
same plane at two distinct points.

18. Angle sum property of a triangle
• Triangle Angle Sum Theorem
• The sum of the measures of the interior
angles of a triangle is 180.

20. • Properties
• We will utilize the following properties to help us
reason through several geometric proofs.
• Reflexive Property
• A quantity is equal to itself.
• Symmetric Property
• If A = B, then B = A.
• Transitive Property
• If A = B and B = C, then A = C.
• Addition Property of Equality
• If A = B, then A + C = B + C.

21. • Angle Postulates
• Angle Addition Postulate
• If a point lies on the interior of an angle, that angle is
the sum of two smaller angles with legs that go
through the given point.
• Consider the figure below in which point T lies on the
interior of ?QRS. By this postulate, we have that ?QRS
= ?QRT + ?TRS. We have actually applied this postulate
when we practiced finding the complements and
supplements of angles in the previous section.

22. • Corresponding Angles Postulate
• If a transversal intersects two parallel lines, the
pairs of corresponding angles are congruent.
• The figure above yields four pairs of
corresponding angles.

23. • Parallel Postulate
• Given a line and a point not on that line, there exists a unique line
through the point parallel to the given line.
• The parallel postulate is what sets Euclidean geometry apart
from non-Euclidean geometry.
• There are an infinite number of lines that pass through point E, but
only the red line runs parallel to line CD. Any other line
through E will eventually intersect line CD.
• There are an infinite number of lines that pass through point E, but
only the red line runs parallel to line CD. Any other line
through E will eventually intersect line CD.

24. • Angle Theorems
• Alternate Exterior Angles Theorem
• If a transversal intersects two parallel lines,
then the alternate exterior angles are
congruent.
• The alternate exterior angles have the same
degree measures because the lines are parallel
to each other.

25. • Alternate Interior Angles Theorem
• If a transversal intersects two parallel lines,
then the alternate interior angles are
congruent.
• The alternate interior angles have the same
degree measures because the lines are parallel
to each other.

26. • Congruent Complements Theorem
• If two angles are complements of the same angle (or of
congruent angles), then the two angles are congruent.
• Congruent Supplements Theorem
• If two angles are supplements of the same angle (or of
congruent angles), then the two angles are congruent.
• Right Angles Theorem
• All right angles are congruent.

27. • Same-Side Interior Angles Theorem
• If a transversal intersects two parallel lines,
then the interior angles on the same side of
the transversal are supplementary.
• The sum of the degree measures of the same-
side interior angles is 180°.

28. • Vertical Angles Theorem
• If two angles are vertical angles, then they
have equal measures.