1. Points, Lines, Planes and Angles

2. Points, Lines and Planes These basic concepts of geometry are theoretical and cannot be precisely defined. They do not exist in the physical world. ConceptDimension Representation A point is a location in space. It has no dimensions, no length, no width, and no thickness. Points A, B, and M A line is a set of connected points. It is assumed to be straight and is determined by two specific points. It has one dimension, an infinite length, but no width or thickness. Line a and line PQ A plane is an infinite large flat surface. It has two dimensions, an infinite length and infinite width, but no thickness. Plane

3. A line segment is a portion of a line between two points called endpoints. A line segment is named using its endpoints. Line segment AB A ray is half a line and its endpoint. It is a part of a line that has only one endpoint and goes on forever in one direction. A ray is named by using the endpoint and some other point on the ray. The endpoint is always mentioned first. Ray PQ

4. Some Basic Concepts of Angles P is said to be the interior of the AOB whereas Q is exterior to the angle. An angle is the union of two rays with a common endpoint. The rays are called sides and the common point is called the vertex. An angle can be named in different ways:

5. Forming an angle by rotation: OB is called the initial side and OA is called the terminal side. Given OA, OB on the same line and extended in the same direction. If OA is rotated counterclockwise about O, we say that BOA is an angle of rotation generated by OA. When the terminal side rotates until it falls on the initial side, We say that it has made one complete revolution.

6. Equal angles are angles formed by the same amount of rotation, and they are also called congruent angles. Angles formed by a counterclockwise rotation are called positive and those formed by a clockwise rotation are called negative angles. Units of measurement: - degree – 1/360 of a complete revolution - minute – 1/60 of a degree - second – 1/60 of a minute The notation for the measure of angle is

7. Classification of Angles Acute Angle has a measure between 0  and 90 . Right Angle has a measure of 90 . A right angle is angle formed by a quarter of a revolution. Obtuse Angle has a measure between 90  and 180 . Straight Angle has a measure of 180 . A straight angle is an angle formed by one half of a revolution. The sides of a straight angle extended in opposite direction, forming a straight line.

8. Reflex Angle has a measure between 180  and 360 . Adjacent Angles two angles with same vertex and a common side between them. Complementary Angles two angles whose some is a right angle. Supplementary Angles two angles whose some is a straight angle. Vertical Angles two nonadjacent angles, each less than a straight angle, formed by the same measure. Vertical angles have the same measure. Angles a,b and c,d are vertical angles.

9. Names of Angles Pairs Formed by a Transversal Intersecting Parallel Lines Parallel lines never meet. The distance between them is always the same. Intersecting lines cross at exactly one point. Perpendicular lines are intersecting lines that form right angle. A line that intersects two other lines at different points is called transversal.

10. Names of Angles Pairs Formed By Transversal Intersecting Parallel Lines NameDescriptionSketchAngle Pair Described Property Alternate interior angles Interior angles that do not have a common vertex on alternate sides of transversal Alternate interior angles have the same measure. Alternate exterior angles Exterior angles that do not have a common vertex on alternate sides of transversal Alternate exterior angles have the same measure. Corresponding angles One interior and one exterior angle on the same side of the transversal Corresponding angles have the same measure.

11. Identify each pair of complementary angles. Exercises Identify each pair of supplementary angles.

12. Identify the angles that are congruent and their measure. Exercises Find the measure of all angles in each figure.

13. Exercises Given m A=(3x+15)  and m B=(2x-5)  find x if a) the two angles are complementary 3x+15+2x-5=90 b) the two angles supplementary 3x+15+2x-5=180 x=18  x=34 

14. Self Check Identify each pair of complementary angles. Identify each pair of supplementary angles. Given m A=100  and m B=(3x-7) . Find x if A and B are supplementary angles. Find the measure of 1 if line l is parallel to line p.