Cutting Tool

Introduction • Geometry – This word comes from the Greek and means measurement of the earth (geo = earth, metry = measurement). • In order to answer basic questions about the earth and its relationship to the sun, the moon, and the planets some fundamental concepts of Geometry were developed in ancient Greece.

• Geometers of the past had to be able to visualize the earth and the heavenly bodies in space. • Some of the ancient methods are still in use today to solve problems in the modern world such as in construction, road building, and even in medicine.

Geometry Standard • analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships; • specify locations and describe spatial relationships using co-ordinate geometry and other representational systems;

• apply transformations and use symmetry to analyze mathematical situations; • use visualization, spatial reasoning, and geometric modeling to solve problems.

WHY GEOMETRY? Reasons• To promote the ability to visualize and mentally manipulate objects in space. This is skill used in a number professions (e.g., surgeon or dentist a carpenter, architect, clothes designer) • Foundational skill in many jobs to see things in the mind’s eye.

Visualization Point Line Ray Plane Endpoints Line Segment

POINT SEGEMENT LINE

RAY

• Line Segment are formed by joining two points (in the shortest possible way or is the part of a line lying between two points on that line. These two points are called endpoints. • A Ray is the part of a line lying on one side of a point on the line.

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1.1 exercises and activities

1.2 Angles • An angle is the union of two line segments with a common endpoint called a vertex. • Used to represent an amount of rotation (turning) about a fixed point in counterclockwise direction.

• Suppose there are two rays with a common endpoint. The two rays and the region between them is called the angle at a point P formed by the two rays. • The smallest amount of counterclockwise rotation about P needed to rotate one of the rays to the position of the other ray.

Are the two angles the same?

Measurement of Angle • Degrees – indicated with a little circle: º . For example 90º. A full circle (to come back where you started) is 360º. Half turn is 180º. • Clockwise turns have a negative measurement.

ACUTE ANGLE

Right Angle

OBTUSE ANGLE

• Right Angle – If the angle formed by the two rays is 90º. • When two lines in a plane meet, they form four angles. • When all four of these angles made by two intersecting lines are 90º, the lines are called perpendicular lines. • Two lines in a plane that do not intersect are called parallel lines.

• Normal Line at a point on surface is a line that passes through that point and is perpendicular to the surface at that point. • Physical Principle of reflection – – Incoming light and reflected light make the same angle with the normal line at the point where the incoming light ray hits the surface

– Normal Light ray lies in the same plane as the normal line and the incoming light ray. – The reflected ray and incoming light ray coincide only when incoming light ray lines up with the normal line

Light Rays Reflecting Off Surfaces Normal Line Incomming light ray

reflected light ray'

60 degrees 60 degrees Reflective Surface