Lines, Angles, & Circles

MATH 102Contemporary Math

S. Rook

Overview

• Section 10.1 in the textbook:– Points & lines– Angles– Circles

Points & Lines

Points & Lines

• Point: a dot (similar to a vertex when we discussed graphs)

• Line segment: two points that are connected (similar to an edge)– Finite

• Ray: like a line segment except ONE end extends infinitely

Points & Lines (Example)

• Line: like a line segment except both ends extend infinitely in both directions– Indicated by arrowheads on both ends of the line

• As stated in the book, these terms have no definitions, but a “feel” to them– They are the basic building blocks in Geometry

Angles

Angles

• Angle: describes the “space” between two rays that are joined at a common endpoint– Recall that a ray has one terminating side and one

non-terminating side

• Can also think about an angle as a rotation about the common endpoint– Start at OA (Initial side)– End at OB (Terminal side)

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Degree Measure

• Degree measure: expresses the size of an angle. Often abbreviated by the symbol °

360° makes one complete revolution• The initial and terminal sides of the angle are the same

180° makes one half of a complete revolution90° makes one quarter of a complete revolution

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Degree Measure (Continued)• Angles that measure:– Between 0° and 90° are known as acute angles– Exactly 90° are known as right angles

• Denoted by a small square between the initial and terminal sides– Between 90° and 180° are known as obtuse angles– Exactly 180° are known as straight angles

• Complementary angles: two angles whose measures sum to 90°

• Supplementary angles: two angles whose measures sum to 180°

• Vertical angles are opposite the same vertex and have the SAME measure

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Angles (Example)

Ex 1: Sketch:

a) An angle that is: i) acute ii) right iii) obtuse iv) straight

b) A pair of angles that are: i) complementary ii) supplementary iii) vertical

Parallel Lines & Transversals

• Consider two parallel lines• Now consider drawing a third

line that intersects the two parallel lines– This third line is called a

transversal and the angles it creates have special & useful properties

– e.g. Lines l and m are parallel and line n is a transversal

Parallel Lines & Transversals (Continued)

• Corresponding angles have the same measure– Given that angles 1 & 5 are corres- ponding

angles, what is another pair of corresponding angles?

• Alternate interior and alternate exterior angles have the same measure– Alternate angles lie on different sides of the transversal– Interior angles lie in between the parallel lines

• What is a pair of alternate interior angles?– Exterior angles lie outside the parallel lines

• What is a pair of alternate exterior angles?

Parallel Lines & Traversals (Example)

Ex 2: Find the measures of angles a, b, and c in each figure given that lines m and n are parallel:

a) b)

c)

Circles

Circles

• Circle: set of all points that are equidistant from a given point called the center

• Radius: a line segment that connects the center to any point on the outside of the circle– The distance of the radius is the SAME no matter which

point on the outside of the circle is used to connect to the center of the circle

• Circumference: the distance around the circle– i.e. “Perimeter”

Circles (Continued)

• An angle having its vertex at the center of a circle is called a central angle

• A central angle intersects two points on the outside of the circle– e.g. Angle ACB– The curved segment between these two points and

inside the central angle is known as an arc• e.g. Arc AB

Finding the Length of an Arc

• We can find the length of an arc using a part to whole relationship:

– The measure of a central angle is a part of the whole measure of a circle (which is by definition?)

– The length of an arc is part of the whole length around a circle (which is called what by definition?)

part measure of central angle length of arc

whole measure of a complete circle length of the complete circle

part measure of central angle length of arc

whole 360 circumference

Circle Measurements (Example)

Ex 3: Solve, given a circle with center C and points A & B both located on the outside of the circle:

a) Find the length of arc AB if the circumference is 24 feet and the measure of angle ACB is 90°

b) Find the circumference if the measure of angle ACB is 30° and arc AB is 100 mm

Summary• After studying these slides, you should know how to do the

following:– Have an intuitive feel for a point, line segment, line, and ray– Name special types of angles based on their degree measure– Find the measure of angles given two parallel lines and at least one

transversal– Define circle terminology– Relate the length of an arc of a circle to its circumference and measure of

its central angle

• Additional Practice:– See problems in Section 10.1

• Next Lesson:– Polygons (Section 10.2)