10.1 lines, angles, circles - mathematics & statisticsmath.utoledo.edu/~dgajews/1180_old/10-1...

10.1 Lines, Angles, Circles

Classical Geometry is the study of points, lines, angles, circles, etc and the geometric figures built out of them.

The ideas and definitions we use today go back to the mathematician Euclid from Alexandria in 300BC in his book Euclid's Elements.

A point is “that which has no part”.

A line has “length but no breadth.”

A plane has “length and breadth only.”

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 6

Points, Lines, and PlanesEuclid: a point is “that which has no part,” a line has “length but no breadth,” and a plane has “length and breadth only.”A point on a line divides the line into three parts—the point and two half lines. A ray is a half line with its endpoint included.A piece of a line joining two points and including the points is called a line segment.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 7

Points, Lines, and Planes

Parallel lines lie on the same plane and have no points in common.

Intersecting lines lie on the same plane and have a single point in common.

Pop Quiz!!!

The type of object is

1) A ray

2) A line

3) A line segment

4) None of the above

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 8

Angles

Two rays having a common endpoint form an angle.

We measure angles in units called degrees. The symbol ° represents the word degrees.

Let's start with an angle where the initial side and terminal side are the same (i.e. go all the way around). What is the measure of that angle?

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 10

Angles

An angle whose measure is between 0° and 90° is called an acute angle.

A right angle has a measure of 90°.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 11

Angles

An obtuse angle has a measure between 90° and 180°.

A straight angle has a measure of 180°.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 12

Angles

Two intersecting lines form two pairs of angles called vertical angles.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 13

Angles

A pair of angles is complementary if the sum of their measures is 90°.

Two angles having an angle sum of 180° are supplementary angles.

Let's see how to prove this.

Pop Quiz!!!

The type of angle is

1) Acute2) Obtuse3) Right4) Straight5) Vertical6) Complementary7) Supplementary8) None of the above

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 17

Angles

Two lines that intersect forming right angles are called perpendicular lines.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 18

Angles●If we intersect a pair of parallel lines with a third line, called a transversal, we form eight angles.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 16

Angles

Angles

• Example: If lines l and m are parallel, find the measure of the other angles.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 18

Angles

• Solution:

(corresponding angles)

(straight angle)

• Example: If lines l and m are parallel, find the measure of angle 9.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 19

Angles

• Solution:

(same side interior angles)

• Example: If lines l and m are parallel, find the measure of angle 2.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 20

Circles

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 21

Circles

An angle that has its vertex at the center of a circle is called a central angle.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 22

Circles

• Example: A circle has a circumference of 12 meters. If central angle ACB has measure of 120°, then what is the length of the arc from A to B?

(continued on next slide)

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 23

Circles

• Solution:

Pop Quiz!!!

The type of object is

1) The center

2) The diameter

3) The radius

4) A central angle

5) None of the above

The same Eratosthenes that found the prime number sieve also was the first person to prove that the earth was round.

He accurately determined the circumference of the earth.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 24

Circles• Example: Use elementary geometry to estimate the circumference of Earth.

(continued on next slide)

• Solution: Assume that lines l and m are parallel and cut by the transversal t. The point C is the center of the circle. Therefore, angles α and β are equal.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 25

Circles

(continued on next slide)

To measure the circumference of Earth, place a vertical pole in the ground and wait until noon when the rays of the Sun and the pole form an angle of 0°.

Suppose at that very moment, a friend 1,000 miles away also has a similar vertical pole, and the Sun’s rays make an angle of 15° with his pole.

© 2010 Pearson Education, Inc. All rights reserved. Section 10.1, Slide 26

Circles