Measuring Angles

• An angle is formed by two rays with the same endpoint.

• The rays are the sides of the angle.• The endpoint is the vertex of the angle.

• The sides of the angle shown here are and . The vertex is B.

B

1

T Q

How to Name• There are three ways to name an angle1. One point only – vertex2. Three points – one point from each side of

the angle and the vertex listed in the middle3. Number – not a degree value, just a number

within the angle

• You could name this angle∠B, TBQ, QBT, or 1∠ ∠ ∠

B

1

T Q

Measuring Angles• We measure the size of an angle using degrees.• A degree results when a circle is divided into 360

equal parts.• Here are some examples of angles and their degree

measurements.

Protractor• A device used to measure angles

• When we use a protractor, we need to line it up correctly.

• Is this ready to measure the angle?

• It was not correct!• Look for the upside down ‘T’ in the middle of

the straight line on your protractor.• This needs to be exactly on the vertex of your

angle.

• It doesn’t matter which way the angle is, you ALWAYS need to line the upside down ‘T’ to the vertex of the angle.

• Now you are ready to read the measurement

• Read from the 0ᴼ and follow the inner set of numbers.

• Once you reach 30ᴼ you need to be careful!

• You then need to look at the 1ᴼ markings on the outer set of numbers

• The angle measures 35ᴼ

Postulate 1-7• The Protractor Postulate• Let and be opposite rays in a plane. , , and all

the rays with endpoint O that can be drawn on one side of can be paired with the real numbers from 0 to 180 so thata) is paired with 0 and is paired with 180b) If is paired with x and is paired with y, then

m COD = |x-y|∠

Classifying Angles

Acute Angles• An acute angle is an angle measuring between

0 and 90 degrees. • The following angles are all acute angles.

Right Angles• A right angle is an angle measuring 90 degrees. • The following angles are both right angles.

• Note the special symbol in the corner. When you see it, you know the measure is 90ᴼ

Obtuse Angles• An obtuse angle is an angle measuring

between 90 and 180 degrees. • The following angles are all obtuse.

Straight Angle

• A straight angle is 180 degrees.

Practice

Practice

Practice

Practice

• Angles with the same measure are congruent angles.

• If m 1 = m 2, then 1 2∠ ∠ ∠ ∠• Angles can be marked alike to show that they

are congruent

Special Angle Pairs• Vertical angles

• Adjacent angles

• Complementary angles

• Supplementary angles

Vertical Angles• Two angles whose sides are opposite rays

• ∠1 and 3 are vertical angles∠• ∠2 and 4 are vertical angles∠

Adjacent Angles• Two coplanar angles with a common side, a

common vertex, and no common interior points

Complementary Angles• Two angles whose measures have a sum of 90• Each angle is called the complement of the

other

• Do not have to be adjacent angles

Supplementary Angles• Two angles whose measures have a sum of

180• Each angle is called the supplement of the

other

Find the missing angle1. Two angles are complementary. One

measures 65 degrees. What is the other?

2. Two angles are supplementary. One measures 140 degrees. What is the other?

3. Find the missing angle

4. Find the missing angle x55

165x

Identifying Angle Pairs

• Identify all complementary angles

• Identify all supplementary angles

• Identify all vertical angles

Postulate 1-8• The Angle Addition Postulate

You CAN conclude that angles are• Adjacent angles• Adjacent supplementary angles• Vertical angles

You CANNOT assume• Angles or segments are congruent• An angle is a right angle• Lines are parallel or perpendicular

Making ConclusionsFrom A Diagram

• What can you conclude from the information in the diagram?

• ∠1 2 by the markings≅∠• ∠2 and 3 are adjacent angles∠• ∠4 and 5 are adjacent supplementary ∠

angles or m 4 + m 5 = 180 by the Angle ∠ ∠Addition Postulate

• ∠1 and 4 are vertical angles∠