1.4  Angles & Intersecting lines

Angles:

An angle is a figure that can be formed in 2-different ways

• The intersection of two lines will form an angle.  The point of intersection for the lines will be the vertex of the angle.
  
• Two rays with a common endpoint. The common endpoint is the angle’s vertex. The rays are the sides of the angle.

The exterior of an angle is the set of all points outside the angle. The interior of an angle is the set of all points between the sides of an angle.  The interior of an angle will be between 0^o and 180^o.

An angle can be named in several different ways: by its vertex, by a point on each ray and the vertex, or by a number.

Example A

Problem: Given the following figure, identify all of the following:      •  Name of the angles      •  Vertex of the angles      •  Sides of the angles

Solution: We can first identify each angle by it's interior and identify the name of the angle, vertex of the angle, and sides of the angle.

Watch the video:

Example B

Problem: Given the following figure:      a. Name three rays in the diagram.      b. Name three angles in the diagram      c. Could PSQ also be referred to as S?

Solution:

Watch the video:

Protractor Postulate

• The points on a semi-circle can be paired in a one-to-one correspondence with the real numbers such that:
  
  • The smallest degree measure is 0^o and the largest degree measure is 180^o.
    
  • The protractor is always placed as follows to measure the angle.

A protractor is a tool used to measure angles. Unlike segments, angles are measured in degrees. One degree is a unit of angle measure that is equal to of a circle.

Now try this...

Angle Addition Postulate: 

• If AXB has a common side with BXC, , then AXC = AXB + BXC
  
• This also means that BXC = AXC - AXB and AXB = AXC - BXC

Example C

Problem: What are the measures of each angle in this figure?

Solution: Put the protractor on the angle.

Watch the video:

Types of Angles from Intersecting Lines:

When two lines intersect, their intersection will form 2-pairs of angle,  If we look at the rays formed by the intersecting lines we can then identify the angles and classify them according to their angle measure.

If two lines intersect like this:  , then you will have these angles:

• An acute angle measures greater than 0° and less than 90°.
• An obtuse angle measures greater than 90° and less than 180°.

If two lines intersect like this: " Perpendicular Lines ", then you will only have this angle:

• A right angle measures exactly 90°. A box drawn at the vertex of an angle shows that it is a right angle, as shown in the diagram.
  

If two lines intersect like this: " Coinciding Lines ", then you will only have this angle:

• A straight angle measures exactly 180°.

Example D

Problem: Given the following angle, identify the angle measure and type of angle.

Solution: First we need to line up the protractor on the angle. We now need to look at where the ray intersects the protractor to find its measure. It looks like the ray is on 94, remember we always measure from the line that is on the zero line on the bottom of the protractor.           So we have an angle measure of 94°.  Since this is a measure that is between 90° and 180°, we have an obtuse angle..

Watch the video:

Angle Bisectors and Congruence Marks

To bisect a figure is to divide it into two congruent parts. An angle bisector is a ray that divides an angle into two congruent angles. Congruent angles have the same measure. They are marked with arc marks.

NOTE:  We can never use a protractor to determine if an angle has been bisected.  We can only determine if a angle is bisected or even congruent by either construction or with algebra.

Example E

Problem:	Solution:	Watch the video:
The measure of ABC = 44°. bisects ABD. The measure of EBF = 23°. Find the measure of CBE.

Practice Problems

Measure each angle and complete each statement.

A.  The blue has an angle measure of ____________ and is a/an ____________ angle.

B.  The red has an angle measure of ____________ and is a/an ____________  angle.

C.  If a 174° is angle bisected, what will angle measure be of the resulting two angles?

Use the following diagram to classify and find the measure of each angle.

D.  AGC

E.  AGB

F.  CGD

G.  BGD

H.  AGD

Assignment:

For this assignment

• Use the Geogebra App below and do all 10 problems.
      
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