1.5 Angle Relationships

Adjacent Angles

Two angles that lie in the same plane, have a common vertex and a common side, but no common interior points Examples: NonExamples: B is the common Vertex

BC

is the common side

Vertical Angles

Two nonadjacent angles formed by two intersecting lines Examples: NonExamples: Vertical angles must be formed by a nice neat “X”

Linear Pairs

A pair of adjacent angles whose noncommon sides are opposite rays.

Examples: NonExamples:

ED

&

EC

form a straight line

ED

&

EC

do not form a straight line

Example 1

Complementary Angles

Two angles whose measures have a sum of 90°

Supplementary Angles

Two angles whose measures have a sum of 180° (These angles do not have to be connected)

Example 2

Draw a picture: What do we know?

Complementary means a sum of 90° 

A

 

B

 90 Difference means subtract 

A

Solve one equation for one of the variables: 

A

 

B

 90  2  12 

A

2 

A m

  

A



A

12     72 36  90 90 If

m



A m



B

 36 ,  ??



B

 

A

 12 

B

  

A

 12   A 

A

Substitute into the other equation & solve

m



B

 90 36  54

Perpendicular Lines

Lines that form right angles



Perpendicular lines intersect to form 4 right angles



Perpendicular lines intersect to form congruent adjacent angles



Segments & rays can be perpendicular to lines or to other line segments & rays



The right angle symbol in the figure indicates that the lines are perpendicular

 

is read as “is perpendicular to” (Perpendicular lines don’t form 90˚ angles; they form right angles, and right angles have a measure of 90 ˚) – this is a nit-picky fact that will be used in proofs

Example 3

Look for an equation to write & solve.

12

y

 10  6

x

 3

x

Too many variables; look for something else 6

x

9 

x

3

x

  90

x

 10 90 12

y

 10  12

y

 100 90

If we want the lines to be perpendicular, they have to make right (90˚) angles.

y

 100 12  25  8 .

3 3 Do the solutions work?

≈ means “approximately equal to” because we rounded the decimal.

What can you assume?

Make a list of things you “think” might be true How many did you come up with? Now double check with the chart below. Mark whether each one from your list can be assumed.

Example 4

HW : Page 41 (4– 10 all, 11 – 35 & 39 odds)