Download skew, trans, and special angles

Learning Objectives
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Identify parallel lines, skew lines and perpendicular lines.
Parallel Lines and Planes
Parallel lines are coplanar (they lie in the same plane) and never intersect.
Below is an example of two parallel lines:
Parallel lines never ______________________________ each other.
Coplanar lines are in the same __________________________.
We use the symbol
!###" !##"
! for parallel, so we describe the figure above by writing MN $ CD .
When we draw a pair of parallel lines, we use an arrow mark ( > ) on the lines to show that the lines are
parallel. Just like with congruent segments, if there are two (or more) pairs of parallel lines, we use one
arrow ( > ) for one pair and two (or more) arrows ( ≫ ) for the other pair.
There are two types of symbols to show that lines are parallel:
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In a geometric statement,!##the
" symbol _____________ is put in between two lines
(“line XY ” is written as XY ) to give the notation for parallel lines.
In a picture, we draw the symbol _____________ on both lines to show that the
lines are parallel to each other.
Reading Check:
What symbols let you know that the lines below are parallel?
Fill in the blanks to make a symbolic statement that the two lines are parallel.
_____ _____
Perpendicular Lines
Perpendicular lines intersect at a
in the
90! right angle. This intersection is usually shown by a small square box
90! angle.
Perpendicular lines meet at a ________________ angle.
The symbol
is used to show that two lines, segments, or rays are perpendicular. In the picture above,
!##"
!!!"
!!!" #!!$
we could write BA ! BC . (Notice that BA is a ray while BC is a line.)
Note that although "parallel" and "perpendicular" are defined in terms of lines, the same definitions
apply to rays and segments with the minor adjustment that two segments or rays are parallel (or
perpendicular) if the lines that contain the segments or rays are parallel (or perpendicular).
If you think about a table, the top of the table and the floor below it are usually in parallel planes.
Skew Lines
The other of relationship you need to understand is skew lines. Skew lines are lines that are non-coplanar
(they do not lie in the same plane) and never intersect.
Skew lines are in different _____________________ and never ______________________.
Segments and rays can also be skew. In the cube below, segment
AB and segment CG are skew:
Reading Check:
In the picture to the right...
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Put arrows on two line segments to show they are parallel
Put a small square box at the intersection of two perpendicular segments
Circle two line segments that are skew.
Learning Objectives
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Identify angles made by transversals: corresponding, alternate interior, alternate
exterior and same-side/consecutive interior angles.
Angles and Transversals
Many geometry problems involve the intersection of three or more lines. Examine the diagram below:
In the diagram, lines g and h are crossed by line l. We have quite a bit of vocabulary to describe this
situation:
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Line l is called a transversal because it intersects two other lines (g and h). The
intersection of line l with g and h forms eight angles as shown.
The area between lines g and h is called the interior of the two lines. The area not
between lines g and h is called the exterior.
Angles !1 and !2 are a linear pair of angles. We say they are adjacent because
they are next to each other along a straight line: they share a side and do not
overlap.
o There are many linear pairs of angles in this diagram. Some examples are
!2 and !3, !6 and !7 , and !8 and !1 .
A ________________________________ is a line that cuts across two or more lines.
The area in between lines g and h is called the __________________________ of the lines.
The area above line g and below line h is called the _______________________ of the lines.
The word _______________________________ means “next to.”
A linear _____________________ is a set of two angles that are adjacent to each other along a straight
line.
The prefix “trans” means “across.”
A transversal cuts across two or more lines.
Other words that have the prefix “trans” are transport (to carry across) and transmit (to send across).
Can you think of some other words that begin with the prefix “trans”?
Reading Check:
What do you remember about linear pairs?
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They are adjacent, which means they are right next to each other and they share a
side.
They are supplementary, which means they add up to 180! .
Supplementary angles sum to ____________________.
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!1 and !3 in the diagram are vertical angles. They are nonadjacent angles
(angles that are notnext to each other) made by the intersection of two lines.
o Other pairs of vertical angles in the diagram on the previous page are
!2 and !8, !4 and !6 and !5 and !7 .
Pairs of ________________________________ angles are across from each other at an intersection of two
lines.
Reading Check:
What do you remember about vertical angles from?
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Vertical angles get their name because they have the same vertex.
You learned that vertical angles are congruent. In other words, vertical angles
have the same measure.
Corresponding angles are in the same position relative to both lines crossed by the transversal. !1 is on
the upper left corner of the intersection of lines g and l. !7 is on the upper left corner of the intersection
of lines h and l. So we say that !1 and !7 are corresponding angles.
Corresponding angles are in the ________________________ position at each intersection.
!3 and !5 are _________________________________ angles because they are both in the bottom right
corner of their intersections.
The word “corresponding” means “matching” or “similar.”
For example, Mexico City corresponds with Washington, D.C., because they are both the capitals of their
countries.
Reading Check:
Fill in the blanks with the angle that corresponds to each of the following angles in the diagram above.
1. Angle 1 and angle ______.
2. Angle 8 and angle ______.
3. Angle 2 and angle ______.
4. Angle 5 and angle ______.
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!3 and !7 are called alternate interior angles. They are in the interior region
of the lines g and h and are on opposite sides of the transversal.
Alternate __________________________ angles are inside lines g and h and on opposite sides of line l, the
transversal.
!4 and !8 are another example of __________________________ interior angles.
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Similarly, !2 and !6 are alternate exterior angles because they are on opposite
sides of the transversal, and in the exterior of the region between g and h.
Alternate exterior angles are on opposite sides of the _____________________________.
!1 and !5 are another example of ____________________ ___________________ angles.
If you alternate between two things, you switch between them.
Do you see how alternate angles switch sides?
One angle is to the right of the transversal and the other angle is to the left of the transversal.
To _______________________________ means “to switch.”
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Finally, !3 and !4 are consecutive interior angles. These are also known as
same-side interior angles. They are on the interior of the region between lines g
and h and are on the same side of the transversal. !8 and !7 are also
consecutive interior angles.
Consecutive interior angles are the same thing as _______________ - ________________ interior angles.
Reading Check:
Look at the picture of the lines being cut by a transversal below. Then, circle the type of angle pair
represented by the angles given.
1.
!1and!5 corresponding
alternate interior
alternate exterior
same-side interior
2.
!2and!4 corresponding
alternate interior
alternate exterior
same-side interior
3.
!3and!5 corresponding
alternate interior
alternate exterior
same-side interior
4.
!3and!4 corresponding
alternate interior
alternate exterior
same-side interior