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G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are
congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints.
Ask ? > Research > Hypothesis > Experiment > Analyze
Question:
What rules and theorems control the measurements of angels when a line
passes through two parallel lines?
Definition
The following terms have standard mathematical definitions next to them.
Student will research the definitions and give a definition of their own based on class
and group discussions. A worksheet will be given to students with following definitions.
Definitions in RED are part of student answers
Parallel lines - coplanar lines that do not intersect. Multiple lines that do not ever touch.
Transversal - Line that intersects 2 coplanar lines. Line that cuts through multiple
parallel lines.
Alternate interior angel - 2 nonadjacent interior angels on opposite sides on transversal
line. Matching angles form a Z shape with cross line in the middle between the parallel
lines.
Alternate external angel - 2 nonadjacent exterior angels on opposite sides of transversal
line. Matching angels on the outside on parallel lines on the opposite sides of cross
line.
Same-side Interior angel - 2 interior angels on the same side of the transversal. Nonmatching angels that will add up to 180.
Corresponding angel - 2 nonadjacent angels on the same side of the transversal - one
is external one is internal. Both angels on one side of transversal and they are equal
form an F.
Vertical Angle - 2 nonadjacent angels formed by a pair of intersecting lines. Opposite on
a line, are equal and form an X
Research
As a class or in small groups students will look at the definitions of these terms
and will come up with a unique paraphrased way of defining the terms. The students
answers may look like the answers in red. Once the students have had time to think
about the definitions we will discuss as a class what examples of these concepts look
like. Draw and example of each term.
Hypothesis
We can figure out the measurement of all of the angels created by a line
intersecting two or more parallel lines if we know the measurement of at least one
angel.
Experiment
Each student will draw two parallel lines on their paper. They will be given a cutout of a pizza slice, all with varying angels. Students will be directed (demonstrate to
class) to use the cut-out to draw a transversal line across their parallel lines. They need
to place on side of the cut-out on one of the parallel lines and use a ruler on the other
side of the cut-out to draw a line through both parallel lines (do not use the crust side of
the pizza). Students will then number each of the angles. Using a protractor students
can measure the angle of the pizza slice cut-out.
Analyze
Based on the measurement of the angel they know (the angle made by the pizza
slice cut-out) students can find the measurements of every angel using the theorems
and definitions discussed earlier.
Assignment
Using the theorems and definitions from earlier record the following conclusions
regarding each angles’ relationship to other angels. Use the terms: alternate internal,
alternate external, same side, corresponding, or vertical.
1⇲ is_______________ to 2⇲
5⇲ is _________________to 8⇲
2⇲ is________________to 3 ⇲
1⇲ is__________________to 5⇲
3⇲ is________________to 4⇲
7⇲ is__________________to 1⇲
4 ⇲ is________________to 5 ⇲
8⇲ is__________________to 2⇲
6⇲ is ________________to 7⇲
4⇲ is__________________ to 6⇲
7⇲ is ________________to 8⇲
3⇲ is__________________to 5⇲
8⇲ is_________________to 4⇲
5⇲ is__________________to 6 ⇲
What are the measurements for the angels?
1⇲
2⇲
3⇲
4⇲
5⇲
6⇲
7⇲
8⇲
What conclusions can we draw about special pairs of angels. Use complete sentences!
Hint:
Alternate internal =
Alternate external =
Same-side + up to 180
Corresponding =
Vertical =