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Sample Daily Wiki Entry
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Today we reviewed the geometry of the coordinate plane (Analytic Geometry—links to
http://www.sparknotes.com/testprep/books/act/chapter10section5.rhtml)
Warm Up
Determine the length of AB .
1.) A: (5, 10) B: (1, 3)
Solution: We can plot these two points and imagine a third point that could be added to create a
right triangle, with the ends of the legs at the points A and B.
The triangle has a base of 4 (found using the distance traveled along the x axis: 5 – 1 = 4).
The height is 7: 10-3 = 7 (going up along the y-axis).
Applying the Pythagorean Theorem (links to: http://www.cut-the-knot.org/pythagoras/index.
shtml):
 AB 2  4 2  7 2
So: AB  16  49  65
We always need to check if this can be simplified further. To do so, we see if 65 has any factors
that are perfect squares:
 Is 22 a factor of 65? No. 65 / 4 is not a whole number
 Is 32 a factor of 65? No. 65/9 is not a whole number
 Don’t need to check 42, since I know 4 isn’t a factor
 Is 52 a factor of 65? No. 65/25 is not a whole number
 Don’t need to check 6 since I know that 22 and 32 don’t work, so clearly (2x3)2 won’t
work
 Is 72 a factor? No. 65/49 is not a whole number.
 Is 82 a factor? Clearly not, since 22 wasn’t.
 I don’t need to check any higher, since all other perfect squares are larger than 64.
Class Activity
Today we practiced applying properties of alternate and corresponding, interior and exterior
angles.
Note: all of this is in our book on pages 363 and 364.
Before we begin to talk about the angles made, we need to name another geometric figure.
Definition: a transversal is a line that intersects coplanar lines at two distinct points
Many people get confused with all of the names of the angles. For each pair of angles, just ask
yourself two questions:
1.) Are the angles on the same side or opposite sides (alternate)?
2.) Are the angles on the outside of the shape (exterior) or on the inside (interior)?
It is easiest to see some examples of these:
Type
Alternate Interior
Description
 Opposite sides of the transversal
 Inside the horizontal lines
Examples
< 3 & <6
<4 & <5
Alternate Exterior


Opposite sides of the transversal
Outside the horizontal lines
<1 & 8
<2 & <7
Corresponding

Same side and position related
to horizontal line (top-top and
bottom-bottom)
<1 & <5
<3 & <7
<2 & <6
<4 & <8
Same side interior

<3 & 5
<4 & 6

Same side of transversal (leftleft and right-right)
Inside the horizontal lines


Same side
Outside the horizontal lines
<1 & <7
<2 & <8
Same side
(corresponding) exterior
When the horizontal lines are intersecting (We know this just means they do not have the same
slope. We call them intersecting even if they do not actually cross each other in the picture), we
do not know any more information about the angles.
However, when the horizontal lines intersect, we can identify some very interesting properties.
Postulate #5 (Corresponding Angles Postulate, 7-1 in book):
If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
Example:
Because of what we know by the Vertical Angle Theorem, we can complete more of this
diagram:
In this step, we actually have used two theorems that Ms. Bigelow said we can easily prove.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs
of the same side interior angles are congruent.
Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the
pairs of same-side interior angles are supplementary.
I know that we are supposed to prove theorems, but I don’t get the proof and I don’t see why we
have to use them anyway. I figured it out above and I think people get it.
Seeing the proof was actually kind of neat, even if I couldn’t do it myself, because it used what
we already know about supplementary and vertical angles.
If the converse is true, then this is actually a useful way to figure out if two lines are parallel.
When hanging pictures on a wall or building anything, this could come in handy. Perhaps we’ll
also use this idea if we ever have to construct parallel lines? I guess we might also have to use
this when solving for variables that are put in the angle measurements.