Lab 3 Part 2: Similar triangles and parallels
The goal of this section is to make a connection between similar triangles and (1) parallel lines and (2) the equation of a line.
Step 1: Download and use the sketch lab03b.gsp
On the first page of this sketch are these directions:
Explain why the points must be collinear.
Step 2: Finding the intersection of two lines
On page 2 of the sketch is constructed a line AB and a line AC and also a line through B perpendicular to line AC.
Now we construct the line m perpendicular to line AB through C.
Move C to a position so that angle BAC is acute.
Here is the giant logical step:
For any line AC where angle BAC is acute, the line AC intersects line m at the point F constructed above.
Then we can say this. For any line n through A, the line n must intersect m unless the line n is perpendicular to line AB.
The reasoning:
If the line is not perpendicular, one of the angles it makes with line AB must be acute, so placing the point C on the line as above, the line n intersects m at F.
Big conclusion: (Euclidean Parallel Property)
Given line m and point A not on m, there is exactly one line through A that is parallel to m. This line is perpendicular to the line AB through A perpendicular to m.
Think this through. Do you see it? If not, ask questions.
More about parallels and angles
In a new sketch, construct two parallel lines m and n and a line p that intersects both of them. Do some measuring. What angles are always equal? What angles are supplementary?
By constructing another line perpendicular to both m and n, find two similar right triangles and use them to explain why the equal angles that you found are in fact equal.
Reverse the process. Read BB Theorem 14 and see why this is true.
Big Connection: Equation of a line and slope
Look over your work above. If the line AB is the x-axis on graph paper, can you see the slope of the line AC in your work? Think about this.