An email from a student

"I was just using your instructions to construct an angle bisector. How do you know that it really cuts the angle in half?"

Dear Student,
There are probably many ways to prove that the angle bisector that you have created is actually bisecting the angle. I can think of two very simple ways. Bisecting means to divide into two equal parts, so if you were to measure from the bisector to one of the angle's rays with your compass, and measured the other side in the exact same location they would have the same measure. Also, if you were to measure the angles of each "half" of your angle the angle measurements would be the same. Thank you for asking this great question!
Sincerly,
Madeleine G.

How to...construct a parallel line to a given line through a point off of the line segment

This construction combines all of the constructions you have learned to construct parallel lines.
There are many ways to construct parallel lines, however this is my favourite and easiest way to do it.

Step 1
Extend a ray from one of the endpoints on you line segment to make an acute angle. Make sure that the ray intersects the given point.





Step 2
Make an arc from the starting point to the given point. Keep this measure. Using this measure, measure from the given point(which is on the ray) and make an arc. Make a point where the arc intersects the ray. Label this point.






Step 3
Measure from the given point to where the first arc that you made intersects the given line segment. Using this measure, go from the last point you just made and make an arc. Make a point where the arc you just made intersects the other arc. Label this point.



Step 4
Connect the point that you just made to the original given point. Label the line.

It doesn't look like the lines are parallel in the picture, but that is only because the surface i was taking the picture on was not flat. That is why it appears to bulge and then get smaller in the bottom left hand corner.

How to...construct a perpendicular line to a line segment with a point that is on the line segment

Constructing a perpendicular line segment with a point that is on the line segment it less complicated than constructing a perpendicular line segment with a point that is not on the line.

Step 1
First, make a circle of any size using the point on your line segment as the center point. Make and label points where the circle intersects the line segment.




Step 2
Next, you measure a little bit more than 1 and a half of your circle's radius from one of the points you just made, and make a very large arc, almost a circle. Now, using this same measure, do the same thing at the other point that you just made.
Make and label points where the two arcs intersect. Connect these last two points that you just made to form a line segment. label the midpoint.

**Don't forget to label all points, including the midpoint**

How to...Construct a perpendicular line segment with a point off of the line segment

Constructing a perpendicular line segment with a point off of a line segment is a little bit more complicated than all of the other constructions you have done. It involves most of the constructions that you have already learned, though.

Step 1
First, measure from the point off of the line segment to a little bit more than that distance, and create an arc. Make a points where this arc intersects the line segment. Label these points.

Step 2
Next, measure from one of the points that you just created a little bit more than half way and make an arc. Now, using this same measurement, go to the other point that you just created and make another arc. make and label a point where these two arcs intersect.

Sorry that i had to change blogs; i lost my password to the other one.