For necessary background information on this lesson, please review the latest posts on the It All Adds Up category.
Today’s lesson consists less of writing proofs and more of the logical thinking that is applied through proofs.
First, let’s go through some vocabulary.
Midpoint: the point that divides the segment into two congruent segments.
Just from the word ‘mid’, we can infer that the two segments it creates are congruent. This is especially useful when solving for variables, as you can set the values of each segment equal to each other.
Segment Bisector: a point, ray, line, line segment, or plane that intersects the segment at its midpoint.
Remember: the difference between a midpoint and a segment bisector is that a SB is what actually does the bisecting, whereas the midpoint is the intersection of the line and the SB, as long as it is specifically a bisector, and not just any line.
The logic required to find the midpoint of a segment is easy to understand. You take the x coordinates, add them together, then divide by 2. Then, you do the same for the y.
The Midpoint Formula looks like this, but it appears funky because this is a blog:
m= (x1+x2)/2 , (y1 + y2)/2
When you substitute in for the variables (the numbers stand for which x coordinate is used, given that you will begin with two separate coordinates), you will result in an x and a y value that you can represent as coordinate (x,y).
- Find the midpoint of a segment if the endpoints are (8,3) and (2,1).
- (1.5,7) and (95,4)?
- (|-8|,|0|) and (-8,0)? Hint: the bars represent absolute value.
Alternatively, we can use the Distance Formula to find the overall length of a segment, or the distance between two points. The distance formula looks like this:
d= √[(x2-x1)^2 + (y2-y1)^2]
Once again, just substitute in your variable values to solve.
- Find the distance of a segment if the endpoints are (3,-1) and (9,0).
- (2,4) and (-2,4)?
- Are these two segments congruent?
- Segment A: (2,6) and (0,3)
- Segment B: (-1,0) and (1,3)
In regards to proofs, you won’t specifically use the Midpoint Formula as a reason, but you will use the “Definition of a Midpoint” to prove that two segments are congruent. You won’t use the Distance Formula as a specific reason either, but we’ll get into this more on future proofs.
Introduction to Angles
In preparation for upcoming proofs and topics, though, we need to know how to work with angles in addition to segments. Here’s some vocabulary:
Ray: each side of an angle, begins with a certain endpoint and extends with no end
Angle: 2 different rays with the same endpoint
Classifying angles should be review, but you can classify angles by their measure: acute (<90), right (90), obtuse (>90) and straight (180).
Just like segments, we can apply the theorem that two smaller angles, when added together, form one larger angle. This is called the Angle Addition Postulate, which states that, using <RST, if point P is in the interior (between rays) of <RST, then the measure of
Also, just for clarity, congruent angles are 2 angles with the same measure
Angle Bisector: a ray that divides an angle into two congruent parts.
- If an angle has a measure of 80 degrees, and is made up of two smaller angles, <1 (x+20) and <2 (2x+30), find the value of x.
- If PT is an angle bisector of <SPU, and
All of this will be used to help us with topics in the future, so you’ll want to refer to this information when writing proofs.
The following information should also be familiar:
Complementary Angles: two angles whose sum is 90 degrees
Supplementary Angles: two angles whose sum is 180 degrees
Supplementary and Complementary angles can be any two angles, even if they aren’t adjacent.
Adjacent Angles: two angles that share a common vertex or side, but have no common interior points. This means that their sides are touching, and the angles do not intersect into each other.
Vertical Angles: two angles ONLY whose sides form two pairs of opposite rays. The angles are congruent.
Linear Pair: two adjacent angles whose non common sides are opposite rays (they are supplementary and form a straight line).
(By the way: opposite rays are rays that share the same endpoint but extend in directly opposite directions).
There are several theorems that go along with each classification, and can be used for various purposes.
I hope you’ll find everything we’ve discussed in this post useful for the future. Thanks so much for reading! Stay tuned to the Post Calendar for when the next post releases.