Brief Explanation of Sections 5.4-5.6
Section 5.4 – Midsegment Theorem
The midsegment of a triangle is a segment you get by connecting the midpoints of two of the sides of
the triangle. (see ex. 1: M is the midpoint of JL and N is the midpoint of KL. MN is the midsegment)
The Midsegment Theorem:
The midsegment is parallel to the 3rd side, and half the length of the 3rd side (see top of pg. 288
and ex. 2).
Section 5.5 – Inequalities in One Triangle
Theorems 5.10 & 5.11 (see pg. 295)
The shortest side of a triangle is across from the smallest angle. The longest side is across from
the biggest angle. Conversely, the smallest angle is across from the shortest side and the largest
angle is across from the longest side (see ex. 1 on pg. 295).
See Pg. 297
In order to be able to make a triangle out of three given length segments, you have to check and
make sure that the two shortest sides add up to more than, not equal to, MORE THAN the 3rd
side.
Ex. 3 on pg. 297
a. Cannot make a triangle because 2+2 is not more than 5
b. Cannot make a triangle because 2+3 is not more than 5
c. Can make a triangle because 2+4 is more than 5
Triangle Inequality (Theorem 5.13 on pg. 297)
Basically to use this, it works like this: if you are given two lengths to use to make a triangle and
want to figure out what the possible lengths of the 3rd side are, take the two you are given,
subtract them to get the shortest possible length and add them to get the longest possible
length (See ex. 4).
You are given side lengths of 10 cm and 14 cm. The shortest the 3rd side could be is just over 4
cm (14-10) and the longest the 3rd side could be is just under 24 (14+10). It is usually written like
this: 4 < 3rd side < 24 or 4 < x < 24. It cannot be exactly 4 or 24 because then the two
shortest sides will not add up to more than the 3rd side.
Brief Explanation of Sections 5.4-5.6
Section 5.6 – Indirect Proof and Inequalities in Two Triangles
The Hinge Theorem (Thm. 5.14) and the Converse of the Hinge Theorem (Thm. 5.15) on Pg. 303
Basically, these are the same as theorems 5.10 and 5.11 from section 5.5, but they are used to
compare lengths in two different triangles, instead of just one. The triangle with the bigger
angle has a longer 3rd side; the triangle with the smaller angle has a shorter 3rd side. Conversely,
the triangle with the longer 3rd side has a bigger angle and the triangle with the shorter 3rd side
has a smaller angle (see pg. 305 #3-#5).
3. m<1 > m<2 because 27 > 26
4. KL < NQ because 45˚ < 47˚
5. DC < FE because 37˚ < 38˚
Indirect Proof
An indirect proof is a proof where you try to draw a reasonable conclusion by assuming the
opposite and working until you reach a point where it is impossible. Often these are used when
you are asked to prove that something cannot happen. So you assume the opposite (that this
thing can happen) and work toward the point of impossibility. This is called a contradiction.
There are three steps to an indirect proof:
1. Assume the opposite of what you are asked to prove
2. Work toward a contradiction
3. Explain what the contradiction tells you (that you were wrong, and the original assumption
was correct)
See Ex. 1 on Pg. 302
Your Assignment:
 Section 5.4: Pg. 290-93 12-18, 26-29
 Section 5.5: Pg. 298-301 6-25
 Section 5.6: Pg. 305-8 7-24, 28, 29 (for 28 & 29 use ex. 4 for help)