Identify a midsegment of a triangle and use it to solve problems. Analyze the relationship between the angles of a triangle and the lengths of the sides Determine allowable lengths for sides of triangles

1. Obj. 23 Triangle Theorems
The student is able to (I can):
• Identify a midsegment of a triangle and use it to solve
problems.
• Analyze the relationship between the angles of a triangle
and the lengths of the sides
• Determine allowable lengths for sides of triangles

2. midsegment
A segment that joins the midpoints of two
sides of a triangle.
O
Points I, C, and E are
midpoints of ∆HOT.
IC, CE, and EI
are midsegments.
H
I
C
E
T

3. Triangle Midsegment Theorem
A midsegment of a triangle is parallel to
a side of the triangle, and its length is
half the length of that side.
O
1
IC HT, IC = HT
2
H
I
C
E
T

4. Examples
Find each measure.
1. FE
FE = 2(LT) = 2(14)
= 28
U
L
14
9
F
2. m∠UFE
m∠UFE = m∠TSE
= 62º
T
S
62º
E
LT and TS
3. UE
UE = 2(9) = 18
are midsegments.

5. Thm 5-5-1
If two sides of a triangle are not congruent,
then the larger angle is opposite the longer
side.
AT > AC → m∠C > m∠T
Thm 5-5-2
If two angles of a triangle are not
congruent, then the longer side is opposite
the larger angle.
m∠C > m∠T → AT > AC
C
A
T

6. Triangle Inequality Theorem
The sum of any two side lengths of a
triangle is greater than the third side
length.
Example:
1. Which set of lengths forms a triangle?
4, 5, 10
7, 9, 12
4 + 5 < 10
7 + 9 > 12

7. Note: To find a range of possible third
sides given two sides, subtract for
the lower bound and add for the
upper bound.
Examples:
2. What is a possible third side for a
triangle with sides 8 and 14?
14 — 8 = 6 lower bound
14 + 8 = 22 upper bound
The third side can be between 6 and 22.

8. 3. What is the range of values for the
third side of a triangle with sides 11 and
19?
19 — 11 = 8
19 + 11 = 30
8 < x < 30
lower bound
upper bound