5.5 Inequalities in Triangles
Can you figure out the puzzle below???
DOM
Domino
Comparison Property of Inequality
a
b
c
Comparison Property of Inequality:
If a = b + c, and c > 0, then a > b.
Proof of Comparison Property of Inequality
Given: a = b + c, c > 0
Prove: a > b
Statements
Reasons
c>0
b+c>b+0
1.
3.
4.
b+c>b
a=b+c
5.
a>b
5.
1.
2.
3.
2.
4.
Theorem
m1  m2  m3 by the Δ Exterior  Theorem.
If we apply the comparison prop of inequality, what do we know?
Angle Theorem: The measure
 Corollary to the Triangle Exterior
of an exterior angle of a triangle is greater than the measure of each
of its remote interior angles.
m1  m2 and m1  m3
Application
Given the figure below, explain why
Statements
Reasons
1.
m1  m3
1.
2.
m1  m2
2.
3. m2  m3
3.
m2  m3 .
Theorem
Theorem 5-10: If two sides of a triangle are not congruent, then
the larger angle lies opposite the longer side.
If XZ  XY , then mY  mZ .
Theorem
List the angles of the following figure in order from smallest to
largest.
Theorem
Theorem 5-11: If two angles of a triangle are not congruent, then
the longer side lies opposite the larger angle.
If mA  mB, then BC  AC.
Sides of a Triangle
List the sides of the following triangle in order from
shortest to longest.
Determine which segment is shortest in the following diagram.
Theorem
Theorem 5-12: Triangle Inequality Theorem: The sum of the
lengths of any two sides of a triangle is greater than the length of
the third side.
XY  YZ  XZ
YZ  ZX  YX
ZX  XY  ZY
Theorem
Can a triangle have sides with the given lengths?
a) 7 ft, 3 ft, 8 ft
b) 10 cm, 6 cm, 3 cm
A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths
possible for the third side.
Lines Review
Find the equation of a line, in y = mx + b form, that goes through the
points (-5, 4) and (5,6).
Find the equation of the perpendicular bisector of the segment joined
by points (3,2) and (-7, 4) in y = mx + b form.
Lines Review
Find the equations of the lines containing the midsegments of the
triangle with vertices A(-3,2), B(5,6), and C(3,-4) in y = mx + b form.
5.5 Inequalities in Triangles
Can you figure out the puzzle below???
HW: p.292 #1-25 (x3), 32, 33, 37
ESGG
SGEG
GEGS
SGGE
Scrambled Eggs