Inequalities Involving Two Triangles

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Lesson 5-5

• Inequalities Involving Two Triangles

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Transparency 5-5
5-Minute Check on Lesson 5-4
Determine whether the given measures can be the
lengths of the sides of a triangle. Write yes or
no. 1. 5, 7, 8 2. 4.2, 4.2, 8.4 3. 3, 6,
10 Find the range for the measure of the third
side of a triangle given the measures of two
sides. 4. 4 and 13 5. 8.3 and 15.6 6.
Which cannot
be the measure of the third side of a triangle if
two sides measure 8 and 17?
Standardized Test Practice
22
C
D
19
26
A
B
15
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Transparency 5-5
5-Minute Check on Lesson 5-4
Determine whether the given measures can be the
lengths of the sides of a triangle. Write yes or
no. 1. 5, 7, 8 yes 2. 4.2, 4.2, 8.4 no 3.
3, 6, 10 no Find the range for the measure of
the third side of a triangle given the measures
of two sides. 4. 4 and 13 9 lt n lt 17 5. 8.3
and 15.6 7.3 lt n lt 23.9 6.
Which cannot be the measure
of the third side of a triangle if two sides
measure 8 and 17?
Standardized Test Practice
22
C
D
19
26
A
B
15
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Objectives

• Apply the SAS Inequality
• Apply the SSS Inequality

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Vocabulary

• Included side the side in common between two
  angles (the end points are the vertexes)
• Included angle the angle formed by two sides
  sharing a common end point (or vertex)

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Theorems

• Theorem 5.13, SAS Inequality, or Hinge Theorem
  If two sides of a triangle are congruent to two
  sides of another triangle and the included angle
  in one triangle has a greater measure than the
  included angle in the other, then the third side
  of the first triangle is longer that the third
  side of the second triangle.
• Theorem 5.14, SSS Inequality If two sides of a
  triangle are congruent to two sides of another
  triangle and the third side in one triangle is
  longer than the third side in the other, then the
  angle between the pair of congruent sides in the
  first triangle is greater than the corresponding
  angle in the second triangle.

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Non Congruent Triangles
Theorem 5.13 SAS Inequality or Hinge Theorem
AB ? XY and AC ? XZ m?A gt m?X, then BC gt YZ
C
Z
A
X
B
Y
Theorem 5.14 SSS Inequality AB ? XY and AC ?
XZ BC gt YZ, then m?A gt m?X
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Write an inequality comparing m?LDM and m?MDN
using the information in the figure.
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Write an inequality finding the range of values
containing a using the information in the figure.
SSS Inequality
Substitution
Subtract 15 from each side.
Divide each side by 9.
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Also, recall that the measure of any angle is
always greater than 0.
Subtract 15 from each side.
Divide each side by 9.
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Write an inequality using the information in the
figure. a. Compare m?WYX and m?ZYW b. Find the
range of values containing n.
Answer 6 lt n lt 25
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HEALTH Doctors use a straight-leg-raising test
to determine the amount of pain felt in a
persons back. The patient lies flat on the
examining table, and the doctor raises each leg
until the patient experiences pain in the back
area. Nitan can tolerate the doctor raising his
right leg 35 and his left leg 65 from the
table. Which foot can Nitan raise higher above
the table?
Assume both of Nitans legs have the same
measurement, the SAS Inequality tells us that the
height of the left foot opposite the 65 angle is
higher than the height of his right foot opposite
the 35 angle. This means that his left foot is
raised higher.
Answer his left foot
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HEALTH Doctors use a straight-leg-raising test to
determine the amount of pain felt in a persons
back. The patient lies flat on the examining
table, and the doctor raises each leg until the
patient experiences pain in the back area. Megan
can lift her right foot 18 inches from the table
and her left foot 13 inches from the table. Which
leg makes the greater angle with the table?
Answer her right leg
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Summary Homework

• Summary
• SAS Inequality In two triangles, if two sides
  are congruent, then the measure of the included
  angle determines which triangle has the longer
  third side.
• SSS Inequality In two triangles, if two sides
  are congruent, then the length of the third side
  determines which triangle has the included angle
  with the greatest measure.
• Homework pg 270 273 3-6, 13-16, 32