HONORS GEOMETRY
5.6. Inequalities in Two Triangles
Do Now:
• If the base of an isosceles triangle is 6 inches, what is the
range of potential leg lengths?
• What is the range of possible perimeters for figure
ABCDE if AC = 7 and DC = 9?
Homework
• Questions?
• Comments?
• Confusions?
• Concerns!?
• ASK ASK ASK!
Hinge Theorem
• If two sides of a triangle are congruent to two sides of
another triangle, and the included angle of the first
triangle is larger than the included angle of the second
triangle, then the third side of the first triangle is longer
than the third side of the second triangle.
Meaning, if we know HA = KC and KD = HB
and <H is smaller than <K…..
Conclusion?
AB is smaller than CD
Converse Hinge Theorem
• If two sides of a triangle are congruent to two sides of
another triangle, and the third side in the first is longer
than the third side in the second triangle, then the
included angle measure of the first triangle is greater
than the included angle measure of the second triangle.
Meaning….. If HA = KC and KD = HB
and CD is larger than AB…..
Conclusion?
<K is larger than <H
So….
• Regular is going from an angle to a side
• CONVERSE is going from a side to the angle
Example One:
Which length is larger? WX or XY? What theorem justifies
this?
Example Two:
• Which angle is larger? Angle 1 or Angle 2? Which
theorem justifies this?
You Try!
• Which angle is larger? 𝑚 < 𝐶𝐴𝐵 or 𝑚 < 𝐷𝐴𝐶? Which
theorem justifies this?
• Which side is larger? AD or CD? Which theorem justifies
this?
Example Three:
• Two groups of snowmobilers leave from the same base
camp. Group A goes 7.5 miles due west and then turns 35
degrees north of west and goes 5 miles. Group B goes
7.5 miles due east and then turns 40 degrees north of
east and goes 5 miles. At this point, which group is farther
from the base camp?
You Try!
• Two groups of skiers leave from the same lodge. Group A
goes 4 miles due east and then turns 70 degrees north of
east and goes 3 miles. Group B goes 4 miles due west
and then turns 75 degrees north of west and goes 3
miles. At this exact moment, which group is farther from
the lodge? Explain your reasoning.
Example Four:
• Determine the range of all possible values for x
Example Five:
• Determine the range of all possible values for x
You Try!
• Determine the range of all possible values for k
Example Six:
• Given: 𝐾𝐿 ≅ 𝑁𝐿
• Prove: 𝐾𝑀 > 𝑁𝑀
Example Seven:
• Given: C is the midpoint of 𝐵𝐷
𝑚<1=𝑚<2
𝑚>3>𝑚>4
• Prove: 𝐴𝐵 > 𝐸𝐷
You Try!
• Given: 𝑚 < 𝑆𝑅𝑇 ≅ 𝑚 < 𝑆𝑇𝑅
𝑇𝑈 > 𝑅𝑈
• Prove: 𝑚 < 𝑇𝑆𝑈 > 𝑚 < 𝑅𝑆𝑈
Practice Problems
• Try some on your own/in your table groups
• As always don’t hesitate to ask me questions if you are
confused :D
Exit Ticket:
• What is the range of possible values for x?