Inequalities for Two Triangles

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Inequalities for Two Triangles

• Lesson 6.5
• Pre-AP Geometry

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SAS Inequality Theorem

• If two sides of one triangle are congruent to
  two sides of another triangle, but the included
  angle of the first triangle is larger than the
  included angle of the second, then the third side
  of the first triangle is longer than the third
  side of the second triangle.

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Example 1

• Given ?ABC D is the midpoint of AC m?1 lt m?2.
  
• What can you deduce?

In the two triangles, you have BD ?BD (Reflexive)
as well as AD ?CD (Definition of a Midpoint).
Since m?1 lt m?2, you can apply the SAS
Inequality Theorem to get BC lt AB.
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SSS Inequality Theorem

• If two sides of one triangle are congruent to
  two sides of another triangle, but the third side
  of the first triangle is longer than the third
  side of the second, then the included angle of
  the first triangle is larger that the included
  angle of the second.

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Example 2

• Given BC ? DC AB gt AD
• Prove m?1 gt m?2

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Practice 1

• Given ?ABC and ?DEF AC ? DF AB ?DE BC
  15 EF 14
• What can you deduce? Name the theorem that
  supports your answer.

m?CAB gt m?FDE SSS Ineq. Thrm
Drag the box to the line to see the solution.
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Practice 2

• Given ?JKL and ?NOP JL ? NP LK ? PO m?JLK
  40
• m?NPO 41.
• What can you deduce? Name the theorem that
  supports your answer.

NO gt JK SAS Ineq. Thrm.
Drag the box to the line to see the solution.
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Written Exercises

• Problem Set 6.5, p. 231 2 12 (even)
• Handout 6-5 Inequalities for Two Triangles
• Handout Inequalities for Two Triangles