http://www.ck12.org Chapter 5. Relationships with Triangles
Guidance
Look at the triangle below. The sides of the triangle are given. Can you determine which angle is the largest? As
you might guess, the largest angle will be opposite 18 because it is the longest side. Similarly, the smallest angle
will be opposite the shortest side, 7. Therefore, the angle measure in the middle will be opposite 13.
Theorem:If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger
than the angle opposite the shorter side.
Converse:If one angle in a triangle is larger than another angle in a triangle, then the side opposite the larger angle
will be longer than the side opposite the smaller angle.
Proof of Theorem:
Given:AC>AB
Prove:m^6 ABC>m^6 C
TABLE5.6:
Statement Reason
1.AC>AB Given- Locate pointPsuch thatAB=AP Ruler Postulate
- 4 ABPis an isosceles triangle Definition of an isosceles triangle
4.m^61 =m^63 Base Angles Theorem
5.m^63 =m^62 +m^6 C Exterior Angle Theorem
6.m^61 =m^62 +m^6 C Substitution PoE
7.m^6 ABC=m^61 +m^62 Angle Addition Postulate
8.m^6 ABC=m^62 +m^62 +m^6 C Substitution PoE
9.m^6 ABC>m^6 C Definition of “greater than” (from step 8)
We have two congruent triangles 4 ABCand 4 DEF, marked below: