5.6. Comparing Angles and Sides in Triangles http://www.ck12.org
Therefore, ifAB=DEandBC=EFandm^6 B>m^6 E, thenAC>DF. Now, let’s adjustm^6 B>m^6 E. Would that
makeAC>DF? Yes. See the picture below.
The SAS Inequality Theorem (Hinge Theorem):If two sides of a triangle are congruent to two sides of another
triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then
the third side of the first triangle is longer than the third side of the second triangle.
SSS Inequality Theorem (also called the Converse of the Hinge Theorem):If two sides of a 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
triangle, then the included angle of the first triangle is greater in measure than the included angle of the second
triangle.
Example A
List the sides in order, from shortest to longest.
First, we need to findm^6 A. From the Triangle Sum Theorem,m^6 A+ 86 ◦+ 27 ◦= 180 ◦. So,m^6 A= 67 ◦. Therefore,
we can conclude that the longest side is opposite the largest angle. 86◦is the largest angle, soACis the longest side.
The next largest angle is 67◦, soBCwould be the next longest side. 27◦is the smallest angle, soABis the shortest
side. In order from shortest to longest, the answer is:AB,BC,AC.
Example B
List the angles in order, from largest to smallest.
Just like with the sides, the largest angle is opposite the longest side. The longest side isBC, so the largest angle is
(^6) A. Next would be (^6) Band finally (^6) Ais the smallest angle.