5.6. Comparing Angles and Sides in Triangles http://www.ck12.org
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:
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.
Example C
List the sides in order, from least to greatest.
Let’s start with 4 DCE. The missing angle is 55◦. Therefore the sides, in order areCE,CD, andDE. For 4 BCD,
the missing angle is 43◦. The order of the sides isBD,CD, andBC. By the SAS Inequality Theorem, we know that
BC>DE, so the order of all the sides would be:BD=CE,CD,DE,BC.
Watch this video for help with the Examples above.