4.5. Third Angle Theorem http://www.ck12.org
4.5 Third Angle Theorem
Here you’ll learn the Third Angle Theorem and how to use it to determine information about two triangles with two
pairs of angles that are congruent.
What if you were given 4 F GHand 4 XY Zand you were told that
(^6) F∼= (^6) X
and
(^6) G∼= (^6) Y
? What conclusion could you draw about^6 Hand^6 Z? After completing this Concept, you’ll be able to make such a
conclusion.
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Findm^6 Candm^6 J.
The sum of the angles in each triangle is 180◦by the Triangle Sum Theorem. So, for 4 ABC, 35 ◦+ 88 ◦+m^6 C= 180 ◦
andm^6 C= 57 ◦. For 4 HIJ, 35◦+ 88 ◦+m^6 J= 180 ◦andm^6 Jis also 57◦.
Notice that we were given thatm^6 A=m^6 Handm^6 B=m^6 Iand we found out thatm^6 C=m^6 J. This can be
generalized into the Third Angle Theorem.
Third Angle Theorem:If two angles in one triangle are congruent to two angles in another triangle, then the third
pair of angles must also congruent.
In other words, for triangles 4 ABCand 4 DEF, if^6 A∼=^6 Dand^6 B∼=^6 E, then^6 C∼=^6 F.
Notice that this theorem does not state that the triangles are congruent. That is because if two sets of angles are
congruent, the sides could be different lengths. See the picture below.