4.5. Third Angle Theorem http://www.ck12.org
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.
Example A
Determine the measure of the missing angles.
From the markings, we know that^6 A∼=Dand^6 E∼=^6 B. Therefore, the Third Angle Theorem tells us that^6 C∼=^6 F.
So,
m^6 A+m^6 B+m^6 C= 180 ◦
m^6 D+m^6 B+m^6 C= 180 ◦
42 ◦+ 83 ◦+m^6 C= 180 ◦
m^6 C= 55 ◦=m^6 FExample B
The Third Angle Theorem states that if two angles in one triangle are congruent to two angles in another triangle,
then the third pair of angles must also congruent. What additional information would you need to know in order to
be able to determine that the triangles are congruent?
In order for the triangles to be congruent, you need some information about the sides. If you know two pairs of
angles are congruent and at least one pair of corresponding sides are congruent, then the triangles will be congruent.
Example C
Determine the measure of all the angles in the triangle: