A kosarad üres
The relation between the sides and the angles of the triangle
A triangle can clearly be defined by
- its three sides;
- two of its sides and the included angle;
- one of its sides and the two angles on it;
- two of its sides and the angle opposite the longer side.
We are going to use this knowledge henceforth.
\latex{ \beta}
\latex{ \gamma}
Figure 32
\latex{ B }
\latex{ F }
\latex{ C }
\latex{ A }
Let us consider an isosceles triangle, and let point F be the mid-point of base BC (Figure 32). The corresponding sides of triangles ABF and AFC are equal in length, thus based on (1) \latex{\angle}ABF = \latex{\angle}FCA, which means that the angles on the base of triangle ABC are equal in size.
So we can say the following:
THEOREM: In a triangle there are equal angles opposite equal sides.
From this theorem all interior angles of a equilateral triangle are \latex{60^{\circ}}.
It can be proven that the converse of the previous theorem is also true.
THEOREM: If two angles of a triangle are equal, then the sides opposite them are also equal.
Let us assume that in triangle \latex{ ABC\, AC\gt AB }.
What can we say about the interior angles opposite sides \latex{ AC } and \latex{ AB ?}
Let point \latex{ D } be the point of side \latex{ AC } for which \latex{ AB = AD } (Figure 33). Then in triangle \latex{ ABD\, AB = AD }, from which \latex{\angle ABD= \angle BDA = \beta _{1}}.
Based on the theorem about the exterior angles of a triangle,
\latex{\beta = \beta _{1} +\beta _{2} \gt \beta _{1} = \beta _{2} + \gamma \gt \gamma }.
As a result if \latex{ AC\gt AB }, then \latex{\beta} > \latex{\gamma}. With this we have proven the following:
\latex{ \beta_{2}}
\latex{ \gamma}
\latex{ \beta_{1}}
\latex{ \beta_{1}}
Figure 33
\latex{ B }
\latex{ D }
\latex{ C }
\latex{ A }
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.
It can similarly be proven that the converse of this theorem is also true:
THEOREM: In any triangle, the side opposite the larger interior angle is longer than the side opposite the smaller interior angle.
