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Congruence of figures
DEFINITION: Two figures are congruent if there is a series of congruent transformations which transforms one of the figures to the other one.
The notation for figure \latex{ A } being congruent to figure \latex{ B } is as follows: \latex{A\cong B}.
Congruence is a relation, and two figures are in relation with each other (are congruent) if there is a suitable congruent transformation between them. Not only the special transformations discussed earlier but also the execution of them finitely many times in succession gives a congruent transformation in the plane.
The following properties of the congruence as a relation directly result from the definition (we set aside their proofs now):
- Every figure is congruent to itself, i.e. \latex{A\cong A}.
- For any figures \latex{A, B,} if \latex{A\cong B}, then \latex{B\cong A}.
- For any figures \latex{A,B,C,} if \latex{A\cong B} and \latex{B\cong C,} then \latex{A\cong C}.
If we want to prove the congruence of two figures based on the definition, then we have to find a suitable congruent transformation which transforms one of the figures to the other one. In several particular cases this is not easy, therefore we try to find such necessary and sufficient conditions with the help of data characterising the figures (length of certain line segments, measure of angles) which ensure the congruence of the considered figures. For example two circles are congruent if their radii are equal.
Henceforth we give conditions for triangles and polygons which are necessary and sufficient for two figures being congruent.
The basic cases of the congruence of triangles
About triangles \latex{ ABC } and \latex{ A’B’C’ } in Figure 89 we know that \latex{ AB = A’B’ }, \latex{ AC = A’C’ } and \latex{CAB\sphericalangle=C’A’B’\sphericalangle}. Is the equality of this data enough for the two triangles being congruent?
In Figure 89 it can be seen that with the help of the translation by and then of the rotation about \latex{\overrightarrow{AA’} }through a suitable angle \latex{\phi} we
Figure 89
\latex{\phi}
\latex{\phi}
\latex{ A }
\latex{ C }
\latex{ B }
\latex{ C' }
\latex{ B' }
\latex{ A' }
transformed triangle \latex{ ABC } to triangle \latex{ A’B’C’ }, so based on the definition the two triangles are congruent.
Note: This time the orientation of the two triangles was the same. If the orientations had been different, then by reflecting one of the triangles about an arbitrary straight line of the plane we could have changed the orientation of the triangles to match.
We have just examined one basic case of the congruence of triangles. The proof of the other basic cases is done in a similar way, based on the definition.
THEOREM: Two triangles are congruent if and only if one of the following conditions applies:
- the length of the corresponding sides are mutually equal (the SSS rule: Side Side Side);
- the length of two sides of each triangle are mutually equal, and the angles included between these are equal (the SAS rule: Side Angle Side);
- the length of one side in each triangle and the two angles on these sides are mutually equal (the AAS rule: Angle Angle Side);
- the length of two and two sides of the triangles are mutually equal, and the angles opposite the longer ones of these sides are equal (the SSA rule: Side Side Angle).
Example 1
Are triangles \latex{ ABC } and \latex{ A’B’C’ } congruent if we know that \latex{ AB = A’B’ }, \latex{ AC = A’C’ },\latex{BCA\sphericalangle=B’C’A’\sphericalangle} and \latex{ AB\lt AC }?
Solution
In Figure 90 it can be seen that in this case it can happen that the two triangles are not congruent, since for the position of vertex \latex{ B' } despite the conditions there are two options.
The example shows well why basic case (4) contains the criteria “the angles opposite the longer ones of these sides are equal”.
Figure 90
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ A' }
\latex{ B' }
\latex{ B }*
\latex{ C' }
\latex{ AB=A'B'=A'B }*
Example 2
Let us verify that two triangles are congruent if the lengths of two sides in each triangle are mutually equal and the medians belonging to one of these sides are also equal.
Solution
By using the notations of figure 91 it can be said that based on the simple case (1) \latex{AFC_{\triangle } \cong A’F’C’_{\triangle }}. However, it implies that \latex{CAB\sphericalangle=C’A’B’\sphericalangle}, i.e. triangles ABC and A’B’C’ satisfy the conditions of basic case (2).
So the two triangles are indeed congruent.
Figure 91
\latex{ C }
\latex{ A }
\latex{ F }
\latex{ B }
\latex{ A' }
\latex{ F' }
\latex{ B' }
\latex{ C' }
Congruence of quadrilaterals, polygons
In the case of triangles the equality of the corresponding sides was sufficient for congruence. In the case of quadrilaterals it can easily be seen that it is not sufficient, since otherwise all the rhombi with a given side length would be congruent. (Figure 92)
In general the following is valid for the congruence of polygons:
THEOREM: Two polygons are congruent if and only if one of the following conditions applies:
- the length of the corresponding sides and the length of the corresponding diagonals are mutually equal;
- the length of the corresponding sides and the measure of the corresponding angles are mutually equal.
Figure 92
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
Exercises
{{exercise_number}}. Prove that two regular triangles are congruent if the following properties are the same in both triangles:
- altitude;
- median;
- radius of the circumscribed circle.
{{exercise_number}}. Verify that two isosceles right-angled triangles are congruent if the following data are the same in both triangles:
- hypotenuse;
- leg;
- radius of the circumscribed circle.
{{exercise_number}}. Show that two right-angled triangles are congruent if
- two and two legs are equal;
- one and one leg in each and the angles opposite these sides are equal;
- their hypotenuses and the altitudes belonging to the hypotenuses are equal.
{{exercise_number}}. Prove that two isosceles triangles are congruent if the following properties are the same in both triangles:
- the base and the angle included between the legs;
- the base and the altitude belonging to it;
- the altitude belonging to the base and the angles on the base.
{{exercise_number}}. Is the following statement true? “Two isosceles triangles are congruent if one side and two angles of each triangle are equal.” (Justify your answer.)
{{exercise_number}}. Is the following statement true? “If a triangle can be divided into two congruent parts, then it is an isosceles triangle.” (Justify your answer.)
{{exercise_number}}. Take a right-angled triangle and construct squares above one of its legs and its hypotenuse, outside the triangle. Prove that (when using the notation of the figure)\latex{ABE_{\triangle } \cong AGC_{\triangle }}.
\latex{ A }
\latex{ G }
\latex{ F }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }