Equality of triangles . Two triangles that have all their sides and angles respectively congruent (equal).
Summary
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- 1 Triangles
- 2 Equality of triangles
- 3 Equality criteria of triangles
- 4 Properties of equality of triangles
- 5 Sources
Triangles
A triangle is a three-sided polygon , therefore it has three vertices and three angles.
Nomenclature:
- In a triangle, the letter used for the vertex is capital: A, B, C.
- The same capital letters, with a “hat”, to name the angle: Â, B, Ĉ, although the Greek letters are also usual: α, β, λ
- The name of each side is expressed with a lowercase letter: a, b, c; is the letter corresponding to the vertex opposite the side. Each side can also be expressed with two capital letters: BC, AC, AB, those of the vertices contained in that side.
Properties of triangles
- The sumof the interior angles is 180º.
- The lengthof each side is less than the sum of the other two.
- It is rigid, in fact, it is the only non-deformable polygon.
Equality of triangles
Two triangles are the same if “we can superimpose one on the other” obtaining coincident figures. In terms of the elements that make up a triangle, this concept can be expressed in the following way:
Definition Two triangles are equal if they have sides and angles equal two to two. If two triangles ABC and A0B0C0 are equal, ABC = A0B0C0 will be written.
Example. Let ABCD be a parallelogram on which one of its diagonals has been drawn:
The ADC and ABC triangles are the same. Based on the definition, it must be verified that the sides and the angles coincide two by two. First, the AC side is common to both triangles. On the other hand, since ABCD is a parallelogram, the parallel sides are equal two by two, this reveals that
DC (in ADC) = AB (in ABC)
AD (in ADC) = BC (in ABC)
There is equality of sides. Next, equality between angles will be demonstrated.
As ABCD is a parallelogram, the opposite angles are equal, this allows us to affirm that ˆD = ˆB (or, if preferred, <ADC = <ABC). For the other angles, note that the sides AD and BC are parallel and the diagonal AC cuts both, by the property of alternate-internal angles it is verified:
Aˆ (in ADC) = Cˆ (in ABC)
Cˆ (in ADC) = Aˆ (in ABC)
Equality criteria of triangles
Two triangles are the same if any of the following criteria is met:
- They have equal sides two to two.
- They have two sides equal and the included angle.
- They have one side and the two adjacent angles equal.
- They have two sides and the angle opposite the longest side respectively equal.
According to the previous criteria, two triangles are equal if we can see three equal elements in them, one of these elements being one side: in any case, when verifying that any two triangles are equal, one of the three criteria will be applied ( depending on the data available) because they require less work than the definition of equality of triangles.
Properties of equality of triangles
- Reflective character: Every triangle is equal to itself.
- Symmetrical character: If one triangle is equal to another, this one is equal to first.
- Transitive character: If one triangle is equal to another and this is equal to a third, the first is equal to the third.
