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Study Notes 013:
By William E. Steinman:
Geometry 6 Triangles:
We have talked about all kinds of figures being congruent. Let’s just take the general statement.
Congruent figures are figures that have the same size and shape.
That covers everything, circles, angles, lines, triangles, etcetera.
So, congruent triangles are triangles that have the same size and shape.
It follows that if two triangles are congruent, their corresponding sides and angles must be congruent.
This is stated as the first principle of congruent triangles thus:
If two triangles are congruent then their corresponding parts are congruent.
For example, take the classic 3 – 4 - 5 right triangle.
The congruence of all parts of these two examples, X and Y, is clear.
They are congruent in every way.
Once again, we seem to belabor the obvious, but this discipline will be useful in situations where things are not so obvious.
We have three other principles of congruent triangles, which also seem to belabor the obvious.
The second principle of congruent triangles is:
If two sides and the included angle of a triangle are congruent to the corresponding parts or another triangle, then the triangles are congruent.
For example, it is clear that the isosceles triangle A with two equal sides and their included angle is congruent with the isosceles triangle B with two equal sides and their included angle. It does not matter which two sides we select.
The third principle of congruent triangles is:
If two angles and the included side of a triangle are congruent to the corresponding parts of anther triangle, then the triangles are congruent.
We cans refer to triangles A and B above to confirm that.
The fourth principle of congruent triangles is:
If three sides of a triangle are congruent to the three sides of another triangle, then the triangles are congruent.
Again, refer to triangles A and B above.
Considering that congruent figures are figures that have the same size and shape four principles for the special cases of isosceles and equilateral triangles.
The definition of an isosceles triangle is a triangle having two equal sides. An equilateral triangle is a triangle with all sides being equal. For example:
The first principle is:
If two sides of a triangle are congruent, the angles opposite the sides are congruent. Refer to the isosceles triangle C below to prove this.
The second principle is:
If two angles of a triangle are congruent, the sides opposite these angles are congruent.
It is the same obvious conclusion. Refer to triangle C above.
The prize for belaboring the obvious goes to the third and fourth principles:
An equilateral triangle is equiangular.
An equiangular triangle is equilateral.
Enough already!