o Line of reflection
o Center of rotation
o Angle-angle condition
o Side-side-side (proportionality) condition
o Side-angle-side (proportionality) condition
o Understand how transformations can be used as a method of understanding congruence and similarity
o Prove that two triangles are similar using appropriate criteria
Another View of Congruence and Similarity
These mathematical manipulations are a more rigorous way of saying that if we cut out of a piece of paper two triangles with exactly the same shape, then those triangles are still the same no matter how we turn, flip, or move those pieces of paper!
Again, because the triangles overlap, they are congruent (we also know the triangles are congruent from the given diagram by the SAS condition).
As with congruence, we can identify several conditions for proving similarity in triangles that does not require us to show that all three angles are congruent. Again, the condition for similarity is that the interior angles of one triangle are congruent with those of another triangle.
The first condition that we can use to prove similarity is the angle-angle condition. Recall that the sum of all the angles in a triangle is always 180°; thus, if two triangles have two angles that are congruent, they must also have a third angle that is congruent, as shown below.
Because each side of the rightmost triangle is half the length of the corresponding side in the leftmost triangle, the two triangles are therefore similar. As such, we can also conclude that the corresponding angles for the two triangles are congruent.
These two triangles are similar because the corresponding labeled sides are proportional (the length of the sides for the rightmost triangle are 5/3 times the lengths of the sides for the leftmost triangle) and the corresponding included angles are congruent. Thus, using any of the above criteria, we can prove that two triangles are congruent.
Practice Problem: Prove that triangles ABD and BCD are similar.
Solution: Let's prove that the triangles are similar using a two-column proof format. Note that the triangles have congruent angles and . We can also separate the triangles for clarity.
2. Given, 8 = 2 ∙ 4
3. Given, 4 = 2 ∙ 2
4. Sides BC and BD, and Statements 2 and 3
sides BD and AB are
5. Triangles ABD and BCD Side-angle-side (proportionality) condition
Thus, we have shown the two triangles to be similar.
Practice Problem: Prove that any two equilateral triangles are similar.
Solution: We know from our study of triangles that an equilateral triangle contains three congruent angles; thus, the measure of each angle in an equilateral triangle is 60°. As a result, by the angle-angle condition, all equilateral triangles must be similar to one another.
Practice Problem: Two isosceles triangles both have an interior angle of 100°. Prove that these triangles are similar.
Solution: Let's start by drawing a diagram (which is not necessarily to scale). We'll call the two isosceles triangles ABC and XYZ.
Because a triangle has only 180° total, the other two angles in each triangle must sum to 80°; furthermore, because the triangles are isosceles, these angles must be congruent to each other. Let's do a two-column proof to show our reasoning step by step.
2. Isosceles triangle
3. Isosceles triangle
4. 180° in a triangle
5. Triangles ABC and XYZ are Angle-angle condition
More generally, we can say that any two isosceles triangles that have congruent obtuse angles must be similar. (The same reasoning applies-all that it required is that both isosceles triangles have an obtuse angle with the same measure.)