Lesson 9

Conditions for Triangle Similarity

  • Let’s prove some triangles similar.

9.1: Math Talk: Angle-Side-Angle As A Helpful Tool

How could you justify each statement?

Triangles P Q R, P prime q prime R prime, and S T U.

Triangle \(P'Q'R'\) is congruent to triangle \(STU\). 

Triangle \(PQR\) is similar to triangle \(STU\). 

Triangles G H I, G prime H prime I prime, and M N O.

Triangle \(G'H'I'\) is congruent to triangle \(MNO\).

Triangle \(GHI\) is similar to triangle \(MNO\).

9.2: How Many Pieces?

For each problem, draw 2 triangles that have the listed properties. Try to make them as different as possible.

  1. One angle is 45 degrees.
  2. One angle is 45 degrees and another angle is 30 degrees.
  3. One angle is 45 degrees and another angle is 30 degrees. The lengths of a pair of corresponding sides are 2 cm and 6 cm.
  4. Compare your triangles with your neighbors’ triangles. Which ones seem to be similar no matter what?
  5. Prove your conjecture.

9.3: Any Two Angles?

Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle.

2 triangles.
  1. Explain how you know the triangles are similar.
  2. How long are the sides labeled \(x\) and \(y\)?


Under what conditions is there an Angle-Angle Quadrilateral Similarity Theorem? What about an Angle-Angle-Angle Quadrilateral Similarity Theorem? Explain or show your reasoning.

Summary

When 2 angles of one triangle are congruent to 2 angles of a second triangle, the 2 triangles are similar. We call this the Angle-Angle Triangle Similarity Theorem.

In the diagram, angle \(A\) is congruent to angle \(D\), and angle \(B\) is congruent to angle \(E\). If a sequence of rigid motions and dilations moves the first figure so that it fits exactly over the second, then we have shown that the Angle-Angle Triangle Similarity Theorem is true.

\(\angle A \cong \angle D, \angle B \cong \angle E\)

Triangles A B C and D E F. Angle A corresponds to angle D, angle B to E.

Dilate triangle \(ABC\) by the ratio \(\frac{DE}{AB}\), so that \(A’B’\) is congruent to \(DE\). Now triangle \(A’B’C’\) is congruent to triangle \(DEF\) by the Angle-Side-Angle Triangle Congruence Theorem, which means there is a sequence of rotations, reflections, and translations that takes \(A’B’C’\) onto \(DEF\).

Two triangles A B C and D E F.

Therefore, a dilation followed by a sequence of rotations, reflections, and translations will take triangle \(ABC\) onto triangle \(DEF\), which is the definition of similarity. We have shown that a dilation and a sequence of rigid motions takes triangle \(ABC\) to triangle \(DEF\), so the triangles are similar.

Glossary Entries

  • similar

    One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

    Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).

    Triangle ABC, rotated and then dilated.