Lesson 10

Other Conditions for Triangle Similarity

  • Let’s prove more triangles are similar.

10.1: Math Talk: Triangle Congruence

Evaluate mentally. Is there enough information to determine if the pairs of triangles are congruent? If so, what theorem(s) would you use? If not, what additional piece of information could you use?

\(\overline{KM} \perp \overline{NL}, \overline{KL} \cong \overline{ML}\)

Triangle K L M. Sides K L and L M have single ticks marks. Point N is on side K M and segment L N intersects side K M at a right angle.

\(\angle E \cong \angle D\)

Circle with 2 segments, A C B and E C D, that cross the circle and intersect at point C. Segments connect E to A and D to B, creating 2 triangles. Angles E and D have arcs and single ticks.

\(\overline{HI} \cong \overline {FG}\)

Quadrilateral I H G F. Segment connecting I to G. Sides I H and F G are marked with single ticks.

\(\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}, \angle DAC \cong \angle BCA\)

2 parallel lines, A B and D C. Segments connecting A to D, A to C, and B to C. Angles D A C and A C B are marked with arcs and single ticks.

10.2: Side-Angle-Side Triangle Similarity?

Andre remembers lots of ways to prove triangles congruent. He asks Clare, “Can we use Angle-Side-Angle to prove triangles are similar?”

Clare: “Sure, but we don’t need the Side part because Angle-Angle is enough to prove triangles are similar.”

Andre: “Hmm, what about Side-Angle-Side or Side-Side-Side? What if we don’t know 2 angles?”

Clare: “Oh! I don’t know. Let’s draw a picture and see if we can prove it.”

Andre: “Uh-oh. If ‘side’ means corresponding sides with the same length, then we’ll only get congruent triangles.”

  1. What could ‘side’ stand for to prove triangles similar?
  2. Draw a diagram that would help you prove the Side-Angle-Side Triangle Similarity Theorem.
  3. Write a proof.

10.3: Side-Side-Side Triangle Similarity

Prove that these 2 triangles must be similar.

Triangle A B C and D E F. Length of B C is a, A C is b, A B is c, E F is k a, D F is K B, and D E is k c.

 



Prove or disprove the Side-Side-Angle Triangle Similarity Theorem. 

Summary

Besides the Angle-Angle Triangle Similarity Theorem, what other criteria are sufficient to prove triangles similar?

When 2 sides of one triangle are proportional to 2 corresponding sides of a second triangle using the same scale factor \(k\), and the pair of angles between these sides are congruent, the triangles are similar by the Side-Angle-Side Triangle Similarity Theorem.

For example, angles \(EDF\) and \(BDC\) are vertical angles and so they are congruent, and there are 2 pairs of corresponding sides with the same scale factor.

Two triangles, E D F and B D C. Angles E D F and B D C are vertical angles. Segment D C labeled k h. Segment D B labeled k j. Segment F D labeled h. Segment E D labeled j.  
Two triangles.

Dilate triangle \(DEF\) using center \(D\) and scale factor \(k\). Since \(\frac{BD}{ED}=\frac{CD}{FD} = k\), \(BD\) is now congruent to \(E'D\), and \(CD\) is congruent to \(F'D\). The dilation did not change the size of the angles. Therefore, triangle \(E'DF'\) is congruent to triangle \(BDC\) by the Side-Angle-Side Triangle Congruence Theorem. This means there is a sequence of rigid motions that takes triangle \(E'DF'\) to triangle \(BDC\). That means triangle \(BDC\) is similar to triangle \(EDF\) because there is a dilation and a sequence of rigid motions that takes one to the other. There wasn’t anything special about these 2 triangles, therefore, any pair of triangles with 2 pairs of sides whose lengths are in the same proportion and with the angle between them congruent must be similar.

We can also show that if all 3 pairs of corresponding sides are proportional and use the same scale factor \(k\), this is sufficient to prove the triangles are similar. We call this the Side-Side-Side Triangle Similarity Theorem.

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.