Similar Triangles

Definition

Two triangles are similar if corresponding angles are congruent and if the ratio of corresponding sides is constant. If the triangles ΔABC and ΔDEF are similar, we can write this relation as ΔABC∼ΔDEF.

The difference between similar and congruent triangles is that similar triangles do not have to be the same size. See Congruent Triangles for information about the congruence cases and the theorem about isosceles triangles. See Summary - Angles for information about related angles.

Triangle Proportionality Theorem

Interactive help to prove the triangle proportionality theorem.

A transversal is a line that intersects two or several lines.

Theorem: A transversal that is parallel to one of the sides in a triangle divides the other two sides proportionally.

Proof: All you need to know in order to prove the theorem is that the area of a triangle is given by \[A=\frac{w\cdot h}{2}\]

where w is the width and h is the height of the triangle. Compare areas three times!

  1. Make a triangle poly1=ΔAED and a triangle poly2=ΔBED. Let poly1 and poly2 denote the areas of the triangles. What is the ratio of poly1 and poly2? Prove your answer!
  2. Make a triangle poly3=ΔDEC. What is the ratio of poly3 and poly1? Prove your answer!
  3. What is the ratio of poly3 and poly2? Prove your answer!
  4. Show that \(\dfrac{a}{b}=\dfrac{c}{d}\)!

Corollary of Proportionality Theorem

Interactive help to prove the triangle proportionality theorem.
Interactive help to prove corollary of proportionality theorem.

Corollary: A transversal that is parallel to a side in a triangle defines a new smaller triangle that is similar to the original triangle.

Proof:

Show that corresponding angles in the two triangles are congruent (equal).

Then show that \[\frac{a+b}{a}=\frac{c+d}{c}\]

Draw another transversal parallel to another side and show that \[\frac{a+b}{a}=\frac{c+d}{c}=\frac{f}{e}\]

Three cases that yield similarity

Side-Angle-Side (SAS)If two sides in a triangle are in the same ratio to two corresponding sides of another triangle, and if the included angles in both triangles are the same, then the triangles are similar.

Proof

Image

Place the point G such that AG=DE.
Draw GH such that GH is parallel to BC.

Now ΔAGH∼ΔABC and ΔDEF∼ΔABC.

Show that AH=DF!

Then the congruence case Side-Angle-Side yields ΔAGH≅ΔDEF and the proof is done.

The proofs of the other two cases are similar.

Side-Side-Side (SSS) If three pairs of corresponding sides are in the same ratio then the triangles are similar.

Angle-Angle-Angle (AA) If the angles in a triangle are congruent (equal) to the corresponding angles of another triangle then the triangles are similar.

Also note that it suffices that two angles are equal (why?).

Angle bisector theorem

The line through C is parallel to AB.

Theorem: In a triangle \(\Delta ABC\) an angle bisectris is drawn at B. The bisectris intersects AC at a point D. The angle bisectris theorem states that:

\[\frac{AB}{AD}=\frac{BC}{CD}\]

Proof: Draw a line parallel to AB through C. Let E be the intersection of the new line and the bisectris. Explain why:

  1. \(\angle ABD = \angle CED\)
  2. triangle \(\Delta BCE\) is an isosceles triangle, and hence why \(BC=CE\).
  3. \(\angle BDA = \angle CDE\)
  4. \(\Delta ABD \sim \Delta CED\)
  5. \(\dfrac{AB}{AD}=\dfrac{CE}{CD}\)
  6. \(\dfrac{AB}{AD}=\dfrac{BC}{CD}\)

Exercises

Exercise 1

AC is parallel to EF. AB is parallel to DF. h, h1, h2 and h3 are perpendicular heights in the triangles. Prove what you see!

Exercise 1.

Exercise 2

The blue points are midpoints on each side. Prove what you see!

Exercise 2.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

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