Introduction


Objects that have the same shape but not the same size are said to be similar.

Similar_figures.png

In mathematics, two polygons are defined to be similar if their corresponding angles are equal in measure and the ratio of their corresponding sides are in proportion. This proportion is known as the similarity ratio.

The mathematical symbol used to denote similarity is ~, e.g. external image %5Cnormalsize%5C%21%5Cbigtriangleup%20ABC%20%5Csim%20%5Cbigtriangleup%20DEF.gif.

A flash program showing similar triangles:

Based on the above definition, think about the following questions:
1. Are congruent triangles always similar?
2. Are similar triangles always congruent?
3. Are circles always similar?
4. Are rectangles always similar?
5. Are two triangles similar if all corresponding angles are congruent?
6. Are two triangles similar if all correpsonding sides are proportional?


Identification of Similar Triangles


To prove that two triangles are similar, it is sufficient to show one (not all) of the following three statements are true:

SSS (for similarity)
Two triangles are similar if the three sets of corresponding sides are in proportion.

Take note that SSS for similar triangles is NOT the same theorem as we used for congruent triangles.
SSS.png
In the above diagram, external image %5Cnormalsize%5C%21%5Cfrac%7BAB%7D%7BDE%7D%3D%5Cfrac%7BBC%7D%7BEF%7D%3D%5Cfrac%7BAC%7D%7BDF%7D%3D2.gif, therefore, external image %5Cnormalsize%5C%21%5Cbigtriangleup%20ABC%20%5Csim%20%5Cbigtriangleup%20DEF.gif.

Use the applet to verify this theorem:

SSS_applet.png

http://www.mathopenref.com/similarsss.html

Think: are two triangles similar if only two sets of corresponding sides are in proportion? Give a counter-example if your answer is "No".

SAS (for similarity)
Two triangles are similar if two sets of corresponding sides are in proportion and their included angles are equal.

Take note that SAS for similar triangles is NOT the same theorem as we used for congruent triangles.
SAS.png
In the above diagram,external image %5Cnormalsize%5C%21%5Cfrac%7BAB%7D%7BDE%7D%3D%5Cfrac%7BAC%7D%7BDF%7D%3D2.gif and external image %5Cnormalsize%5C%21%5Cangle%20A%20%3D%20%5Cangle%20D.gif, therefore, external image %5Cnormalsize%5C%21%5Cbigtriangleup%20ABC%20%5Csim%20%5Cbigtriangleup%20DEF.gif.

Use the applet to verify this theorem:
SAS_applet.png
http://www.mathopenref.com/similarsas.html

Think: In triangle ABC and triangle DEF, if AB : DE = AC : DF and angle C = angle F, are the two triangles similar? Give a counter-example if your answer is "No".

AA
Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
AA.png
In the above diagram, external image %5Cnormalsize%5C%21%5Cangle%20A%20%3D%20%5Cangle%20D.gif and external image %5Cnormalsize%5C%21%5Cangle%20C%20%3D%20%5Cangle%20F.gif, therefore, external image %5Cnormalsize%5C%21%5Cbigtriangleup%20ABC%20%5Csim%20%5Cbigtriangleup%20DEF.gif.

Use the applet to verify this theorem:
AA_applet.png
http://www.mathopenref.com/similaraaa.html

Think: why are two congruent sets of angles sufficent to prove similarity instead three congruent sets?

Given two overlapping triangles ABC and ADE, where BC is parallel to DE, are they similar? Explain your answer.
overlapping_triangles.png

Recap


Watch the following video for a review (or preview if you have not learnt about congruent triangles) on congruent triangles and similar triangles.

http://www.youtube.com/watch?v=OEp7YK6WEXE


Quiz

Take 5 minutes to complete the following. Please include your class and index number before the name. (E.g. 2I426 Jerry Yong)

http://www.proprofs.com/quiz-school/story.php?title=similar-triangles