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

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. .

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.

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.

## Introduction

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

similar.In mathematics, two polygons are defined to be similar if their

corresponding angles are equal in measureand theratio 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. .

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.

In the above diagram, , therefore, .

Use the applet to verify this theorem:

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.

In the above diagram, and , therefore, .

Use the applet to verify this theorem:

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".AATwo triangles are similar if two angles of one triangle are congruent to two angles of another triangle.

In the above diagram, and , therefore, .

Use the applet to verify this theorem:

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.## 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

your class and index numberbefore thename. (E.g.2I426 Jerry Yong)http://www.proprofs.com/quiz-school/story.php?title=similar-triangles