Section 8.3Proving Triangles Similar

By: Asad Ashraf

What is Similarity

• Similar figures are figures in which the shape is exactly the same, but the size is not.

• A Dilation is an enlargement of a figure. It is still similar, however.

• A Reduction is a reduction of a figure. This is also similar.

• **Remember that similar figures are not necessarily congruent**

Proving Triangles Similar

• There are several methods to prove triangles similar.

• These are very similar to the methods of proving triangles congruent.

• The methods are AAA~, AA~, SAS~, and SSS~.

AAA~ Postulate

• AAA~ Postulate - If three angles of one triangle are congruent to three angles of a 2nd triangle then the triangles are similar.

AA~ Theorem

• AA~ if 2 angles of one triangle are congruent to two angles of a 2nd triangle then the triangles are similar.

• This is proved by the No-Choice Theorem and AAA~ Theorem combined.

SSS~ Theorem

• SSS~ If 3 sides of one triangle are proportional to three sides of a 2nd triangle then the two triangles are similar.

SAS~ Theorem

• SAS ~ If two sides of one triangle are proportional to the corresponding two sides of a 2nd triangle and their included angles are congruent, then the triangles are similar.

Sample Problems

• Click Here• SSS~ and SAS~ Problems

Sample ProblemsAre these triangles ~?

Answer: Yes they are because Included angles were given congruent and the ratios of the sides are congruent as well.

Practice Problems

• Prove that an acute angle of one right angle is congruent to the vertex angle of an isosceles triangle, they are similar.

• Always, Sometimes, Never– If 2 triangles are similar, they are congruent ____– If 2 triangles are congruent, they are similar ____– 2 rectangles are similar if neither is a square ____– 2 right triangles are similar____

Answers: A, S, S, S

Practice Problems

• Are any 2 isosceles triangles similar? Show all work.

Practice Problems

• Are these triangles similar? Why or why not?

Answer: NO

Practice Problems

• Given: XS and RY are Altitudes of RTS

• Prove: Triangle TSX is similar to Triangle TYR

Works Cited

• Our Book• “Similar Triangles”. Mathwarehouse.com. May

27,2008.<http://www.mathwarehouse.com/geometry/similar/triangles/index.html>.

• “Chapter 8”. Teacherweb.org. 2003. May 27,2008.<http://teacherweb.ftl.pinecrest.edu/wingjoa/My%20Webs/Geometry/chpt8.htm#Test%20Review%20-%20%20Chapter%208.1-4>.