Download Unit 7 Section 2 – Similar Triangles

Module 8 Lesson 3 – Similar Triangles
We are going to cover three parts in this section.
By the end of this section you will be able to;
1. Identify Similar Triangles by AA Similarity, SSS Similarity, and SAS Similarity,
2. Use the Special Segments Theorems to solve problems.
Similar Triangles
There are three ways to identify if two triangles are similar.
Angle-Angle (AA~) Similarity Postulate – If two angles of one triangle are congruent to two angles of another triangle,
then the triangles are similar.
Example:
Side-Side-Side (SSS~) Similarity Theorem – If the measures of the corresponding sides of two triangles are
proportional, then the triangles are similar.
Example:
.
Side-Angle-Side (SAS~) Similarity – If the measures of two sides of a triangle are proportional to the measures of two
corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
Example:
BC  AB and B  E, so ABC ~ DEF .
EF DE
4 1 6 1
 ;  and B  E, so ABC ~ DEF
8 2 12 2
Example 1:
A.
B.
C.
Example A – demonstrates
the SAS Similarity Postulate
because the two sides of the
first triangle are proportional
to the two sides of the second
triangle and the included angles
are congruent.
Example B – AA Similarity
because the two angles of the
the first triangle are congruent
to the two angles of the second
triangle. (If you look closely you
notice the vertical angles in this
image and remember that vertical
angles are congruent.
Example C – SSS Similarity
because the corresponding
sides of the two triangles are
proportional. When looking at a
set of triangles, to set up your
own proportions, you place the
smallest side over the smallest
side and largest over largest,
etc. Then test to see if the
ratios are equivalent.
=
Now You Try
For questions 1 – 6, if triangles can be proved similar, write the postulate or theorem that would be used to prove it and write
the similarity statement for numbers 2 and 3.
1.
2.
3.
T
4.
5.
6.
Parts of Similar Triangles
Special Segments of Similar Triangles Theorems
Altitude
If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the
corresponding sides.
Example:
Altitudes Examples: If the two triangles are similar, find x.
A.
B.
Angle Bisector
If two triangles are similar, then the measures of the corresponding angle bisectors of the triangles are proportional to the
measures of the corresponding sides.
Example:
Angle Bisector Example: If the two triangles are similar, find x.
Median
If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the
corresponding sides.
Example:
Median Examples: If the two triangles are similar, find x.
A.
B.
Now You Try: If the two triangles are similar, find x or MN.
1.
3.
2.
Find MN.