Download 6.3 Use Similar Polygons / 6.4 Similar Triangles by AA

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6.3 Use Similar Polygons / 6.4 Similar Triangles by AA
Goal  Use proportions to identify similar polygons / Use the AA Similarity Postulate.
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Similar polygons Two polygons are similar polygons if (1) corresponding angles are congruent
and
(2) corresponding side lengths are proportional.
Same shape, different size
Scale factor of two similar polygons - the ratio of the lengths of two corresponding sides
Example Use similarity statements
In the diagram, ∆ ABC  ∆ DEF.
a. List all pairs of congruent angles.
b. Check that the ratios of corresponding side lengths are equal.
Solution
a. A  ____,  B   ____,  C   ____
AB
=
=
DE
BC
EF
CA
=
FD
Example - Given ∆PQR  ∆XYZ, list all pairs of congruent angles.
Example Find the scale factor
Determine whether the polygons are similar. (You need to know if all ratios are = )
If they are, write a similarity statement.
Example Use similar polygons
In the diagram, ∆BCD  ∆RST. Find the value of x.
Checkpoint In the diagram, FGHJ  LMNP.
1. What is the scale factor of LMNP to FGHJ?
2. Find the value of x.
THIS IS THE SCALE FACTOR
FROM ∆
ABC TO ∆ DEF
THEOREM: PERIMETERS OF SIMILAR POLYGONS
If two polygons are similar, then the ratio of their perimeters is equal to
the ratios of their corresponding side lengths.
Checkpoint In the diagram, ∆PQR  ∆WXY.
Find the perimeter of ∆WXY.
POSTULATE: ANGLE-ANGLE (AA) SIMILARITY POSTULATE
If two angles of one triangle are congruent to two angles of another triangle, then
the two triangles are similar.
Remember, angles
are buy 2 get 1 free!
Examples Use the AA Similarity Postulate
Determine whether the triangles are similar.
If they are, write a similarity statement.
Ex.Show that ∆BCD ~ ∆EFD.
Indirect Measurement To estimate the diameter of the sun, you punch a tiny hole into a piece of paper and
hold the paper so the sun shines through the hole onto a screen with a surface perpendicular to the direction of
the sun. Using the information in the figure, estimate the diameter of the sun. Explain how you can deduce this.