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Similarity
Chapter 6
6.1 – Use Similar Polygons
• Similar Polygons
• Two polygons are similar if corresponding angles are congruent
and corresponding sides length are proportional
• Notation used is a tilde ( ~ )
• Example 1:
• ΔRST ~ ΔXYZ
• RS=20, ST=30, TR=25
XY=12, YZ=18, ZX=15
a) List all pairs of congruent angles
b) Check that the ratios of corresponding side lengths are equal
c) Write the ratios of the corresponding side lengths in a
statement of proportionality
Scale Factor
• Statement of proportionality
• Any pair of ratios form a true proportion
• Scale factor
• Ratio of the lengths of two corresponding sides of two similar
polygons
• Example 3
• ΔDEF ~ ΔMNP
• Find the value of x
• GP #2-3, page 359
Perimeters
• Ratio of lengths in similar polygons is the same as the scale
factor, this is also true for the perimeters of the polygons
• Theorem 6.1 – 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
• If 𝐾𝐿𝑀𝑁~𝑃𝑄𝑅𝑆, 𝑡ℎ𝑒𝑛
𝐾𝐿+𝐿𝑀+𝑀𝑁+𝑁𝐾
𝑃𝑄+𝑄𝑅+𝑅𝑆+𝑆𝑃
=
𝐾𝐿
𝑃𝑄
=
𝐿𝑀
𝑄𝑅
=
𝑀𝑁
𝑅𝑆
=
𝑁𝐾
𝑆𝑃
Example 4
• A town is building a new swimming pool. An Olympic pool is
rectangular with length 50 meters and width 25 meters. The
new pool will be similar in shape, but only 40 meters long.
a.
b.
Find the scale factor of the new pool to an Olympic pool
Find the perimeter of an Olympic pool and the new pool
• GP #4-6, page 360
6.3 – Prove Triangles Similar
by AA
• Postulate 22 – 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
• Example 1
• Example 2
Indirect Measurement
• A useful way to find measurements indirectly is by using
similar triangles
• Example 3
• A flagpole casts a shadow that is 50 feet long. At the same time, a
woman standing nearby who is five feet four inches tall casts a
shadow that is 40 inches long.
• How tall is the flagpole to the nearest foot?
GP #4-5, page 377
6.4 – Prove Triangles Similar
by SSS and SAS
• In addition to using congruent corresponding angles, to show
that two triangles are similar, you can use proportional
corresponding side lengths
• Theorem 6.2 – Side-Side-Side (SSS) Similarity Theorem
• If the corresponding side lengths of two triangles are
proportional, then the triangles are similar
𝐼𝑓
• Example 1
• Example 2
𝐴𝐵 𝐵𝐶 𝐶𝐴
=
=
𝑅𝑆 𝑆𝑇 𝑇𝑅
𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶~∆𝑅𝑆𝑇
SAS Similarity Theorem
• Theorem 6.3 – Side-Angle-Side (SAS) Similarity Theorem
• If an angle of one triangle is congruent to an angle of a second
triangle and the lengths of the sides including these angles are
proportional, then the triangles are similar
𝑍𝑋
𝑋𝑌
𝐼𝑓 < 𝑋 =< 𝑀 𝑎𝑛𝑑
=
, 𝑡ℎ𝑒𝑛 ∆𝑋𝑌𝑍~∆𝑀𝑁𝑃
𝑃𝑀 𝑀𝑁
• Example 4 (on board)
• Tell what method you would use to show that the triangles are
similar
GP #1-2, page 383 & #3-4, page 385
6.5 – Use Proportionality
Theorems
• The Midsegment theorem is a special case of the Triangle
Proportionality theorem and its converse
• Theorem 6.4 – Triangle Proportionality Theorem
• If a line parallel to one side of a triangle intersects the other two
sides, then it divides the two sides proportionally
𝑅𝑇 𝑅𝑈
𝐼𝑓 𝑇𝑈||𝑄𝑆, 𝑡ℎ𝑒𝑛
=
𝑇𝑄 𝑈𝑆
Theorem 6.5
• Theorem 6.5 – Converse of Triangle Proportionality Theorem
• If a line divides two sides of a triangle proportionally, then it is
parallel to the third side
𝑅𝑇 𝑅𝑈
𝐼𝑓
=
, 𝑡ℎ𝑒𝑛 𝑇𝑈||𝑄𝑆
𝑇𝑄 𝑈𝑆
• Example 1
• GP #1-2, page 392
Theorems
• Theorem 6.6
• If three parallel lines intersect two transversals, then they divide
the transversals proportionally
• Theorem 6.7
• If a ray bisects an angle of a triangle, then it divides the opposite
side into segments whose lengths are proportional to the lengths
of the other two sides
Examples
• Example 3
• Example 4 (on board)
• Use the given information to find the length of segment RS
• GP #3-4, page 393