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7.9: SAS Similarity

  • Page ID
    5806
  • Triangles are similar if two pairs of sides are proportional and the included angles are congruent.

    SAS Similarity Theorem

    By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if you know only that two pairs of sides are proportional and their included angles are congruent, that is enough information to know that the triangles are similar. This is called the SAS Similarity Theorem.

    SAS Similarity Theorem: If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

    f-d_b6480e8de58638d9acdadbb2d2c9e2f9533bbc87a3f86ce202078519+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    If \(\dfrac{AB}{XY}=\dfrac{AC}{XZ}\) and \(\angle A\cong \angle X\), then \(\Delta ABC\sim \Delta XYZ\).

    What if you were given a pair of triangles, the lengths of two of their sides, and the measure of the angle between those two sides? How could you use this information to determine if the two triangles are similar?

    Example \(\PageIndex{1}\)

    Determine if the following triangles are similar. If so, write the similarity theorem and statement.

    f-d_3c5ec26759a40dc1d524b1c5af8864d5e87135063ce6f4e75d37af4d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Solution

    We can see that \(\angle B\cong \angle F\) and these are both included angles. We just have to check that the sides around the angles are proportional.

    \(\begin{aligned} \dfrac{AB}{DF} &=\dfrac{12}{8}=\dfrac{3}{2} \\ \dfrac{BC}{FE}&=\dfrac{24}{16}=\dfrac{3}{2} \end{aligned}\)

    Since the ratios are the same \(\Delta ABC\sim \Delta DFE\) by the SAS Similarity Theorem.

    Example \(\PageIndex{2}\)

    Determine if the following triangles are similar. If so, write the similarity theorem and statement.

    f-d_495a286c1a8e3a9f2c5297e632ccf96b7e68ec4d05e42e277f4ef460+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Solution

    The triangles are not similar because the angle is not the included angle for both triangles.

    Example \(\PageIndex{3}\)

    Are the two triangles similar? How do you know?

    f-d_ba359cb1394e27c050f010f12d43492c473d92e7b5f5310b91c3ee4a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    We know that \(\angle B\cong \angle Z\) because they are both right angles and \(\dfrac{10}{15}=\dfrac{24}{36}\). So, \(\dfrac{AB}{XZ}=\dfrac{BC}{ZY}\) and \(\Delta ABC\sim \Delta XZY\) by SAS.

    Example \(\PageIndex{4}\)

    Are there any similar triangles in the figure? How do you know?

    f-d_d2288153b234f479d5ec0f9baa221b7a3b806b25b312c61ad4ae2c1b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    \(\angle A\) is shared by \(\Delta EAB\) and \(\Delta DAC\), so it is congruent to itself. Let’s see if \(\dfrac{AE}{AD}=\dfrac{AB}{AC}\).

    \(\begin{aligned} \dfrac{9}{9+3}&=\dfrac{12}{12+5} \\ \dfrac{9}{12}&=\dfrac{3}{4}\neq \dfrac{12}{17}\qquad \text{ The two triangles are not similar. }\end{aligned}\)

    Example \(\PageIndex{5}\)

    From Example 4, what should \(BC\) equal for \(\Delta EAB\sim \Delta DAC\)?

    Solution

    The proportion we ended up with was \(\dfrac{9}{12}=\dfrac{3}{4}\neq \dfrac{12}{17}\). AC needs to equal 16, so that \(\dfrac{12}{16}=dfrac{3}{4}\). \(AC=AB+BC\) and \(16=12+BC\). \(BC\) should equal 4.

    Review

    Fill in the blanks.

    1. If two sides in one triangle are _________________ to two sides in another and the ________________ angles are _________________, then the triangles are ______________.

    Determine if the following triangles are similar. If so, write the similarity theorem and statement.

    1. f-d_a4a69765e7c78f4cdb880fdf5743d9b9a9a0cbcba6f9a1b414a52ff7+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{6}\)

    Find the value of the missing variable(s) that makes the two triangles similar.

    1. f-d_0f9b303cda32b94496979b576545470668baec69d07b1e9cbab0f75b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. f-d_323edf503c8ca93dfc4bf307df28e89973c6aa803279a696d709d5ac+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    3. f-d_0bf5ce4b3576915654e454c3ad6fd3c1aba19316ad7f1c480a603b25+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)

    Determine if the triangles are similar. If so, write the similarity theorem and statement.

    1. \(\Delta ABC\) is a right triangle with legs that measure 3 and 4. \(\Delta DEF\) is a right triangle with legs that measure 6 and 8.
    2. \(\Delta GHI\) is a right triangle with a leg that measures 12 and a hypotenuse that measures 13. \(\Delta JKL\) is a right triangle with legs that measure 1 and 2.
    3. f-d_2873eacf3d8a3972a1f262545bac5bd89e875b215b5194d146f0e8a0+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    4. f-d_2c00c88531a025e29492cd7a0671548293e34155b54de673b673424e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{11}\)
    5. f-d_87e6d837b5fff24ace0cab92bdae3ba30b137bafbd4275146b1e0207+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
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      Figure \(\PageIndex{13}\)
    7. \(\overline{AC}=3\)

    \(\overline{DF}=6\)

    f-d_a479f34b11071c9e3f33d5c4232f51d562ec991421ae1f96aa5ecfea+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{14}\)
    1. f-d_16bca2e2b112b6705ac3486634b8e1d118f8159781f0a6fe9e8edc85+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
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      Figure \(\PageIndex{16}\)
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      Figure \(\PageIndex{17}\

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 7.7.

    Resources

    Vocabulary

    Term Definition
    AA Similarity Postulate If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
    Congruent Congruent figures are identical in size, shape and measure.
    Dilation To reduce or enlarge a figure according to a scale factor is a dilation.
    SAS SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known.
    SAS Similarity Theorem The SAS Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.
    Similarity Transformation A similarity transformation is one or more rigid transformations followed by a dilation.

    Additional Resources

    Interactive Element

    Video: Congruent and Similar Triangles

    Activities: SAS Similarity Discussion Questions

    Study Aids: Polygon Similarity Study Guide

    Practice: SAS Similarity

    Real World: Triangle Similarity