What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

In today’s geometry lesson, you’re going to learn about the triangle similarity theorems, SSS (side-side-side) and SAS (side-angle-side).

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

In total, there are 3 theorems for proving triangle similarity:

  1. AA Theorem
  2. SAS Theorem
  3. SSS Theorem

Let’s jump in!

How do we create proportionality statements for triangles? And how do we show two triangles are similar?

Being able to create a proportionality statement is our greatest goal when dealing with similar triangles. By definition, we know that if two triangles are similar than their corresponding angles are congruent and their corresponding sides are proportional.

AA Theorem

As we saw with the AA similarity postulate, it’s not necessary for us to check every single angle and side in order to tell if two triangles are similar. Thanks to the triangle sum theorem, all we have to show is that two angles of one triangle are congruent to two angles of another triangle to show similar triangles.

But the fun doesn’t stop here. There are two other ways we can prove two triangles are similar.

SAS Theorem

What happens if we only have side measurements, and the angle measures for each triangle are unknown? If we can show that all three sides of one triangle are proportional to the three sides of another triangle, then it follows logically that the angle measurements must also be the same.

In other words, we are going to use the SSS similarity postulate to prove triangles are similar.

SSS Theorem

Or what if we can demonstrate that two pairs of sides of one triangle are proportional to two pairs of sides of another triangle, and their included angles are congruent?

This too would be enough to conclude that the triangles are indeed similar. As ck-12 nicely states, using the SAS similarity postulate is enough to show that two triangles are similar.

But is there only one way to create a proportion for similar triangles? Or can more than one suitable proportion be found?

Triangle Similarity Theorems

Just as two different people can look at a painting and see or feel differently about the piece of art, there is always more than one way to create a proper proportion given similar triangles.

And to aid us on our quest of creating proportionality statements for similar triangles, let’s take a look at a few additional theorems regarding similarity and proportionality.

1. If a segment is parallel to one side of a triangle and intersects the other two sides, then the triangle formed is similar to the original and the segment that divides the two sides it intersects is proportional.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Proportional Segment Theorem

2. If three parallel lines intersect two transversals, then they divide the transversals proportionally.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Proportional Transversal Theorem

3. The corresponding medians are proportional to their corresponding sides.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Corresponding Medians

4. If a ray bisects an angle or a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Ray Bisecting a Triangle Creating Proportional Sides

5. The perimeters of similar polygons are proportional to their corresponding sides.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Perimeter of Similar Polygons

Together we are going to use these theorems and postulates to prove similar triangles and solve for unknown side lengths and perimeters of triangles.

Triangle Theorems – Lesson & Examples (Video)

1 hr 10 min

  • Introduction SSS and SAS Similarity Postulates
  • 00:00:19 – Overview of Proportionality Statements for Segments Parallel to a Side of a Triangle
  • 00:15:24 – Find the value of x given similar triangles (Examples #1-6)
  • 00:28:42 – Given three parallel lines cut by two transversals, find the value of x (Example #7)
  • 00:31:36 – Overview of SSS and SAS Similarity Postulates and Similarity Theorems
  • Exclusive Content for Member’s Only
  • 00:35:37 – Determine whether the triangles are similar, and create a similarity statement (Examples #8-12)
  • 00:51:37 – Find the unknown value given similar triangles (Examples #13-18)
  • 01:02:36 – Find the unknown value or create the proportion for finding perimeter (Examples #19-21)
  • 01:10:16 – Given similar triangles, find the perimeter (Examples #22-23)
  • Practice Problems with Step-by-Step Solutions
  • Chapter Tests with Video Solutions

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What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

Triangles are similar if two pairs of sides are proportional and the included angles are congruent.

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.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
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.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
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.

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
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?

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
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?

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
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.

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. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{6}\)

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

  1. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{7}\)
  2. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{8}\)
  3. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    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. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{10}\)
  4. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{11}\)
  5. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{12}\)
  6. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{13}\)
  7. \(\overline{AC}=3\)

\(\overline{DF}=6\)

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
Figure \(\PageIndex{14}\)
  1. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{15}\)
  2. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{16}\)
  3. What triangle similarity theorem states that if two triangles have proportional sides then they are similar?
    Figure \(\PageIndex{17}\

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

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.

Video: Congruent and Similar Triangles

Activities: SAS Similarity Discussion Questions

Study Aids: Polygon Similarity Study Guide

Practice: SAS Similarity

Real World: Triangle Similarity

What triangle similarity theorem states that if two triangles have proportional sides then they are similar?

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What triangle similarity theorem states that if two triangles have proportional sides then they are similar?