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7. Similarity
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Geometry

7.03 Justifying triangle similarity

Lesson
Practice
Lesson

Concept summary

Two triangles can be shown to be similar using several different theorems involving their angles and sides.

Angle-Angle similarity (AA\sim) theorem

If two angles in a triangle are congruent to two corresponding angles in another triangle, then the triangles are similar.

A larger triangle A B C, and a smaller triangle X Y Z. Angles A and Z are congruent, as well as angles B and Y.
Side-Angle-Side similarity (SAS\sim) theorem

If an angle in 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.

Two triangles. The larger triangle has 2 sides of length x and y. The angle formed by the two sides is marked. The smaller triangle has 2 sides of length a x and a y. The angle formed by the two sides is marked. The marked angles on the top and bottom triangles are congruent.
Side-Side-Side similarity (SSS\sim) theorem

If the lengths of the corresponding sides of a two triangles are proportional, then the triangles are similar.

Two triangles. The larger triangle has sides of length x, y, and z. The smaller triangle has sides of length a x, a y, and a z.

Another way to show that two figures are similar is to provide a sequence of similarity transformations that map one figure onto the other. Since similarity transformations preserve side ratios and angle measure, any combination of translations, reflections, rotations, and dilations will always produce similar figures.

-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y

\triangle{ABC}\sim\triangle{DEF} because a rotation about the origin, and a dilation will map them onto each other.

Worked examples

Example 1

Consider the two triangles.

Two triangles of different sizes are drawn. For each triangle, two angles are labelled a degrees and b degrees respectively.
a

Explain whether or not there is enough information to determine that the two triangles are similar.

Approach

We can use the given information about the angles in the two triangles to determine whether we can show that they are similar.

Solution

In this case, we have two pairs of congruent angles given, so there is enough information to show that the triangles are similar.

b

State which similarity theorem can be used to show the two triangles are similar.

Solution

The angle-angle (AA) similarity theorem.

Example 2

Determine whether the pair of triangles are similar. If so, write a similarity statement and justify with a similarity postulate or theorem. If not, explain why not.

Triangle M L N is drawn. Point K lies on J M and Point L lies on J N. Segment K L is drawn. The following are the lengths of the segments: J K, 2; J L, 3; K M, 4; and L N, 6.

Approach

First, we should consider whether the given information is enough to establish similarity between the two triangles. In this case, we are given some side lengths, but also note that there is an angle common to both triangles.

Solution

Based on the given lengths, we can see that \dfrac{JK}{JM}=\dfrac{JL}{JN}. The included angle, \angle J belongs to both \triangle JMN and \triangle JKL.

Therefore, \triangle JMN \sim \triangle JKL by the side-angle-side (SAS) similarity theorem.

Outcomes

G.CO.A.1

Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.

G.SRT.A.1

Use properties of dilations given by a center and a scale factor to solve problems and to justify relationships in geometric figures.

G.SRT.A.2

Define similarity in terms of transformations. Use transformations to determine whether two figures are similar.

G.SRT.B.3

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.

G.MP1

Make sense of problems and persevere in solving them.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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