Ways To Prove Triangles Similar

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

Jenn (B.S., M.Ed.) of Calcworkshop® introducing triangle similarity theorems

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

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

AA Theorem
SAS Theorem
SSS Theorem

Let's jump in!

How do nosotros create proportionality statements for triangles? And how do we show ii triangles are like?

Beingness 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'due south not necessary for u.s.a. to check every single bending and side in order to tell if two triangles are similar. Cheers to the triangle sum theorem, all we have to show is that two angles of one triangle are congruent to ii angles of another triangle to show similar triangles.

Simply the fun doesn't end here. There are ii other ways nosotros tin prove ii triangles are like.

SAS Theorem

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

In other words, we are going to employ the SSS similarity postulate to show triangles are similar.

SSS Theorem

Or what if we tin demonstrate that two pairs of sides of ane 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 like. As ck-12 nicely states, using the SAS similarity postulate is enough to show that two triangles are similar.

Only is in that location merely one way to create a proportion for similar triangles? Or can more than 1 suitable proportion be establish?

Triangle Similarity Theorems

Merely as two different people tin look at a painting and run into or feel differently well-nigh the piece of art, in that location is always more than i fashion to create a proper proportion given like triangles.

And to help the states on our quest of creating proportionality statements for similar triangles, permit'southward have a expect at a few additional theorems regarding similarity and proportionality.

i. If a segment is parallel to i side of a triangle and intersects the other ii sides, then the triangle formed is similar to the original and the segment that divides the 2 sides it intersects is proportional.

proportional segment theorem

Proportional Segment Theorem

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

proportional transversal theorem

Proportional Transversal Theorem

iii. The respective medians are proportional to their corresponding sides.

corresponding medians

Respective Medians

iv. If a ray bisects an bending or a triangle, then information technology divides the contrary side into segments whose lengths are proportional to the lengths of the other two sides.

ray bisecting a triangle creating proportional sides

Ray Bisecting a Triangle Creating Proportional Sides

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

perimeter of similar polygons

Perimeter of Similar Polygons

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

Triangle Theorems – Lesson & Examples (Video)

1 60 minutes 10 min

Introduction SSS and SAS Similarity Postulates
00:00:xix – 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 #ane-6)
00:28:42 – Given iii parallel lines cut by two transversals, find the value of x (Instance #7)
00:31:36 – Overview of SSS and SAS Similarity Postulates and Similarity Theorems
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00:35:37 – Determine whether the triangles are like, and create a similarity argument (Examples #eight-12)
00:51:37 – Discover the unknown value given like 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, observe the perimeter (Examples #22-23)
Practise Problems with Step-by-Footstep Solutions
Chapter Tests with Video Solutions