What theorem can be used to prove that the two triangles are similar?

What theorem can be used to prove that the two triangles are similar?

Two triangles are similar if they have:

  • all their angles equal
  • corresponding sides are in the same ratio

But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough.

There are three ways to find if two triangles are similar: AA, SAS and SSS:

AA

AA stands for "angle, angle" and means that the triangles have two of their angles equal.

If two triangles have two of their angles equal, the triangles are similar.

So AA could also be called AAA (because when two angles are equal, all three angles must be equal).

SAS

SAS stands for "side, angle, side" and means that we have two triangles where:

  • the ratio between two sides is the same as the ratio between another two sides
  • and we we also know the included angles are equal.

If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.

What theorem can be used to prove that the two triangles are similar?

In this example we can see that:

  • one pair of sides is in the ratio of 21 : 14 = 3 : 2
  • another pair of sides is in the ratio of 15 : 10 = 3 : 2
  • there is a matching angle of 75° in between them

So there is enough information to tell us that the two triangles are similar.

Using Trigonometry

We could also use Trigonometry to calculate the other two sides using the Law of Cosines:

In Triangle ABC:

  • a2 = b2 + c2 - 2bc cos A
  • a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
  • a2 = 441 + 225 - 630 × 0.2588...
  • a2 = 666 - 163.055...
  • a2 = 502.944...
  • So a = √502.94 = 22.426...

In Triangle XYZ:

  • x2 = y2 + z2 - 2yz cos X
  • x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
  • x2 = 196 + 100 - 280 × 0.2588...
  • x2 = 296 - 72.469...
  • x2 = 223.530...
  • So x = √223.530... = 14.950...

Now let us check the ratio of those two sides:

a : x = 22.426... : 14.950... = 3 : 2

the same ratio as before!

Note: we can also use the Law of Sines to show that the other two angles are equal.

SSS

SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.

If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

What theorem can be used to prove that the two triangles are similar?

In this example, the ratios of sides are:

  • a : x = 6 : 7.5 = 12 : 15 = 4 : 5
  • b : y = 8 : 10 = 4 : 5
  • c : z = 4 : 5

These ratios are all equal, so the two triangles are similar.

Using Trigonometry

Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:

In Triangle ABC:

  • cos A = (b2 + c2 - a2)/2bc
  • cos A = (82 + 42 - 62)/(2× 8 × 4)
  • cos A = (64 + 16 - 36)/64
  • cos A = 44/64
  • cos A = 0.6875
  • So Angle A = 46.6°

In Triangle XYZ:

  • cos X = (y2 + z2 - x2)/2yz
  • cos X = (102 + 52 - 7.52)/(2× 10 × 5)
  • cos X = (100 + 25 - 56.25)/100
  • cos X = 68.75/100
  • cos X = 0.6875
  • So Angle X = 46.6°

So angles A and X are equal!

Similarly we can show that angles B and Y are equal, and angles C and Z are equal.

7861, 7862, 7863, 7864, 7865, 7866, 7867, 7868, 7869, 7870, 7871, 7872

Copyright © 2022 Rod Pierce

Solution:

When two triangles are put on each other and are exactly the same in shape and size. they are called congruent triangles. The symbol of congruence is '≅'.

The congruent triangles are exactly the same with respect to their shape and size. Thus, all corresponding parts of congruent triangles are also congruent.

Let △ ABC and △ PQR be two triangles, if ∠A = ∠P, AB = PQ and  ∠B = ∠Q

What theorem can be used to prove that the two triangles are similar?

Then △ ABC ≅ △ PQR by Angle Side Angle (ASA) rule.

Similarly, let △ ABC and △ XYZ be two triangles, if  BC = YZ, ∠C = ∠Z and  AC = XZ

What theorem can be used to prove that the two triangles are similar?

Then △ ABC ≅ △ XYZ by Side Angle Side (SAS) rule.

Similarly, let △ ABC and △ DEF be two triangles, if  AC = DE, BC = FE and AB = DF

What theorem can be used to prove that the two triangles are similar?

Then △ ABC ≅ △ DEF by Side Side Side (SSS) rule.

Similarly, let △ ABC and △ LMN be two right triangles, if  ∠B = ∠M = 90º, AC = LN (Hypotenuse) and AB = LM (side)

What theorem can be used to prove that the two triangles are similar?

Then △ ABC ≅ △ LMN by Right-Angle Hypotenuse Side (RHS) rule.

Thus, the congruence of the triangle can be proved by ASA, SAS, SSS, and RHS rules.

Summary:

The congruence of the triangle can be proved by ASA, SAS, SSS, and RHS rules.

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 theorem can be used to prove that the two triangles 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 theorem can be used to prove that the two triangles are similar?

Proportional Segment Theorem

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

What theorem can be used to prove that the two triangles are similar?

Proportional Transversal Theorem

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

What theorem can be used to prove that the two triangles 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 theorem can be used to prove that the two triangles are similar?

Ray Bisecting a Triangle Creating Proportional Sides

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

What theorem can be used to prove that the two triangles 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 theorem can be used to prove that the two triangles are similar?