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How to prove that two triangles are similar?

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How to prove that two triangles are similar?  [#permalink]

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New post 19 Jun 2018, 14:01
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Hello - this is a topic that I struggle with.

How do you prove that two triangles are similar? What are all the ways?

One I can think of is that two of the angles are equal in both triangles, but are there other ways to prove this for the purposes of the GMAT?

Thanks!
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Re: How to prove that two triangles are similar?  [#permalink]

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New post 19 Jun 2018, 21:57
gmat5ever wrote:
Hello - this is a topic that I struggle with.

How do you prove that two triangles are similar? What are all the ways?

One I can think of is that two of the angles are equal in both triangles, but are there other ways to prove this for the purposes of the GMAT?

Thanks!


Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.


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Re: How to prove that two triangles are similar?  [#permalink]

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New post 25 Jun 2018, 14:58
gmat5ever wrote:
Hello - this is a topic that I struggle with.

How do you prove that two triangles are similar? What are all the ways?

One I can think of is that two of the angles are equal in both triangles, but are there other ways to prove this for the purposes of the GMAT?

Thanks!


Bunuel covered the technical details, but this may be useful to you as well. It's a GRE article, but all of the math applies on the GMAT as well! It goes over three common diagrams where two triangles are similar - if you spot one of these, you can know immediately that the triangles are similar, without having to calculate or prove anything, which can save you some time.

https://www.manhattanprep.com/gre/blog/ ... triangles/
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Re: How to prove that two triangles are similar? &nbs [#permalink] 25 Jun 2018, 14:58
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