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Angle-angle triangle similarity criterion

Use dilations and rigid transformations to show why a pair of triangles with at least two pairs of congruent corresponding angles must be similar.

What does it mean for triangles to be similar?

Definition 1
What does similar mean in geometry?
Choose 1 answer:

Proving that triangles are similar

Using the properties of translations, rotations, and reflections, we can show that two triangles are congruent when we only know a few of their measurements. How much information do we need to know that two triangles are similar?

Two pairs of angles and two pairs of sides?

Triangle A B C and triangle D E F. Side A B is five. Side B C is nine. Side D E is two point five. Side E F is four point five. Angle A and angle D are congruent. Angle B and angle E are congruent. Triangle A prime B prime C prime had side A prime B prime is two point five. Angle B prime and angle E are congruent. Side B prime C prime is at four point five.
How could we justify that ABC is congruent to DEF?
Choose 2 answers:

What kind of transformations map ABC onto DEF?
ABC is congruent to DEF, so we can map ABC onto DEF using only
.

Triangle A B C and triangle D E F. Side A B is five. Side B C is nine. Side D E is two point five. Side E F is four point five. Triangle A prime B prime C prime had side A prime B prime is two point five. Side B prime C prime is at four point five. Point P is at the top right.
Complete the sequence of transformations to justify that ABC is similar to DEF.
DEF is the image of ABC after:
  1. Dilate by a scale factor of
    from point P.
  2. Translate along the directed line segment
    .
  3. Rotate about point
    by mCEF.
  4. Reflect over line
    .

How low can we go?

Based on our transformations above, we can be sure that two triangles are similar if they have 2 pairs of congruent corresponding angles and 2 pairs of corresponding sides with equal ratios. Could we show that the triangles were similar with less information? How much less?

Two pairs of angles and a pair of sides?

Triangle G H I and triangle J K L. Angle G and angle J are congruent. Angle H and angle K are congruent. Side G H is twelve. Side K J is eighteen. Triangle G prime H prime I prime had side G prime H prime is eighteen. Angle G prime is congruent to angle J. Angle H prime is congruent to angle K.
Complete the justification that GHI is similar to JKL.
  1. GHI is the image of GHI after a dilation by a scale factor of
    .
  2. We can use rigid transformations map GHI to JKL because GHIJKL by
    congruence.
  3. We can map GHI to JKL with a sequence of rigid transformations and dilations, so GHIJKL.

Just two pairs of angles?

Two triangles M N O and P Q R. Angle M and angle P are congruent. Angle N and Angle Q are congruent.
Complete the justification that MNO is similar to PQR.
StatementReason
1MP and NQGiven
2Dilate MNO by a scale factor of
.
3MM and NNDilation preserves angle measures.
4MP and NQTransitive property of congruence
5MN=
We defined the dilation that way.
6MNOPQRAngle-side-angle congruence
7There is a sequence of rigid transformations that map MNO to PQR.Definition of
8MNOPQRThere is a sequence of rigid transformations and dilations that map MNO to PQR.

Yes! We can show that two triangles are similar even if all we know is that they have two pairs of congruent corresponding angles.

Dig deeper

Here are some more questions to take your thinking deeper. Please share your responses in the comments.
  • How could you prove the angle-angle (AA) similarity criterion using the angle-angle-side (AAS) congruence criterion instead of the angle-side-angle (ASA) congruence criterion?
  • What would be the difference between the side-side-side similarity criterion and the side-side-side congruence criterion?
  • Is there a similarity criterion using only angles for quadrilaterals?

Want to join the conversation?

  • piceratops ultimate style avatar for user Anthony
    My responses for the last three questions:

    1. I'm assuming there wouldn't be much difference in the AAS? Dilate a side not between two angles to the scale of the corresponding side on the similar triangle. AAS states they are now congruent since those sides are now equal and the angles were already congruent.
    2.The difference would be the SSS similarity criterion requires the ratio of all corresponding sides be equal while SSS Congruence requires all the corresponding sides be equal.
    3.No, because a rectangle always has four 90 degree angles but not all rectangles have the same ratio of their lengths. A square and a rectangle with different lengths for its' width and length, for example.
    (35 votes)
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  • blobby green style avatar for user andrea.309728
    What would be the difference between the side-side-side similarity criterion and the side-side-side congruence criterion? If 3side of one triangle are congruent to tree side of a second triangle then the two triangle are congruent

    Is there a similarity criterion using only angles for quadrilaterals?If two angels of one triangle are congruent to two angles of another triangle the the triangle are similar
    (11 votes)
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    • leaf green style avatar for user kubleeka
      The SSS similarity criterion says that two triangles are similar if their three corresponding side lengths are in the same ratio. That is, if one triangle has side lengths a, b, c, and the other has side lengths A, B, C, then the triangles are similar if A/a=B/b=C/c. These three ratios are all equal to some constant, called the scale factor.

      Two triangles are congruent, by the SSS congruence criterion, if they are similar and the scale factor happens to be 1. That is, that a=A, b=B, and c=C.

      There are no similarity criteria for other polygons that use only angles, because polygons with more than three sides may have all their angles equal, but still not be similar. Consider, for example, a 2x1 rectangle and a square. Both have four 90º angles, but they aren't similar.
      (10 votes)
  • boggle yellow style avatar for user simonob1997
    1. How could you prove the angle-angle (AA) similarity criterion using the angle-angle-side (AAS) congruence criterion instead of the angle-side-angle (ASA) congruence criterion?

    We can use the angle-angle-side (AAS) congruence criterion instead of the angle-side-angle (ASA) congruence criterion because to prove angle-angle (AA) similarity we only need two angles. If we can show that two corresponding angles are congruent, then we know we're dealing with similar triangles.

    2. What would be the difference between the side-side-side similarity criterion and the side-side-side congruence criterion?

    Side-side-side (SSS) similarity criterion:
    The ratio between all of the sides are going to be the same.

    e.i.: for the triangle ABC and triangle XYZ the following is true:
    AB/XY = BC/YZ = AC/XZ

    Side-side-side (SSS) congruence criterion:
    The corresponding sides are congruent.

    e.i.: for the triangle ACB and triangle DBC the following is true:
    the segment AB is congruent to the segment CD, and the segment AC is congruent to the segment BD.

    3. Is there a similarity criterion using only angles for quadrilaterals?

    No, there is not a similarity criterion using only angles for quadrilaterals. This is because some figures can have all corresponding pairs of angles congruent and still not be similar.

    For example, all angles in a rectangle are 90 degrees, but a 3-by-4 rectangle is not similar to a 3-by-5 rectangle.

    Feel free to give me any feedback or critiques!
    (14 votes)
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  • aqualine ultimate style avatar for user INKLING NOW
    Can I get help? I did all the work but do not get it. Please help me!
    (7 votes)
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    • female robot grace style avatar for user loumast17
      Cross multiply is a term used when you have one fraction equaling another. so something like x/5 = 2/3. When you cross multiply you multiply both sides by the denominators of both fractions.

      x/5 = 2/3
      5 * 3 * x/5 = 2/3 * 3 * 5
      3x = 10


      Congruent is kind of a way of saying equal. You may want to look into a more in depth explanation, but in this instance it means the triangles have the same angle measures and side lengths.


      Similar is very much like congruent. Congruent means that the angle measures are equal, but side lengths don't have to be. So if something is congruent to another, they are also similar. If two things are similar, you have to check if the sides are equal as well to determine if they are congruent.
      (9 votes)
  • blobby green style avatar for user Peanut butter Parker
    1. If a pair of triangles are congruent because of AAS they are similar because if two angles are congruent they are also similar.
    2. If all three sides are similar in a pair of triangles they are ratios. But if they are congruent, they are all the same length.
    3. Ummmm, unless AAAA is a postulate, then no.
    (7 votes)
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  • blobby green style avatar for user riverajose524
    I would prove two triangles are similar using angle-angle-side congruency postulate, by showing two triangles are congruent, if they are, they are also similar.

    side-side-side similarity tells you two triangles are similar if two corresponding angles are similar but side-side-side congruency tells you two triangles are congruent if all three corresponding sides are congruent.

    I'm going to assume to assume that there isn't a similarity criterion for quadrilaterals using just angles.
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      We aren't told to prove similarity, but to prove the AA criterion for similarity.

      We can do this using the AAS congruence criterion in pretty much the same way the ASA criterion was used in the article.

      The only differences are that in step 2 we dilate △𝑀𝑁𝑂 by scale factor 𝑄𝑅∕𝑁𝑂, which in step 5 means 𝑁′𝑂′ = 𝑄𝑅.
      Then in step 6 we use the AAS congruence criterion to show
      △𝑀′𝑁′𝑂′≅ △𝑃𝑄𝑅

      – – –

      SSS similarity: the ratio between the lengths of corresponding sides is constant.
      SSS congruency: corresponding sides are congruent.

      – – –

      To prove that there is no "angles-only" similarity criterion for quadrilaterals, let's first remind ourselves what similarity means:
      Two figures are similar iff there exists a sequence of rigid transformations and dilations that maps one figure to the other.

      Rigid transformations and dilations preserve angle measures.
      Thus, in order for two figures to be similar, corresponding angles must be congruent.

      Now consider quadrilateral 𝐴𝐵𝐶𝐷.
      Let 𝐸 be a point on 𝐴𝐵, and 𝐹 be a point on 𝐶𝐷,
      such that 𝐸𝐹 is parallel to 𝐵𝐶.

      Between the two quadrilaterals 𝐴𝐵𝐶𝐷 and 𝐴𝐸𝐹𝐷 corresponding angles are congruent, but there is no sequence of rigid transformations and dilations that will map 𝐴𝐵𝐶𝐷 to 𝐴𝐸𝐹𝐷.

      Therefore, the two quadrilaterals are not similar even though their corresponding angles are congruent.

      Hence, we can not rely on angles alone to establish similarity for quadrilaterals.
      (7 votes)
  • male robot hal style avatar for user RN
    My answer for the three points at the end:

    i.) If we were to use AAS instead of ASA, we would have a corresponding side for both triangles, and by definition the pair of corresponding sides are congruent(this would be given) , and we already have two given congruent angles, so AAS would state that they are congruent and therefore similar. I am not too sure about it, but this is what conclusion I came to.

    ii.) The difference between the SSS similarity postulate and the SSS congruence postulate is that: SSS for similarity refers to the ratios of corresponding sides that are of some equal value K, whereas for the SSS congruence postulate we have three pairs of corresponding sides that are equivalent in length.

    iii.) I don't exactly think so, but I might be wrong. Quadrilaterals are of different shapes and sizes, so ratios might differ, and mapping shapes onto each other limited to the domain of rigid transformations and dilation's would be seemingly wrong. Again I'm only guessing.
    (5 votes)
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  • aqualine ultimate style avatar for user maliha.tart
    How do I determine what is the scale factor?
    (2 votes)
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    • blobby green style avatar for user Zionel
      You may determine the scale factor based on whats being asked. For example if you are trying to find what scale factor is used to bring (ABC) to JKL and JKL has larger side lengths, you would divide JK by AB to get the scale factor for bringing ABC to JKl. Vice versa
      (4 votes)
  • blobby green style avatar for user Grayson
    sss all three pairs of corresponding sides are equal
    (2 votes)
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  • aqualine seedling style avatar for user hummus
    The difference would be side-side-side similarity criterion is the ratios and side-side-side congruence criterion is the congruence of sides.
    (3 votes)
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