Angle-angle triangle similarity criterion (article) | Khan Academy (2024)

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

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  • Anthony

    4 years agoPosted 4 years ago. Direct link to Anthony's post “My responses for the last...”

    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.

    (38 votes)

  • andrea.309728

    4 years agoPosted 4 years ago. Direct link to andrea.309728's post “What would be the differe...”

    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)

    • kubleeka

      4 years agoPosted 4 years ago. Direct link to kubleeka's post “The SSS similarity criter...”

      Angle-angle triangle similarity criterion (article) | Khan Academy (6)

      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)

  • simonob1997

    a year agoPosted a year ago. Direct link to simonob1997's post “*1. How could you prove t...”

    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!

    (15 votes)

  • INKLING NOW

    4 years agoPosted 4 years ago. Direct link to INKLING NOW's post “Can I get help? I did all...”

    Can I get help? I did all the work but do not get it. Please help me!

    (7 votes)

    • loumast17

      4 years agoPosted 4 years ago. Direct link to loumast17's post “Cross multiply is a term ...”

      Angle-angle triangle similarity criterion (article) | Khan Academy (12)

      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.

      (11 votes)

  • Peanut butter Parker

    9 months agoPosted 9 months ago. Direct link to Peanut butter Parker's post “1. If a pair of triangles...”

    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.

    (8 votes)

  • riverajose524

    a year agoPosted a year ago. Direct link to riverajose524's post “I would prove two triangl...”

    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.

    (3 votes)

    • Jerry Nilsson

      a year agoPosted a year ago. Direct link to Jerry Nilsson's post “We aren't told to prove s...”

      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.

      (9 votes)

  • RN

    4 years agoPosted 4 years ago. Direct link to RN's post “My answer for the three p...”

    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)

  • maliha.tart

    a year agoPosted a year ago. Direct link to maliha.tart's post “How do I determine what i...”

    How do I determine what is the scale factor?

    (3 votes)

    • Zionel

      a year agoPosted a year ago. Direct link to Zionel's post “You may determine the sca...”

      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

      (5 votes)

  • tmthslzr

    9 months agoPosted 9 months ago. Direct link to tmthslzr's post “I am confused, nothing in...”

    I am confused, nothing in the three videos, "Intro to triangle similarity", "Triangle similarity postulates/crit...", and "Angle-angle triangle similarity cri..." mentioned dilations or transformations etc. Why are these questions following up those three videos? I was directed to this page, "Introduction to triangle similarity lesson(Opens in a new window)" from the "Getting ready for right triangles and trigonometry" page. I am feeling very confused to have the above set of questions out of the blue...

    (3 votes)

    • 🅗🅐🅝🅝🅐🅗 😜

      9 months agoPosted 9 months ago. Direct link to 🅗🅐🅝🅝🅐🅗 😜's post “Go to Unit 3 and check ou...”

      Go to Unit 3 and check out the lessons there, that's were most of the proofs are.
      I hope this helps. :)

      (3 votes)

  • jeylid

    2 years agoPosted 2 years ago. Direct link to jeylid's post “So is similarly determine...”

    So is similarly determined mainly by the <‘s if more then two are the same

    (3 votes)

Angle-angle triangle similarity criterion (article) | Khan Academy (2024)
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