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Lesson 6. Similar and Congruent Triangles. Definition of Similar Triangles. Two triangles are called similar if they both have the same three angle measurements. The two triangles shown are similar. Similar triangles have the same shape but possibly different sizes.

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lesson 6

Lesson 6

Similar and Congruent Triangles

definition of similar triangles
Definition of Similar Triangles
  • Two triangles are called similar if they both have the same three angle measurements.
  • The two triangles shown are similar.
  • Similar triangles have the same shape but possibly different sizes.
  • You can think of similar triangles as one triangle being a magnification of the other.
similar triangle notation

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Similar Triangle Notation
  • The two triangles shown are similar because they have the same three angle measures.
  • The symbol for similarity is Here we write:
  • The order of the letters is important: corresponding letters should name congruent angles.
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  • Let’s stress the order of the letters again. When we write note that the first letters are A and D, and The second letters are B and E, and The third letters are C and F, and We can also write:
proving triangles similar

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Proving Triangles Similar
  • To prove that two triangles are similar you only have to show that two pairs of angles have the same measure.
  • In the figure,
  • The reason for this is that the unmarked angles are forced to have the same measure because the three angles of any triangle always add up to
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When trying to show that two triangles are similar, there are some standard ways of establishing that a pair of angles (one from each triangle) have the same measure:
  • They may be given to be congruent.
  • They may be vertical angles.
  • They may be the same angle (sometimes two triangles share an angle).
  • They may be a special pair of angles (like alternate interior angles) related to parallel lines.
  • They may be in the same triangle opposite congruent sides.
  • There are numerous other ways of establishing a congruent pair of angles.
example

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Example
  • In the figure,
  • Show that
  • First, note that because these are alternate interior angles.
  • Also, because these are alternate interior angles too.
  • This is enough to show the triangles are similar, but notice the remaining pair of angles are vertical.
proportions from similar triangles

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Proportions from Similar Triangles
  • Suppose
  • Then the sides of the triangles are proportional, which means:
  • Notice that each ratio consists of corresponding segments.
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Example
  • Given that if the sides of the triangles are as marked in the figure, find the missing sides.
  • First, we write:
  • Then fill in the values:
  • Then:

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Example

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  • In the figure, are right angles, and Find
  • First note that since and since the triangles share angle C.
  • Let x denote AB. Then:

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Example
  • In the figure, and Find
  • There are a lot of triangles in the figure. We should select two that seem similar and whose sides involve the segments in which we’re interested:
  • Note that since they intercept the same arc
  • Also, because they are vertical. So,
definition of congruent triangles

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Definition of Congruent Triangles
  • Two triangles are congruent if one can be placed on top of the other for a perfect match (they have the same size and shape).
  • In the figure, is congruent to In symbols:
  • Just as with similar triangles, it is important to get the letters in the correct order. For example, since A and D come first, we are saying that when the triangles are made to coincide, A and D will coincide.
cpctc

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CPCTC
  • Corresponding parts of congruent triangles are congruent (CPCTC).
  • What this means is that if then:
  • Other corresponding “parts” (like medians) are also congruent.
proving triangles congruent
Proving Triangles Congruent
  • To prove that two triangles are congruent it is only necessary to show that some corresponding parts are congruent.
  • For example, suppose that in and in that
  • Then intuition tells us that the remaining sides must be congruent, and…
  • The triangles themselves must be congruent.

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SAS
  • In two triangles, if one pair of sides are congruent, another pair of sides are congruent, and the pair of angles in between the pairs of congruent sides are congruent, then the triangles are congruent.
  • For example, in the figure, if the corresponding parts are congruent as marked, then
  • We cite “Side-Angle-Side (SAS)” as the reason these triangles are congruent.
slide16

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SSS
  • In two triangles, if all three pairs of corresponding sides are congruent then the triangles are congruent.
  • For example, in the figure, if the corresponding sides are congruent as marked, then
  • We cite “side-side-side (SSS)” as the reason why these triangles are congruent.
slide17

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ASA
  • In two triangles, if one pair of angles are congruent, another pair of angles are congruent, and the pair of sides in between the pairs of congruent angles are congruent, then the triangles are congruent.
  • For example, in the figure, if the corresponding parts are congruent as marked, then
  • We cite “angle-side-angle (ASA)” as the reason the triangles are congruent.
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AAS
  • In two triangles, if one pair of angles are congruent, another pair of angles are congruent, and a pair of sides not between the two angles are congruent, then the triangles are congruent.
  • For example, in the figure, if the corresponding parts are congruent as marked, then
  • We cite “angle-angle-side (AAS)” as the reason the triangles are congruent.
slide19

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HL
  • In two right triangles, if one pair of legs are congruent and the hypotenuses are congruent, then the triangles are congruent.
  • For example, in the figure, if the corresponding parts are congruent as marked, then
  • We cite “hypotenuse-leg (HL)” as the reason why these triangles are congruent.
example20
Example
  • is isosceles with
  • Prove that the angle bisector of bisects
  • Draw the angle bisector and let denote the point where it intersects
  • We first show that
  • We already have one pair of sides congruent and one pair of angles congruent as marked in the figure.
  • Note also that (the two triangles share this segment). So, the triangles are congruent by SAS.
  • So, by CPCTC.

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example21
Example
  • In the figure, a line segment is drawn from the center of the circle to the midpoint of a chord. Prove that this line segment is also perpendicular to the chord.
  • First, draw
  • Note that because they are both radii.
  • Also, (this side is shared by both triangles).
  • So, by SSS.
  • So, by CPCTC.
  • So, since these angles are supplementary they have to each measure

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example22
Example
  • In the figure,
  • Prove that
  • Draw
  • Note that because these are alternate interior angles.
  • Note that because these are alternate interior angles.
  • Note that
  • So, by ASA.
  • So, by CPCTC.

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