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Lesson 6Similar and Congruent Triangles

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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.

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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.

A B

C

E D

F

45

45

80

80

55

55

.

ABC DEF

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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:

A B

C

E D

F

45

45

80

80

55

55

ABC DEF

.A D

.B E

.C F ACB DFE or BAC EDF or CAB FDE but not BCA DFE

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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

A B

C

P

T

D

110

110

30

30

40

40

.ACB PTD

180

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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.

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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.

A B

C

D E

.AB DE.ABC EDC

ABC EDC

BAC DEC

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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. A B

CD E

F.ABC DEF

AB AC BC

DE DF EF

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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:

A B

CD E

F,ABC DEF

68

7

12 9

10.5

AB AC BC

DE DF EF

12

7 8 6

AB BC

12 217 10.5

8 2AB 12

6 98

BC

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Example

• In the figure, are right angles,

and Find

• First note that since and since the triangles share angle C.

• Let x denote AB. Then: A B C

D

E

and DAC EBC 12,DA

9,EB 6.BC

12

6

9

x

.AB

DAC EBC DAC EBC

12 6 or

9 6

DA AC x

EB BC

12

So, 6 6 8. So, 2.9

x x

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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,

11,AC 12,BD 5.DP

A B

C D

P

.CP

and APC BPD CAP DBP

.CDAPC BPD

.APC BPD 11

So, or .12 5

AC CP CP

BD DP

11 55 7So, 5 4 .

12 12 12CP

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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.

A

B CD

E F

ABC.DEF .ABC DEF

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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.

A

B CD

E F

ABC DEF AB DE A D

AC DF B E

BC EF C F

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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.

ABCDEF

A

B C

D

E F

AB DEand AC DFand A D

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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.

A

B C

D

E F

ABC DEF

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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.

A

BC

D

EF

ABC DEF

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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.

A B

C

D E

F

ABC DEF

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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.

A B

C

D E

F

ABC DEF

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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.

A

BC

D

EF

ABC DEF

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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.

A

B C

ABC .AB ACA

.BCD.BC

D

.ABD ACD

AD AD

BD CD

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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

A

B

PM

and .PA PBPA PB

PM PM

APM BPM AMP BMP

90 .

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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.

A B

CD

and .AB CD AD CB and .AB CD AD CB

.ACBAC DCA

DAC BCA

.AC ACABC CDA

and AB CD AD CB