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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. - You can think of similar triangles as one

triangle being a magnification of the other.

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.

- Lets 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

- 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

- 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

- 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

- Suppose
- Then the sides of the triangles are proportional,

which means - Notice that each ratio consists of corresponding

segments.

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

8

6

7

12

9

10.5

Example

D

- In the figure, are right

angles, and

Find - First note that since

and since the triangles share

angle C. - Let x denote AB. Then

E

12

9

x

C

6

A

B

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 were

interested - Note that since they

intercept the same arc - Also, because they are

vertical. So,

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

- 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

- 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.

B

C

A

E

F

D

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.

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.

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.

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.

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.

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

C

B

D

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

P

M

B

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.

C

D

A

B

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