SECTION 9-3

• The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem

Slide 9-3-1

THE GEOMETRY OF TRIANGLES: CONGRUENCE, SIMILARITY, AND THE PYTHAGOREAN THEOREM

• Congruent Triangles• Similar Triangles • The Pythagorean Theorem

Slide 9-3-2

CONGRUENT TRIANGLES

Slide 9-3-3

A

B

C

Triangles that are both the same size and same shape are called congruent triangles.

D

E

FThe corresponding sides are congruent and corresponding angles have equal measures. Notation: .ABC DEF

CONGRUENCE PROPERTIES - SAS

Slide 9-3-4

Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

CONGRUENCE PROPERTIES - ASA

Slide 9-3-5

Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

CONGRUENCE PROPERTIES - SSS

Slide 9-3-6

Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

EXAMPLE: PROVING CONGRUENCE (SAS)

Slide 9-3-7

Given: CE = EDAE = EB

Prove: ACE BDE

STATEMENTS REASONS

1. CE = ED 1. Given

2. AE = EB 2. Given

3. 3. Vertical Angles are equal

4. 4. SAS property

A

B

D

E

C

ACE BDE

Proof

CEA DEB

EXAMPLE: PROVING CONGRUENCE (ASA)

Slide 9-3-8

Given:

Prove: ADB CDB

STATEMENTS REASONS

1. 1. Given

2. 2. Given

3. DB = DB 3. Reflexive property

4. 4. ASA property

A

B

D

C

Proof

ADB CBD ABD CDB

ADB CDB

ADB CBD ABD CDB

EXAMPLE: PROVING CONGRUENCE (SSS)

Slide 9-3-9

Given: AD = CDAB = CB

Prove: ABD CDB

STATEMENTS REASONS

1. AD = CD 1. Given

2. AB = CB 2. Given

3. BD = BD 3. Reflexive property

4. 4. SSS property

A

B

D C

ABD CDB

Proof

IMPORTANT STATEMENTS ABOUT ISOSCELES TRIANGLES

Slide 9-3-10

If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold.

1. The base angles A and C are equal.

2. Angles ABD and CBD are equal.

3. Angles ADB and CDB are both right angles. A C

B

D

SIMILAR TRIANGLES

Slide 9-3-11

Similar Triangles are pairs of triangles that are exactly the same shape, but not necessarily the same size. The following conditions must hold.

1. Corresponding angles must have the same measure.

2. The ratios of the corresponding sides must be constant; that is, the corresponding sides are proportional.

ANGLE-ANGLE (AA) SIMILARITY PROPERTY

Slide 9-3-12

If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar.

EXAMPLE: FINDING SIDE LENGTH IN SIMILAR TRIANGLES

Slide 9-3-13

is similar to .ABC DEF

Find the length of side DF.

A

B

CD

E

F

Solution

16

24

32

8

Set up a proportion with corresponding sides:

EF DF

BC AC

8

16 32

DF Solving, we find that DF = 16.

GOUGO’S THEOREM

Slide 9-3-14

If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then

That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.

2 2 2.a b c

leg a

leg b

hypotenuse c

EXAMPLE: USING THE PYTHAGOREAN THEOREM

Slide 9-3-15

Find the length a in the right triangle below.

Solution2 2 2a b c

39

36

a

2 2 236 39a 2 1296 1521a

2 225a 15a

CONVERSE OF THE PYTHAGOREAN THEOREM

Slide 9-3-16

If the sides of lengths a, b, and c, where c is the length of the longest side, and if

then the triangle is a right triangle.

2 2 2 ,a b c

EXAMPLE: APPLYING THE CONVERSE OF THE PYTHAGOREAN THEOREM

Slide 9-3-17

Is a triangle with sides of length 4, 7, and 8, a right triangle?

Solution2 2 24 7 8 ?

16 49 64 ?

65 64No, it is not a right triangle.