Download Similar Triangles

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
Triangles that are both the same size and same
shape are called congruent triangles.
B
E
A
D
F
C
The corresponding sides are congruent and
corresponding angles have equal measures.
Notation: ABC  DEF.
Slide 9-3-3
CONGRUENCE PROPERTIES - SAS
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.
Slide 9-3-4
CONGRUENCE PROPERTIES - ASA
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.
Slide 9-3-5
CONGRUENCE PROPERTIES - SSS
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.
Slide 9-3-6
EXAMPLE: PROVING CONGRUENCE
(SAS)
Given:
Prove:
C
CE = ED
AE = EB
ACE  BDE
A
B
E
D
Proof
STATEMENTS
REASONS
1. CE = ED
2. AE = EB
3. CEA  DEB
4. ACE  BDE
1.
2.
3.
4.
Given
Given
Vertical Angles are equal
SAS property
Slide 9-3-7
EXAMPLE: PROVING CONGRUENCE
(ASA)
Given:
Prove:
ADB  CBD
ABD  CDB
ADB  CDB
B
A
C
D
Proof
STATEMENTS
REASONS
1. ADB  CBD
2. ABD  CDB
3. DB = DB
4. ADB  CDB
1.
2.
3.
4.
Given
Given
Reflexive property
ASA property
Slide 9-3-8
EXAMPLE: PROVING CONGRUENCE
(SSS)
Given:
Prove:
B
AD = CD
AB = CB
ABD  CDB
A
Proof
STATEMENTS
REASONS
1.
2.
3.
4.
1.
2.
3.
4.
AD = CD
AB = CB
BD = BD
ABD  CDB
D
C
Given
Given
Reflexive property
SSS property
Slide 9-3-9
IMPORTANT STATEMENTS ABOUT
ISOSCELES TRIANGLES
If ∆ABC is an isosceles triangle with AB = CB,
and if D is the midpoint of the base AC, then
the following properties hold.
B
1. The base angles A and C are equal.
2. Angles ABD and CBD are equal.
3. Angles ADB and CDB are both
A
D
right angles.
C
Slide 9-3-10
SIMILAR TRIANGLES
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.
Slide 9-3-11
ANGLE-ANGLE (AA) SIMILARITY
PROPERTY
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.
Slide 9-3-12
EXAMPLE: FINDING SIDE LENGTH IN
SIMILAR TRIANGLES
ABC is similar to DEF.
Find the length of side DF.
8
F
B
32
Set up a proportion with
corresponding sides:
16
24
D
Solution
EF DF

BC AC
8 DF

16 32
E
C
A
Solving, we find that DF = 16.
Slide 9-3-13
GOUGO’S THEOREM
If the two legs of a right triangle have lengths a
and b, and the hypotenuse has length c, then
a 2  b2  c 2 .
That is, the sum of the squares of the lengths of
the legs is equal to the square of the hypotenuse.
leg a
hypotenuse c
leg b
Slide 9-3-14
EXAMPLE: USING THE PYTHAGOREAN
THEOREM
Find the length a in the right triangle below.
Solution
2
2
2
a b  c
39
a
36
a2  362  392
a2  1296  1521
2
a  225
a  15
Slide 9-3-15
CONVERSE OF THE PYTHAGOREAN
THEOREM
If the sides of lengths a, b, and c, where c is
the length of the longest side, and if
a b  c ,
then the triangle is a right triangle.
2
2
2
Slide 9-3-16
EXAMPLE: APPLYING THE CONVERSE
OF THE PYTHAGOREAN THEOREM
Is a triangle with sides of length 4, 7, and 8, a
right triangle?
Solution
?
4 7 8
2
2
2
?
16  49  64
65  64
No, it is not a right triangle.
Slide 9-3-17