Unit 4 Congruent Triangles

1
Unit 4Congruent Triangles

• Identify the corresponding parts of congruent
  figures
• Prove two triangles are congruent
• Apply the theorems and corollaries about
  isosceles triangles

2
Lecture 1 (4-1)

• Objectives
• Identify the corresponding parts of congruent
  figures

3
Congruent Figures

• Exactly the same size and shape.

C
B
A
D
4
Definition of Congruency

• Two figures are congruent if corresponding
  vertices can be matched up so that
• 1. All corresponding sides are congruent
• 2. All corresponding angles are congruent.

5
?ABC ? ?DEF

• This is an abbreviated way to refer to the
  definition of congruency with respect to
  triangles.
• C orresponding
• P arts of
• C ongruent
• T riangles are
• C ongruent.

A
E
C
B
F
D
6
Lecture 2 (4-2)

• Objectives
• Learn about ways to prove triangles are congruent

7
Postulate 12 (SSS)

• If three sides of one triangle are congruent to
  the corresponding parts of another triangle, then
  the triangles are congruent.

E
B
C
D
A
F
8
Postulate 13 (SAS)

• If two sides and the included angle are congruent
  to the corresponding parts of another triangle,
  then the triangles are congruent.

E
B
C
D
A
F
9
Postulate 14 (ASA)

• If two angles and the included side of one
  triangle are congruent to the corresponding parts
  of another triangle, then the triangles are
  congruent.

E
B
C
D
A
F
10
Lecture 3 (4-3)

• Objectives
• Use congruent triangles to prove other things

11
?ABC ? ?DEF

• This is an abbreviated way to refer to the
  definition of congruency with respect to
  triangles.
• C orresponding
• P arts of
• C ongruent
• T riangles are
• C ongruent.

A
E
C
B
F
D
12
Lecture 4 (4-4)

• Objectives
• Apply the theorems and corollaries about
  isosceles triangles

13
Isosceles Triangle

• By definition, it is a triangle with two
  congruent sides.

Base
14
Theorem 4-1

• The base angles of an isosceles triangle are
  congruent.

B
A
C
X
15
The Corollaries

• An equilateral triangle is also equiangular.
• An equilateral triangle has angles that measure
  60.
• The bisector of the vertex angle of an isosceles
  triangle is the perpendicular bisector of the
  base.

See It!
16
Theorem 4-2

• If two angles of a triangle are congruent, then
  it is isosceles.

B
A
C
X
17
Lecture 5 (4-5)

• Objectives
• Learn two new ways to prove triangles are
  congruent

18
Proving Triangles ?

• We can already prove triangles are congruent by
  the ASA, SSS and SAS. There are two other ways
  to prove them congruent

19
Theorem 4-3 (AAS)

• If two angles and the not-included side of one
  triangle are congruent to the corresponding parts
  of another triangle, then the triangles are
  congruent.

E
B
C
D
A
F
20
The Right Triangle
hypotenuse
right angle
21
Theorem 4-4 (HL)

• If the hypotenuse and leg of one right triangle
  are congruent to the corresponding parts of
  another right triangle, then the triangles are
  congruent.

B
E
C
D
A
F
22
Five Ways to Prove ? ?s

• All Triangles
• ASA
• SSS
• SAS
• AAS
• Right Triangles Only
• HL

23
Lecture 6 (4-6)

• Objectives
• Construct a proof using more than one pair of
  congruent triangles.

24
More Than One Pair of ?s ?

• Given X is the midpt of AF CD
• Prove X is the midpt of BE

D
A
X
E
B
F
C
25
Lecture 7 (4-7)

• Objectives
• Define altitudes, medians and perpendicular
  bisectors.

26
Median of a Triangle

• A segment connecting a vertex to the midpoint of
  the opposite side.

27
Altitude of a Triangle

• A segment drawn a vertex perpendicular to the
  opposite side.

28
Perpendicular Bisector

• A segment (line or ray) that is perpendicular to
  and passes through the midpoint of another
  segment.

29
Angle Bisector

• A ray that cuts an angle into two congruent
  angles.

30
Theorem 4-5/6

• A point lies on the perpendicular bisector of a
  segment only if it is equidistant from the
  endpoints.

31
Theorem 4-7/8

• A point lies on the bisector of an angle only if
  it is equidistant from the sides of the angle.