Why Study Chapter 3?
Geometry
Knowledge of triangles is a key application for:
• Support beams
• Theater
• Kaleidoscopes
• Painting
• Car stereos
• Rug design
• Tile floors
• Gates
1
Geometry
•
•
•
•
•
•
•
•
•
Baseball field
Sign making
Architecture
Sailboats
Fire and lifeguard towers
Sports
Airplanes
Bicycles
Surveying
2
Geometry
•
•
•
•
•
•
Canyon
Snowboarding
Advertising (logos)
Kites
Chess
Stenciling
3
Section 3.1
Congruent Figures
Geometry
• Definitions
 Congruent: having the same size and shape.
 Congruent triangles: all pairs of corresponding parts are
congruent.
Corresponding parts
– If triangles ABC and DEF are congruent, then what parts
must match up? Refer to page 111 in text.
A
C
# ABC # DEF
B
D
F
4
E
Problem
Geometry
• Is ABC  FED ? Explain your answer.
• Refer to page 112 in the text.
A
C
D
B
F
5
E
Definitions
Geometry
• Plane: a two-dimensional figure usually
represented by a shape that looks like a wall or
floor even though the plane extends without end
• Polygon: a closed plane figure with the following
properties:
1) It is formed by three or more line segments
called sides.
2) Each side intersects exactly two sides, one at
each endpoint, so that no two sides with a
common endpoint are collinear.
6
A Triangle is a Polygon with Three Sides
Geometry
•Congruent Polygons  all pairs of
corresponding parts are congruent
7
Section 3.1
Congruent Figures
Geometry
• Reflection
 when a figure has a mirror line
 Refer to page 112 in text.
8
Section 3.1
Congruent Figures
Geometry
• Other Types of Correspondences
 Slide: where a copy of the figure has been shifted by some set
amount
 Refer to page 113 in text.
A
B
P
Q
C
E
R
D
T
ABCDE  PQRST
9
S
Section 3.1
Congruent Figures
Geometry
• Other Types of Correspondences
 Rotational: when the figure has been rotated around a common
point
 Refer to page 113 in text.
R
# RAT # BAY
T
A
Y
B
10
Section 3.1
Congruent Figures
Geometry
• These correspondences can be combined!
 Try a reflection and a rotation on the figure below.
11
Section 3.1
Congruent Figures
Geometry
• Reflexive property
 any segment or angle is congruent to itself.
Since the triangles overlap,
this angle is reflexive to
triangles!
12
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
• Introduction
 Included sides and angles
To be included means to be flanked by or trapped between
– Thus, the points of a line segment are included between
its endpoints.
– In a triangle, sides can be included by angles and angles
can be included by sides.
M
List the inclusions
in triangle MRZ.
Z
R
13
Refer to page 115
in text.
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
• Side Side Side (SSS) Postulate
 SSS: If three sides of one triangle are congruent to three sides of
another triangle then the triangles are congruent
A
C
D
B
F
14
E
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
W
Given: XW  XZ
WY  ZY
X
5
7
6
8
Prove: # XWY # XZY
Z
15
Y
V
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
Given:
FA  FB
F
AD  BE
12
B
A
G is the midpoint of DE
Prove: DFG  EFG
D
Statements
G
Reasons
16
E
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
A
Given: AB bisectsCD
CD bisectsAB
C
AC  BD
P
1
2
Prove: ACP  BDP
D
Statements
Reasons
17
B
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
• Angle Side Angle (ASA) Postulate
 ASA: If two angles and the included side of one triangle are
congruent to two angles and the included side of another triangle,
then the triangles are congruent.
A
C
D
B
F
18
E
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
W
Given: WYV  ZYV
XY bisects WXZ
X
Prove: XWY  XZY
5
7
6
8
Z
Statements
Reasons
19
Y
V
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
Given: PQ  AB
P
PQ Bisects APB
12
Prove: APQ  BPQ
A
Statements
Reasons
20
3 4
Q
B
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
W
Given: XY bisects WXZ
YX bisects WYZ
Prove: XWY  XZY
X
5
7
6
8
Z
Statements
Reasons
21
Y
V
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
Given:
FA  FB
F
AD  BE
m1  m2
mD  mE
12
B
A
Prove: DFG  EFG
D
Statements
Reasons
22
G
E
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
Given:
RS  XY
P
RS  PQ
1  4
R
5 6
Q
Prove: PRS  QRS
2 3
1
X
Statements
Reasons
23
4
S
Y
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
• Side Angle Side (SAS) Postulate
 SAS: If two sides and the included angle of one triangle are
congruent to two sides and the included angle of another triangle,
then the two triangles are congruent.
A
C
D
B
F
24
E
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
A
Given: AB bisects CD
CD bisects AB
C
P
Prove: ACP  BDP
1
2
D
Statements
Reasons
25
B
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
P
12
Given: PQ  AB
Q is the midpont of AB
Prove: APQ  BPQ
A
Statements
Reasons
26
3 4
Q
B
Section 3.2
3 Ways to Prove Triangles Congruent
Geometry
W
Given: WY  ZY
VYW  VYZ
X
5
7
6
8
Prove: XWY  XZY
Z
Statements
Reasons
27
Y
V
Section 3.3
Circles and CPCTC
Geometry
If you have proven # XWY # XZY via SSS can you
state that  5 is congruent to  7? WHY?
W
X
5
7
6
8
Y
V
Z
• CPCTC Principle:
 Corresponding Parts of Congruent Triangles are Congruent
You MUST prove the triangles congruent FIRST!!!
28
Section 3.3
Circles and CPCTC
Geometry
• Circle: The set of all points in a plane that are a
given distance from a given point in the plane.
.
.
.
P • Segment OP is a radius
• Segments OP and OR are radii
O
• Remember your formulas?
 Area of a circle
 Circumference of a circle
R • Theorem: All radii of
a circle are congruent.
A r 2
C  2 r
29
Section 3.4
Beyond CPCTC
Geometry
• Median: A line segment drawn
from any vertex of the triangle to
the midpoint of the opposite side.
How many medians does a triangle
have? (3)
• A median divides into two
congruent segments, or bisects
the side to which it is drawn.
30
Section 3.4
Beyond CPCTC
Geometry
• Altitude: A line segment drawn
from any vertex of the triangle to
the opposite side, extended if
necessary, and perpendicular to
that side.
How many altitudes does a triangle
have? (3)
• An altitude of a triangle forms
right (90º) angles with one of the
sides.
31
Section 3.4
Beyond CPCTC
Geometry
• Auxiliary Lines: A line introduced into a
diagram for the purpose of clarifying a proof.
A
R
U
C
B
D
S
T
• Postulate: Two points determine a line (or ray
or segment).
32
Solving Proofs: Remember the Steps
Geometry
• 1. Draw the diagram.
• 2. Carefully read the problem and mark the diagram.
• 3. Place a question mark (?) in the area that you
need to prove.
• 4. Create a flow diagram.
• 5. Use one given at a time and draw as much
information as possible from that given.
• (Disregard information that is not needed.)
• 6. Sequentially list the statements and reasons.
• 7. The last statement should be the prove statement.
• 8. Check the proof. It should follow a logical order.
33
Section 3.4
Practice Proof
Geometry
• Given: CFD  EFD
FD is an altitude
Prove:
FD is a median
C
D
F
E
34
Geometry
• Given:
O
GJ  HJ
• Prove: G  H
J
G
.
O
H
35
Geometry
• Given: TW is a median
ST = x + 40
SW = 2x + 30
WV = 5x – 6
• Find: SW, WV, and ST
T
S
V
W
36
Test Tomorrow
Geometry
• Study lessons 3.1 to 3.4 and class notes
• Define the following:
1. Reflexive Property
2. SSS, SAS, and ASA Postulates
3. CPCTC, radius, radii, and diameter
4. Formulas for the area and circumference
of a circle
5. Median, altitude, auxiliary lines
• Study PowerPoint slides 1-36
37
Section 3.5
Overlapping Triangles
Geometry
38
Section 3.6
Types of Triangles
Geometry
• Scalene Triangle: a triangle in which no two sides
are congruent.
39
Section 3.6
Types of Triangles
Geometry
• Isosceles Triangle: a triangle in which at least two
sides are congruent.
Congruent sides are called legs
Non-congruent side is called the base
Angles included between leg and base are called base angles
Vertex
Leg
Base
Base Angle
40
Section 3.6
Types of Triangles
Geometry
• Equilateral Triangle: a triangle in which all sides
are congruent.
41
Section 3.6
Types of Triangles
Geometry
• Equiangular Triangle: a triangle in which all
angles are congruent.
42
Section 3.7
Angle-Side Theorems
Geometry
• Theorem: If two sides of a triangle are
congruent, then the angles opposite the sides
are congruent.
• Theorem: If two angles of a triangle are
congruent, the sides opposite the angles are
congruent.
Ways to Prove that a Triangle is Isosceles:
1. If at least two sides of a triangle are congruent
2. If at least two angles of a triangle are congruent
43
Section 3.7
Angle-Side Theorems
Geometry
• Theorem: If two sides of a triangle are not
congruent, then the angles opposite them are
not congruent, and the larger angle is opposite
the longer side.
• Theorem: If two angles of a triangle are not
congruent, then the sides opposite them are not
congruent, and the longer side is opposite the
larger angle.
44
Section 3.8
The HL Postulate
Geometry
• HL Postulate: If there exists a correspondence
between the vertices of two right triangles such
that the hypotenuse and a leg of one triangle
are congruent to the corresponding parts of the
other triangle, the two right triangles are
congruent.
45