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1
GeometryWhy Study Chapter 3?
Knowledge of triangles is a key application for:• Support beams• Theater• Kaleidoscopes• Painting• Car stereos• Rug design• Tile floors• Gates
2
Geometry
• Baseball field• Sign making• Architecture• Sailboats• Fire and lifeguard towers• Sports • Airplanes• Bicycles• Surveying
3
Geometry
• Canyon• Snowboarding• Advertising (logos)• Kites• Chess• Stenciling
4
Geometry
Section 3.1Congruent Figures
• 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 B
D
F E
ABC DEF# #
5
GeometryProblem
• Is ? Explain your answer.• Refer to page 112 in the text.
ABC FED
A
C B
D
F E
6
GeometryDefinitions
• 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.
7
GeometryA Triangle is a Polygon with Three Sides
•Congruent Polygons all pairs of
corresponding parts are congruent
8
Geometry
Section 3.1Congruent Figures
• Reflection when a figure has a mirror line Refer to page 112 in text.
9
Geometry
Section 3.1Congruent Figures
• 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
C
DE
P Q
R
ST
ABCDE PQRST
10
Geometry
Section 3.1Congruent Figures
• Other Types of Correspondences Rotational: when the figure has been rotated around a common
point Refer to page 113 in text.
R
T
AY
B
RAT BAY# #
11
Geometry
Section 3.1Congruent Figures
• These correspondences can be combined! Try a reflection and a rotation on the figure below.
12
Geometry
Section 3.1Congruent Figures
• Reflexive property any segment or angle is congruent to itself.
Since the triangles overlap, this angle is reflexive to triangles!
13
Geometry
Section 3.23 Ways to Prove Triangles Congruent
• 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
RZ
List the inclusions in triangle MRZ.
Refer to page 115 in text.
14
Geometry
Section 3.23 Ways to Prove Triangles Congruent
• 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 B
D
F E
15
Geometry
Section 3.23 Ways to Prove Triangles Congruent
XW XZ
WY ZY
W
Z
Y VX
57
68
Given:
Prove: XWY XZY# #
16
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: FA FB
AD BE
G is the midpoint of DE
Prove: DFG EFG
DG
E
F
12
A B
Statements Reasons
17
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: AB bisects CD CD bisects AB AC BD Prove: ACP BDP
A C
B
1
2 P
D
Statements Reasons
18
Geometry
Section 3.23 Ways to Prove Triangles Congruent
• 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 B
D
F E
19
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: WYV ZYV
XY bisects WXZ
Prove: XWY XZY
W
Z
Y VX
57
68
Statements Reasons
20
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: PQ AB
PQ Bisects APB Prove: APQ BPQ
P
12
34A
QB
Statements Reasons
21
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: XY bisects WXZYX bisects WYZ
Prove: XWY XZY
W
Z
Y VX
57
68
Statements Reasons
22
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: FAFB
ADBE
m1 m2 mD mE Prove: DFG EFG
DG
E
F
12
A B
Statements Reasons
23
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: RS XY
RS PQ
1 4 Prove: PRS QRS
P R Q5 6
X S Y
1
23
4
Statements Reasons
24
Geometry
Section 3.23 Ways to Prove Triangles Congruent
• 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 B
D
F E
25
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: AB bisects CD CD bisects AB Prove: ACP BDP
A C
B
1
2 P
D
Statements Reasons
26
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given:
Prove: APQ BPQ
Statements Reasons
Q is the midpont of
PQ AB
AB
P
12
3 4A
QB
27
Geometry
Section 3.23 Ways to Prove Triangles Congruent
Given: Prove: XWY XZY
WY ZY
VYW VYZ
W
Z
Y VX
57
68
Statements Reasons
28
Geometry
Section 3.3Circles and CPCTC
If you have proven via SSS can you state that 5 is congruent to 7? WHY?
• CPCTC Principle: Corresponding Parts of Congruent Triangles are Congruent
You MUST prove the triangles congruent FIRST!!!
W
Z
Y VX
57
68
XWY XZY# #
29
Geometry
Section 3.3Circles and CPCTC
• Circle: The set of all points in a plane that are a given distance from a given point in the plane.
O. P
R.. • Segment OP is a radius
• Segments OP and OR are radii
• Remember your formulas? Area of a circle Circumference of a circle
2
2
A r
C r
• Theorem: All radii of a circle are congruent.
30
Geometry
Section 3.4Beyond CPCTC
• 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.
31
Geometry
Section 3.4Beyond CPCTC
• 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.
32
Geometry
Section 3.4Beyond CPCTC
• Auxiliary Lines: A line introduced into a diagram for the purpose of clarifying a proof.
• Postulate: Two points determine a line (or ray or segment).
A
C
B D
R
S
T
U
33
GeometrySolving Proofs: Remember the Steps
• 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.
34
Geometry
Section 3.4Practice Proof
• Given:
is an altitude
Prove:
is a median
CFD EFD
FD
FD
D
E
C
F
35
Geometry
• Given:
• Prove:
O
GJ HJ
G H
.O
G H
J
36
Geometry
• Given: is a median
ST = x + 40
SW = 2x + 30
WV = 5x – 6• Find: SW, WV, and ST
TW
S
T
VW
37
GeometryTest Tomorrow
• 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
38
Geometry
Section 3.5Overlapping Triangles
39
Geometry
Section 3.6Types of Triangles
• Scalene Triangle: a triangle in which no two sides are congruent.
40
Geometry
Section 3.6Types of Triangles
• Isosceles Triangle: a triangle in which at least two sides are congruent.
Congruent sides are called legsNon-congruent side is called the baseAngles included between leg and base are called base angles
Leg
Base Angle
Vertex
Base
41
Geometry
Section 3.6Types of Triangles
• Equilateral Triangle: a triangle in which all sides are congruent.
42
Geometry
Section 3.6Types of Triangles
• Equiangular Triangle: a triangle in which all angles are congruent.
43
Geometry
Section 3.7Angle-Side Theorems
• 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
44
Geometry
Section 3.7Angle-Side Theorems
• 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.
45
Geometry
Section 3.8The HL Postulate
• 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.