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Transcript
Geometry Definitions, Postulates, and Theorems
Key
Chapter 4: Congruent Triangles
Section 4.1: Apply Triangle Sum Properties
Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems. 13.0 Students prove relationships between angles in polygons by using properties of
complementary, supplementary, vertical, and exterior angles.
B
Triangle – A polygon formed by three segments joining three non-collinear points.
A triangle can be classified by its sides and then by its angles.
A
C
*Classifying Triangles by Sides:
Scalene Triangle – A triangle with no congruent sides.
Vertex Angle
Isosceles Triangle – A triangle with at least two congruent sides.
Legs – The congruent sides of an isosceles triangle, when only
two sides are congruent.
Leg
Leg
Base Angle
Base
Base Angle
Base – The third side (non-congruent side) of an isosceles triangle.
Equilateral Triangle – A triangle with three congruent sides.
*Classifying Triangles by Angles:
Acute Triangle – A triangle with three acute angles.
45
50
85
Right Triangle – A triangle with one right angle.
Leg
Hypotenuse
Legs – The sides that form the right angle.
Hypotenuse – The side opposite the right angle.
Leg
Obtuse Triangle – A triangle with one obtuse angle.
100
30
50
Equiangular Triangle – A triangle with three congruent acute angles.
(over)
Ex. Classify the triangles by their sides and angles.
a)
b)
c)
5
3
120
4
Isosceles
Equilateral
Obtuse
Equiangular/Acute
Scalene
Right
Vertex (plural: vertices) – Each of the three points joining the sides of a triangle.
B
Adjacent Sides of an Angle – Two sides that share a common vertex.
A
C
Opposite Side from an Angle – The side that does not form the angle.
Interior angles – When the sides of a triangle are extended, additional angles
are formed. The original angles are the interior angles.
Exterior angles – When the sides of a triangle are extended, additional angles
are formed. The angles that form linear pairs with the interior
angles are the exterior angles.
***Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the three interior angles of a triangle is 180 .
B
85
350
0
C
m
A
E
670
D
700
B=
***Theorem 4.2 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two non-adjacent interior angles.
1
F
m
1=
Corollary To The Triangle Sum Theorem – The acute angles of a right triangle
are complementary (add up to equal 90 ).
Y
X
Z
(over)
Ex. A triangle has the given vertices. Graph the triangle
and classify it by its sides. Then determine if it is a
right triangle. Yes, right triangle
A( 5, 4), B (2, 6), C (4, 1)
B
A
C
Isosceles
Ex. Find the value of x and y.
Then classify the triangle by its angles.
Ex. Find the value of x.
Then classify the triangle by its angles.
x0
600
500
Linear Pair
y
x
Ex. Find the angle measures of the numbered angles.
620
1
2
3
1100
600
4
Ex. Find the values of x and y.
y0 x0
680
(2x-18)0
720
Section 4.2: Apply Congruence and Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
Two geometric figures are congruent if they have exactly the same size and shape, like placing one
figure perfectly onto another figure.
A
B
D
C
Y
Congruent Figures – All the parts of one figure are congruent to the
corresponding parts of the other figure.
In congruent polygons, the corresponding sides
and the corresponding angles are congruent.
V
X
Congruence Statements – When writing a congruence statement for
two polygons, always list the corresponding
vertices in the same order!
W
Ex. GIVEN: ABC
DEF
B
F
D
A
Corresponding Angles –
B
A
C
E
C
AB
Corresponding Sides –
BC
CA
Ex. Write a congruency statement.
T
S
Q
Ex. ABCD
L
9 cm
A 910
B
1130
R
P
JKHL. Find the value of x and y.
J
4x–3 cm
0
D 86
(5y–12)0 K
C
H
U
***Theorem 4.3 – Third Angle Theorem
IF two angles of one triangle are congruent to two angles of another
triangle, THEN, the third angles are also congruent.
Ex.
B
F
D
K
Ex.
G
55 J
50
50
A
C
E
F 55
I
H
(over)
***Theorem 4.4 – Properties of Congruent Triangles
Reflexive Property of Congruent Triangles –
For any triangle ABC: ABC
ABC.
Symmetric Property of Congruent Triangles –
IF ABC
DEF, then DEF
ABC.
Transitive Property of Congruent Triangles –
IF ABC
DEF, and DEF
JKL, then ABC
Ex. Find the values of x and y.
(8 x 2 y)0
(6 x y)0
290
1090
JKL.
Section 4.3: Prove Triangles Congruent by SSS
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
***Side-Side-Side (SSS) Congruence Postulate –
IF three sides of one triangle are congruent to three sides of a second
triangle, THEN the two triangles are congruent.
*Information about the angles is not needed.
B
E
A
C
If Side AB
D
F
,
Side BC
, and
Side CA
,
then
by
Ex. Is the congruence statement true?
Explain your reasoning.
WXY
.
Ex. Is the congruence statement true?
Explain your reasoning.
YZW
KJL
X
MJL
L
K
Y
W
M
J
Z
B
Ex. Write a proof.
Given: AD CD , AB CB
Prove: ABD
CBD
A
Statements
1. Given
2.
2. Reflexive
ABD
CBD
C
Reasons
1.
3.
D
3.
(over)
Structural Support – A diagonal support added to a figure helps make
the figure stable. The diagonal support forms
triangles with fixed side lengths. By the SSS
Congruence Postulate, these triangles cannot
change shape and so the figure is stable.
Ex. Determine whether the figure is stable. Explain your answer.
a)
b)
B
c)
X
Given: AE FC, BE BF, AB BC
Given: WX YX
Z is the midpoint of WY
Prove:
AFB
CEB
W
A
F
E
1.
AE
FC, BE
BF, AB
Z
Y
Prove:
WXZ
YXZ
C
BC
1. Given
1. WX YX
1. Given
Z is the midpoint of WY
2. EF = EF
2. ______________
2. _____________
2. ____________
3. AE + EF = AF
3. _____________
3. WZ ZY
3. _____________
EF + FC = EC
4. FC + EF = AF
_______________
4. Substitution
_______________
4. ______________ 4. ____________
5. AF = EC
5. ______________
6. ______________
6. Def. of Congruent
Segments
7. _______________
7. SSS
Section 4.4: Prove Triangles Congruent by SAS and HL
Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are
congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
***Side-Angle-Side (SAS) Congruence Postulate –
IF two sides and the included angle of one triangle are congruent
to two sides and the included angle of a second triangle,
THEN the two triangles are congruent.
B
E
If Side AB
Angle
A
C
D
F
,
A
, and
Side AC
,
then
by
.
Ex. Do you have enough information to prove the triangles are congruent by SAS?
a)
b)
A
Ex. Write a proof.
Given: AC EC , DC
ECD
Prove: ACB
D
C
BC
B
Statements
Reasons
1.
1. Given
2.
2. Vertical Angles are Congruent
3.
ACB
ECD
E
3.
(over)
Ex. Write a proof.
M
Given: AB PB , MB AP
Prove: MBA
MBP
A
Statements
Reasons
1.
1. Given
2.
2.
3.
MBA
MBP
P
lines form rights angles
3.
4.
5.
B
4. Reflexive Property
MBA
MBP
5.
Right Triangles: Legs – The sides adjacent to the right angle.
Hypotenuse – The side opposite the right angle.
***Hypotenuse-Leg (HL) Congruence Theorem –
IF the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of a second right triangle,
THEN the two triangles are congruent.
A
D
C
B
F
E
B
Ex. Write a proof.
Given: AB BC , BD
CBD
Prove: ABD
AC
A
Statements
Reasons
1.
1. Given
2.
2.
3.
BDA
BDC
4.
5.
lines form rights angles
3.
4. Reflexive Property
ABD
CBD
5.
D
C
Section 4.5: Prove Triangles Congruent by ASA and AAS
Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are
congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
***Angle-Side-Angle (ASA) Congruence Postulate –
IF two angles and the included side of one triangle are congruent
to two angles and the included side of a second triangle,
THEN the two triangles are congruent.
B
E
If Angle
A
Side AC
A
C
D
F
Angle
,
, and
C
,
then
by
.
***Angle-Angle-Side (AAS) Congruence Postulate –
IF two angles and a non-included side of one triangle are congruent
to two angles and the corresponding non-included side of a second triangle,
THEN the two triangles are congruent.
B
A
E
C
D
If Angle
A
,
Angle
C
, and
F
Side BC
,
then
by
.
Ex. Is it possible to prove the triangles are congruent? If so, state the postulate or theorem used.
a)
/
b)
/
c)
/
d)
/
(over)
X
Ex. Write a proof.
Given: WZ bisects XZY and
Prove: WZX
WZY
XWY
Z
W
Y
Statements
Reasons
1.
1. Given
2.
2. Definition of an angle bisector
3. ZW
4.
ZW
WZX
3.
WZY
4.
Ex. Write a proof.
Given:
Prove:
C
B, D
BDM
C FM
F , M is the midpoint of D F
Statements
Reasons
1.
1. Given
2.
2. Definition of a midpoint
3.
BDM
C
B
C FM
3.
D
M
F
Section 4.6: Use Congruent Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
***Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.) –
1. Prove two triangles are congruent with SSS, SAS, HL, ASA, or AAS.
2. Then, conclude that the corresponding parts of these congruent triangles
are congruent as well.
The triangles below are congruent by SAS. Since the triangles are congruent, we know that:
A
X
A
C
B
C
Y
Z
Ex. Write a proof.
AC
H
Given: HJ II LK , JK II HL
Prove: LHJ
JKL
J
L
Statements
K
Reasons
1.
1. Given
2.
2. Alternate Interior Angles are Congruent
3. JL
JL
3.
4.
LHJ
JKL
4.
5.
LHJ
JKL
5.
Ex. Write a proof.
M
Given: MS II TR , MS
TR
R
A
Prove: A is the midpoint of MT
S
Statements
T
Reasons
1.
1. Given
2.
2. Alternate Interior Angles are Congruent
3.
MAS
TAR
3.
4.
4. CPCTC
5. A is the midpoint of MT
5.
(over)
P
Ex. Write a proof.
Given: MP bisects
Prove: LP
LMN , LM
NM
NP
L
N
M
Statements
Reasons
1.
2.
1. Given
NMP
LMP
2.
3.
4.
3. Reflexive Property
NMP
5. LP
LMP
4.
NP
5.
Ex. Which triangles can you show are congruent in order to prove the statement? What postulate or
theorem would you use?
a)
A
b) SW
C
B
A
D
TY
S
C
W
X
Y
T
Section 4.7: Use Isosceles and Equilateral Triangles
Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles. 12.0 Students find and use measures of sides and of interior and exterior
angles of triangles and polygons to classify figures and solve problems.
Vertex Angle
Leg
Leg
Legs – The two congruent sides in an isosceles triangle.
Vertex Angle – The angle formed by the legs in an isosceles triangle.
Base – The third side of the isosceles triangle.
Base
Angle
Base
Base Angles – The two angles adjacent to the base in an isosceles triangle.
Base
Angle
***Theorem 4.7 – Base Angles Theorem
IF two sides of a triangle are congruent,
THEN the angles opposite them are congruent.
30
x
***Theorem 4.8 – Converse of Base Angles Theorem
IF two angles of a triangle are congruent,
THEN the sides opposite them are congruent.
2x-4
2x+2
x+5
Corollary To Base Angles Theorem
IF a triangle is equilateral,
THEN it is equiangular.
Corollary to the Converse of Base Angles Theorem
IF a triangle is equiangular,
THEN it is equilateral.
Section 4.8: Perform Congruence Transformations
Standards: 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations,
translations, and reflections.
Transformation – An operation that moves or changes a geometric figure in some way
to produce a new figure.
Image – The new figure produced.
*Three Types of Transformations:
A
B
Pre-Image
Image
B'
B
C
A
C'
A'
Translation – Moves every point of a figure the same distance in the same
direction.
B'
C A'
Translate down 6
Translate right 10
C'
Reflection – Uses a line of reflection to create a mirror image of the original figure.
Rotation – Turns a figure about a fixed point, called the center of rotation.
Rotate 90 degrees clockwise
Congruence Transformation – Changes the position of the figure without
changing its size or shape.
Translate A Figure In The Coordinate Plane – Moving an object a given
distance right or left and up or down.
y
*Coordinate Notation for a Translation
B
You can describe a translation by the notation
( x, y)
( x a, y b)
which shows that each point ( x, y) of a figure is
translated horizontally a units and vertically b units
Ex. Figure ABC D has the vertices
A( 4, 2), B ( 2, 5), C ( 1,1), and D ( 3, 1) .
Sketch ABC D and its image after the translation
( x, y) ( x 5, y 2) .
Right 5
Down 2
A
C
B'
A'
x
C'
D
D'
(over)
Usually
Reflect A Figure In The Coordinate Plane – The line of reflection is always
the x-axis or the y-axis.
y
*Coordinate Notation for a Reflection
Reflection in the x-axis: ( x, y) ( x, y)
Multiply the y-coordinate by -1.
Reflection in the y-axis: ( x, y) ( x, y)
Multiply the x-coordinate by -1.
x
Ex. Use a reflection in the x-axis to draw the other half
of the figure.
Rotate A Figure In The Coordinate Plane – The center of rotation is the origin.
The direction of rotation can be either clockwise or counterclockwise.
The angle of rotation is formed by rays drawn from the center of rotation through corresponding points
on the original figure and its image.
900 clockwise rotation
600 counterclockwise rotation
Ex. Graph PQ and RS . Tell whether RS is a rotation
y
P
of PQ about the origin. If so, give the angle and
direction of rotation.
a) P (2, 6), Q (5,1), R(6, 1), S (1, 2)
b) P (4, 2), Q (3, 3), R( 2, 4), S ( 3, 3)
Q
S
x
R