Download 4 notes - Blackboard

Chapter 4:
Congruent Triangles
(page 116)
Congruent Figures
4-1:
(page 117)
CONGRUENT: figures having the same
and
.
A
X
example:
B
C
Y
Z
Two triangles are congruent if and only if their vertices can be matched up so that
corresponding parts (angles & sides) of the triangles are
.
Corresponding Vertices:
Corresponding Angles
Therefore, ∆ ABC ≅ ∆ XYZ
Corresponding Sides
… also,
,
,
,
,
Since congruent triangles have the same shape, their corresponding angles are
ANGLES determine the
.
Since congruent triangles have the same size, their corresponding sides are
SIDES determine the
.
.
Corresponding Parts of Congruent Triangles are Congruent
.
examples: Write congruence statements based on the given information.
(1)
∆ DEF ≅ ∆ TSR
(2)
(3)
ABCDE ≅ VWXYZ
(4)
D
O
S
G
O
R
C
C
E
A
K
T
SOCKER is a regular polygon
Assignment: Written Exercises, pages 120 & 121: 1-23 odd #’s
4-2:
Some Ways to Prove Triangles Congruent
is opposite ∠A and ∠B is opposite
.
A
∠A is the included angle between
&
.
AB is the included side between
&
.
B
Class Activity
(page41)
C
Draw, as accurately as possible, a triangle based on the following description.
(1)
∆ ABC, AB = 3 cm, BC = 5 cm,
and AC = 6 cm.
(2)
∆ DEF, DE = 3 cm, m∠E = 60º,
and EF = 4 cm.
(3)
∆ XYZ, m∠X = 30º, XY = 4 cm,
and m∠Y = 50º.
(4)
∆ UVW, m∠U = 30º, m∠V = 50º,
and m∠W = 100º.
NOTES:
(1)
If two triangles are congruent, then you know
also congruent.
(2)
Based on prior exercises,
that two triangles are congruent.
Postulate 12
pairs of corresponding parts are
pairs of congruent corresponding parts will guarantee
SSS Postulate
If three sides on one triangle are congruent to three sides of another triangle, then the
triangles are
Postulate 13
.
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the triangles are
Postulate 14
.
ASA Postulate
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
.
Complete the following proofs.
H
(1)
I
Given: HI || GJ ; HG || IJ
1
3
Prove: ∆ GHJ ≅ ∆ IJH
4
2
Proof:
G
J
Statements
Reasons
1.
1.
2. ∠ 1 ≅ ∠ 2
∠3≅ ∠4
2.
3.
3. Reflexive Property
4.
4.
O
(2)
Given: OK bisects ∠ MOT ; OM ≅ OT
1 2
Prove: ∆ MOK ≅ ∆ TOK
M
Proof:
T
K
Reasons
Statements
1.
1.
2. ∠ 1 ≅ ∠ 2
2.
3.
3. Reflexive Property
4.
4.
Assignment: Written Exercises, pages 124 & 126: 1-16 ALL #’s
Prepare for Quiz on Lessons 4-1 & 4-2
Using Congruent Triangles
4-3:
(page 127)
CPCTC means
.
Ways to Prove Triangles Congruent:
A Way to Prove Two Segments or Two Angles Congruent:
(1)
Identify 2 triangles in which the 2 segments or angles are corresponding parts.
(2)
Prove that the 2 triangles are congruent.
(3)
State that the 2 parts are congruent, using the reason ____ ____ ____ ____ ____ .
NOTE: It will be helpful to plan these proofs by reasoning backward.
K
(1)
R
Given: ∠1 ≅ ∠R
∠2 ≅ ∠N
1
M
MR ≅ MN
2
Prove: KR ≅ PN
N
P
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
(2)
Given: AB ≅ CB ; ∠1 ≅ ∠ 2
A
Prove: BD bisects AC
B
1
2
D
Proof:
C
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
T
(3)
Given: RQ ≅ QS ; RT ≅ TS
Prove: TQ ⊥ RS
Proof:
R
Statements
Q
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Assignment: Written Exercises, page 130 : 1 & 2
S
The Isosceles Triangle Theorems
4-4:
(page 134)
ISOSCELES TRIANGLE: a triangle with at least
sides congruent.
LEGS (of an isosceles triangle): the
sides.
BASE (of an isosceles triangle): the
side.
A
vertex angle:
legs:
&
base angles:
&
base:
B
THEOREM 4-1
C
THE ISOSCELES TRIANGLE THEOREM
If two sides of a triangle are congruent, then the angles
those sides are congruent.
(ie. Base angles of an isosceles triangle are congruent.)
A
Given: AB ≅ AC
Prove: ∠B ≅ ∠C
Proof:
B
C
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
Corollary 1
An equilateral triangle is also
.
Corollary 2
An equilateral triangle has three
Corollary 3
The bisector of the vertex angle of an isosceles triangle is
angles.
to the base at its midpoint.
example:
THEOREM 4-2
If two angles of a triangle are congruent, then the sides
those angles are congruent.
A
Given: ∠B ≅ ∠C
Prove: AB ≅ AC
Proof:
B
C
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
Corollary
An equiangular triangle is also
.
examples: Find the value of “x”.
(1)
40º
40º
3x-5
x+21
x=
(2)
xº
70º
x=
(3)
62º
xº
x=
Assignment: Written Exercises, pages 137 to 139: 1-10 ALL #’s
Other Methods of Proving Triangles Congruent
4-5:
THEOREM 4-3
(page 140)
AAS THEOREM
If two angles and a
side of one triangle are
congruent to the
parts of another triangle,
then the triangles are congruent.
A
Given: ∆ ABC & ∆ DEF
∠B ≅ ∠E
C
∠C ≅ ∠F
B
D
AC ≅ DF
Prove: ∆ ABC ≅ ∆ DEF
F
E
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
NOTE:
Two-column proofs may be shortened by writing them in
form which emphasize the
steps in the proof.
RIGHT TRIANGLE: a triangle with one
angle.
HYPOTENUSE: in a right triangle, the side opposite the
LEGS (of a right triangle): the other two
angle.
.
A
hypotenuse:
legs:
&
C
THEOREM 4-4
B
HL THEOREM
If the hypotenuse and a leg of one
triangle are congruent
to the corresponding parts of another
triangle, then the
triangles are congruent.
Given: ∆ ABC & ∆ DEF
B
E
∠C & ∠F are right angles
AB ! DE
BC ! EF
Prove: ∆ ABC ≅ ∆ DEF
A
Proof: “see paragraph form with key steps”
C
D
F
Ways to Prove ANY Two Triangles Congruent:
Postulate
Postulate
Postulate
Theorem
Ways to Prove Two RIGHT Triangles Congruent (Look at the Classroom Exercises on page 143, #14.):
Theorem
Method
Method
Method
examples:
State which congruence method(s) can be used to prove the triangles congruent.
If no method applies, write none.
(1)
(2)
________________
________________
(3)
(4)
Y
W
X
________________
Z
________________
(5)
(6)
A
P
C
D
E
∠A ≅ ∠C
AE = DC
B
Q
________________
R
S
________________
Assignment: CLASSROOM Exercises, page 142: 1 to 13 ALL #’s
Prepare for Quiz on Lessons 4-4 & 4-5
T
Using More than One Pair of Congruent Triangles
4-6:
Sometimes it is impossible to prove a pair of triangles congruent directly.
You may first need to prove another pair of triangles ______________.
Use the congruent corresponding parts to prove the original triangles ______________.
There is often more than one correct ________________.
This depends upon which triangles you choose to prove ________________ first.
NOTE:
Plan the proof by reasoning backward.
Use a paragraph proof that focuses on the key ideas or a key step proof.
Given: X is the midpoint of BD
AC ⊥
B
BD
A
C
X
Prove: ∠ABC ≅ ∠ADC
D
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
Assignment: Written Exercises, pages 148 & 149: 1 to 4 ALL #’s
(page 146)
4-7:
Medians, Altitudes, and Perpendicular Bisectors
MEDIAN (of a triangle): a segment from a vertex to the
B
A
of the opposite side.
B
C
A
Y
B
C
Y
X
(page 152)
A
C
Y
Z
X
Z
X
Z
Name given to point of intersection for the 3 medians:
ALTITUDE (of a triangle): the perpendicular segment from a
containing the opposite side.
B
A
B
C
Y
A
B
C
Y
X
to the line
Z
A
C
Y
X
Name given to point of intersection for the 3 altitudes:
Z
X
Z
PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment that is perpendicular
to the segment at its
.
B
A
B
C
Y
A
B
C
A
Y
X
Z
C
Y
X
Z
X
Z
Name given to point of intersection for the 3 perpendicular bisectors:
BISECTOR of an ANGLE: the ray that divides the angle into two
adjacent angles.
B
A
B
C
Y
A
B
C
A
Y
X
Z
C
Y
X
Z
Name given to point of intersection for the 3 angle bisectors:
X
Z
THEOREM 4-5
If a point lies on the perpendicular bisector of a segment, then the point is
from the endpoints of the segment.
Given: Line l is the perpendicular bisector of BC .
A is on l
A
Prove: AB = AC
B
X
C
l
Proof: To prove this theorem, the following triangles must be proven congruent …
∆ ___ ___ ___ ≅ ∆ ___ ___ ___ , by
THEOREM 4-6
.
If a point is equidistant from the endpoints of a segment, then the point
lies on the
bisector of the segment.
Given: AB = AC
Prove: A is on the perpendicular bisector of
BC .
A
B
C
Proof: To prove this theorem, the following triangles must be proven congruent.
∆ ___ ___ ___ ≅ ∆ ___ ___ ___ , by
Theorem 4-6 is the
A point is on the ⊥−-bisector of a segment
.
of Theorem 4-5 and can be combined into a biconditional.
it is equidistant from the endpoints of the segment.
DISTANCE from a POINT to a LINE (or plane): the length of the
segment from the point to the line (or plane).
R
t
THEOREM 4-7
If a point lies on the bisector of an angle, then the point is
from the sides of the angle.
!!!"
Given: BZ bisects !ABC
A
PX ! BA
X
PY ! BC
Z
P
Prove: PX = PY
B
Y
C
Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
THEOREM 4-8
If a point is equidistant from the sides of an angle, then the point lies on
the
of the angle.
Given: PX ! BA
A
PY ! BC
X
PX = PY
!!!"
P
Prove: BP bisects !ABC
B
Y
C
Key Step Proof:
Statements
Reasons
1.
1.
2.
2.
3.
3.
Theorem 4-8 is the
A point is on the bisector of an angle
of Theorem 4-7 and can be combined into a biconditional.
it is equidistant from the sides of the angle.
Assignment: Written Exercises, pages 156 & 157: 7 to 13 ALL #’s, 19
Prepare for Test on Chapter 4