Download 4-1 Classifying Triangles

4-1 Classifying Triangles
Objective: Classify triangles by their angle measures and side lengths.
Use triangle classification to find angle measures and side lengths.
We can classify triangles by Angle Measures
and by Side Lengths.
By Angles:
By Sides:
Acute: If ALL angles are less than 90
Scalene: If NO side lengths are congruent
Obtuse: If ONE angle is more than 90
Isosceles:
If 2 sides or 3 lengths are congruent
Equilateral: If ALL sides lengths are congruent
Right:
If ONE angle is equal to 90
Equiangular: If ALL angles are equal
Use the figures to classify each triangle by its angle measures.
1.
∆ADC-Right Triangle
∆ADB-Acute Triangle
∆BDC-Obtuse Triangle
2.
∆EGH-Right
∆EFH-Obtuse
∆FGH-Equiangular
Use the figures to classify each triangle by its side lengths.
.
3.
∆EGH-Scalene
∆EFH-Isosceles
∆FGH-Scalene
4.
∆ABC-Equilateral
∆ADB-Scalene
∆ADC-Scalene
Find the side lengths of ∆JKL.
5.
X = 8.5
JK =23.3
JL=44.5
KL =23.3
Find the lengths side lengths of equilateral ∆FGH.
6.
y= ___7_______
GF = ____17______
FH = ____17______ GH = __17_____
___
7.
Total =18+18+18=54 420/54= 7
4-2 Angles of Triangles
Find the measure of interior and exterior angles of triangles.
Apply theorems about the interior and exterior angles of triangles.
Objective:
Triangle Sum Theorem: The sum of the angle measures of a triangle is 180
Third Angles Theorem:
If two angles of one triangle is congruent to two angles in another triangle, then the third pair of angles are congruent.
Exterior Angle: An angle formed by
2
1 side of the triangle and the extension of an adjacent side.
5 1
Interior Angle: An angle formed by two sides of a triangle
3
4
Remote Interior Angle:
An interior angle that is not adjacent to the exterior angle
Ext. Angle = Remote int. angle + Remote int. angle COMPLEMENTARY 60 DEGREES
Angle 5 has two Remote interior angles: 2 and 3; What is Angle 4 ?
Remote inter angles?
*The measure of each angle of an equiangular triangle is 60 degrees
Exterior Angle Theorem: Ext. Angle = Remote int. angle + Remote int. angle
*Acute angles of a right triangle are Complementary
Exercises: Find the numbered angles in each picture.
1)
= 120
*Triangle Sum Theorem
4)
109
29
71
2)
= 115
3)
*Exterior Angle Theorem
5)
= 46
=60
=120
*Both Theorems
= 46
= 84
MORE EXAMPLES FOR YOU TO TRY
Exercises: Find the numbered angles in each picture.
6)
7)
= 27
= 27
9)
= 125
= 55
= 95
= 55
=70
=
=
=
10)
11)
60
A
B
A
12)
C
D
B
80
80
J
K
F
141
G
H
I
4-3 Triangle Congruence
Use properties of congruent triangles.
Prove triangles are congruent by using the definition of congruence.
C
D
Objective:
Congruent Triangles:
ORDER MATTERS!!!
Example I: Name all the congruent, corresponding parts of
CAT
DOG .
Example II: Polygon LMNP
5) Given
7)
and
, find
.
.
.
and
.
Find x and m<DBC
x=
m<DBC=
BAC
DOG, given
polygon
Name the congruent and corresponding parts of each triangle.
1)
2)
3)
.
.
.
.
.
4. Given
If
CAT and
6) Given
Find x and m<F
x=
m<F=
8) ACB
.
9) JKM
10) PQR
Proof:
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
4-4 Proving Congruence
Apply SSS and SAS to construct triangles and to solve problems.
Prove triangles are congruent by using SSS and SAS.
Triangle Rigidity:
SSS:
Objective:
SAS:
Included Angle:
1) Refer to the figure.
a) Name the sides of
included angle.
LMN for which
L is the
b) Name the sides of LMN for which N is the included angle.
c) Name the sides of LMN for which M is the included angle.
Examples: Determine what type of congruence for each triangle. If no congruence, write
“no congruence.”
2)
3)
4)
Examples: Determine what type of congruence for each triangle. If no congruence, write
“no congruence.”
6)
5)
7)
9)
8)
11)
10)
12)
13)
Show that the triangles are congruent for the given value of the variable
14) ∆MNO
CPCTC -
∆PQR, when x = 5
15)∆STU
∆VWX, when y = 4
17) Given: T is the midpoint of SV
16)
Given: BCAD , BCAD
Prove: ABDCDB
PROVE THE FOLLOWING!
T is the midpoint of RU
Prove: ∆RTS ∆UTV
17)
19) Given:
ABCB
DBbits sec
Prove:
ABC
ADCD
4-5 Proving Congruence
Objective: Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove
triangles are congruent by using ASA, AAS, and HL.
AAS:
ASA:
Included Side:
HL:
Examples: Determine what type of congruence for each triangle. If no congruence,
write not possible.
Some Important Things You Should Remember:
Things you can assume in a diagram
collinear
coplanar
straight angle
Angle Addition Postulate
adjacent angles
Things you cannot assume in a diagram
congruent segments
congruent angles
angle bisector
perpendicular or right angle
midpoint
parallel
* You can draw conclusions about position but not size! *
Draw and label ABC and DEF. Indicate which additional pair of corresponding parts
needs to be congruent for the triangles to be congruent by the AAS Theorem.
1)
2)
3)
5)
6)
8)
9)
4)
7)
10)
11)
Directions: What corresponding parts must be congruent in order to prove that the
triangles are congruent by the ASA Postulate? Write the congruence statement.
13)
12)
14)
15)
17)
16)
18)
BCAD
AD
Statements
Reasons
Statements
Reasons
19)
20)
Statements
RTEF
Reasons
Statements
Reasons
Statements
Reasons
21)
22)
23)
Prove:
Statements
Reasons
4-4 to 4-6 Overlapping Triangles
Triangles sometimes overlap. You may find it helpful to
redraw the two triangles separately
to better visualize the congruent pieces.
Examples: Draw out the two overlapping triangles. Determine what type of
congruence for each pair of overlapping triangles. If no congruence, write not
possible.
Y and W are right angles.
1) Given:
WXZ
YZX
Y
W
XZ
2) Given:
A
C
D
E
B
3) Given: QP TP
P
Q
R
S T
Examples: Name a pair of overlapping congruent triangles. State which congruence method
can be used to prove the triangles congruent. If no triangles can be proved congruent, write none.
It may help to draw out the two overlapping triangles separately.
X
W
XZ YW XA YA
4) Given:
5)
M
X YX Y
Z
A
Y
A
D
6) Given: AC DB
AB BC DC BC ;
;
BC
J K
7) Given:
Prove:
JAB
KBA
KAB
JBA
J K
Statements
1
AB
Reasons
1)
2)
1)
3)
2)
3)
4)
4)
JAB
KBA
KAB
JBA
JAB
KBA
AB AB
CPCTC
JK
ASA Postulate
Reflexive Property
Given
4-7 Intro to Coordinate Proof
Coordinate Proof:
Tips for positioning figures in the coordinate plane!
* Use the origin as a vertex, keeping the figure in Quad 1
* Center the figure at the origin
* Center a side of the figure at the origin
*Use one or both axes as sides of the figure
Find the missing coordinates of each triangle.
1)
3)
2)
Choose from
the
following
statements and
reasons:
6)
5)
7)
8)
Example: Position each figure in the coordinate plane.
9) A square with a side length of 6 units
10) A right triangle with side
lengths of 2 and 4 units
Given the vertices, classify a triangle by its side lengths,
and determine if it is a right triangle.
F (3, 4) A (6,
6) N (7, -2)
-To classify a
by its sides, use the
_____________________________
- To determine if a is a right
triangle, calculate ________of the
_______
Review:
What are the five ways to prove triangles congruent?
Does CPCTC go before or after two triangles are proved
congruent?
4-8 Isosceles and Equilateral Triangles
Objective: Prove theorems about isosceles and equilateral triangles.
Apply properties of isosceles and equilateral triangles.
A
Legs:
jk
Fjf
Base:
Vertex Angle:
Base Angles:
B
C
Isosceles Triangle Theorem:
Converse of Isosceles Triangle Theorem:
Exercises: Refer to the figure.
11.
12.
13.
14.
15.
16.
*A triangle is equiangular if and only if it is equilateral.*
Examples: Find x.
Statements
Reasons
17.