Lesson 4.1Classifying Triangles
Today, you will learn to…* classify triangles by their sides and
angles* find measures in triangles
We classify triangles by their sides and angles.
SIDES ANGLESEquilateralIsoscelesScalene
EquiangularAcuteObtuseRight
A
BC
_____ is opposite A.CB
_____ is opposite B.AC
_____ is opposite C.AB
Identify the side opposite the given angle.
Theorem 4.1
Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is
________180°
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are
_________________.complementary
Theorem 4.2Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent
interior angles.
4
1
2 3
m1 + m2 = m4
4
1
2 3
m1 + m2 + m3 = 180˚ m4 + m3 = 180˚
m1 + m2m4
m1 + m2 = m4 Sum of
nonadjacent interior s
= ext.
Lesson 4.2Congruence and
Triangles
Today, you will learn to…* identify congruent figures and corresponding parts* prove that 2 triangles are congruent
Figures are congruent if and only if all pairs of
corresponding angles and sides are congruent.
Def. of Congruent Figures
Statement of CongruenceΔ ABC Δ XYZ vertices are written in corresponding order
AB
BC
AC
XY
YZ
XZ
A
B
C
X
Z
Y
XZ
YZ
XY
2. Find all missing measures.
ABC DEF
B E
F
DB
A C
35˚
108.2 5.7
8.2
10
5.755˚
35˚
55˚
?
?
?
?
?
?
?
3. In the diagram, ABCD KJHL. Find x and y.
A
C
L
K
J
H
B
D9 cm
(4x – 3) cm(3y)˚
85˚
93˚
75˚
x = 3 y = 254x-3 = 3y =9 75˚
70˚
Theorem 4.3
Third Angles TheoremIf 2 angles of one triangle are
congruent to 2 angles of another triangle, then…
the third angles are also
congruent.
D
O GC
A T
70˚
60˚
60˚
?
?
E
F
G
J
H
58°
58°
5. Decide if the triangles are congruent. Justify your reasoning.
ΔEFG Δ______J H G
Vertical Angles Theorem
Third Angles Theorem
W X
YZ
M
6.
1) WX YZ , WX | | YZ,M is the midpoint of WY and XZ2) 3)4)5) ΔWXM ΔYZM
1 2
3
4
5 6
1 6 2) Alt. Int. s Theorem
1) Given
3 4 3) Vertical Angles Th.WM MY and ZM MX4) Def. of midpoint
5) Def. of figures
and 2 5
7. Identify any figures you can prove congruent & write a congruence statement.
A B
CD
Reflexive Property Alt. Int. Th.
Third Angle Th.ACD C AB
Theorem 4.4
Properties of Congruent Triangles
Reflexive
Symmetric
Transitive
ABC ABC If ABC XYZ,
then XYZ ABCIf ABC XYZ
and XYZ MNO then ABC
MNO
Lesson 4.3Proving Triangles
are CongruentToday, you will learn to…* prove that triangles are congruent* use congruence postulates to solve
problems
SSS Experiment
Using 3 segments, can you ONLY create 2 triangles that are
congruent?
Side-Side-Side Congruence Postulate
X
Y
Z
A C
B
If Side AB XY Side AC XZ Side BC YZ,then ΔABC ΔXYZ
by SSS
If 3 pairs of sides are congruent, then the two triangles are congruent.
Given: LN NP and M is the midpoint of LPProve: ΔNLM ΔNPM
N
LM
P
2.
Def of midpointLM MPReflexive PropertyNM NM
NLM NPM SSS Congruence
3. Show that ΔNPM ΔDFE by SSS if N(-5,1), P (-1,6), M (-1,1), D (6,1), F (2,6), and E (2,1).
N
P
MD
F
E
NM = MP = NP =
DE = EF = DF =
45
45
41
41(- 5 – - 1)2 + (1 – 6)2(6 – 2)2 + (1 – 6)2
Using 2 congruent segments and 1 included angle, can you ONLY
create 2 triangles that are congruent?
SAS Experiment
Side-Angle-Side Congruence Postulate
If Side AB XY Angle B Y Side BC YZ,then ΔABC ΔXYZ
by SAS X
Y
Z
A C
B
If 2 pairs of sides and their included angle are congruent, then the two
triangles are congruent.
Given: W is the midpoint of VY and the midpoint of ZXProve: ΔVWZ ΔYWX
11.
X
Z
W
Y
V
VW WY and ZW WX Def. of midpointVWZ YWX Vertical Angles ThVWZ YWX SAS Congruence
12.Given: AB PB , MB APProve: ΔMBA ΔMBP
M
PBA
MB MB Reflexive Property
ABM & PBM are right s Def of lines
MBA MBP SAS Congruence
ABM PBM All right s are
Lesson 4.4Proving Triangles
are CongruentToday, you will learn to…* prove that triangles are congruent* use congruence postulates to solve
problems
Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent?
ASA Experiment
Angle-Side-Angle Congruence Postulate
If Angle B Y, Side BC YZ, Angle C Zthen ΔABC ΔXYZ
by ASA X
Y
Z
A C
B
If 2 pairs of angles and the included sides are congruent, then the two
triangles are congruent.
5. Does the diagram give enough info to use ASA Congruence?
AB C
DΔ ABD Δ by ASA
Third Angles Theorem
Reflexive Property
ACD
6. Does the diagram give enough info to use ASA Congruence?
A
B C
D
yes, Δ ACB ______ by ASA
Δ CAD
Alt. Int. Angles Theorem
Reflexive Property
9. Determine whether the triangles are congruent by ASA.
L K
GH
J
HJG _ _ _ by ASA
Vertical Angles Theorem
Alt. Int. Angles Theorem
KJL
Angle-Angle-Side Congruence Theorem
If Angle B Y Angle C Z Side AB XYthen ΔABC ΔXYZ
by AAS X
Y
Z
A C
B
If 2 pairs of angles and a pair of nonincluded sides are congruent, then
the two triangles are congruent.
14. Does the diagram give enough info to use AAS Congruence?
AB C
D
ABD _ _ _ by AAS ACDACDACD
Reflexive Property
16. Determine whether the triangles are congruent by AAS.
L K
GH
J
HJG _ _ _ by AAS
Vertical AnglesTheorem
Alt. Int. AnglesTheorem
K JL
SSA Experiment
Using 2 sides and 1 angle that is NOT included, can you ONLY create two
triangles that are congruent?
NO
AAA ExperimentUsing 3 angles, can you ONLY create
two triangles that are congruent?NO
All of the angles are , but the s are NOT
Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ? 17. CB ZY , AC XZ
SAS Congruence
C
A
B
X
YZ
C Z
Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ?18. CB ZY , AC
XZSSS Congruence
C
A
B
X
YZ
AB XY
Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ?19. CB ZY , C
ZSAA Congruence
C
A
B
X
YZ
A X
Lesson 4.5Corresponding Parts of Congruent Triangles are
CongruentToday, you will learn to…* use congruence postulates to solve
problems CPCTC
1. Given: AB || CD , BC || DA Prove: AB CD B C
DA1 2 Alt. Int. Angles Theorem, 3 4
1
2
3
4
Reflexive Property BD BD ASA ABD CDB
AB CD CPCTC
4. Given: A is the midpoint of MTA is the midpoint of SR Prove: MS || TR
M
S
A
T
R
Def. of midpointVertical Angles Theorem
MA AT
SAS SAM R AT
MS | | TR CPCTC
SAM RATSA AR
Alt. Int. Angles ConverseM T
Triangle Congruence?
SASSSA
AASASA
SSS AAA
2 angles & 1 side?
2 sides & 1 angle?
3 sides or 3 angles?
You can ONLY use CPCTC after you use one of these!
Lesson 4.6Isosceles, Equilateral, and Right Triangles
Today, you will learn to…* use properties of isosceles, equilateral, and right triangles
Students need rulers and protractors.
Theorem 4.6
Base Angles Theorem
If 2 sides of a triangle are congruent, then … the
angles opposite them are congruent.
Theorem 4.7
Base Angles ConverseIf 2 angles of a triangle are congruent, then the sides
opposite them are congruent.
Corollaries to Theorem 4.6/4.7(hint: don’t write these yet)
If a triangle is equilateral, then it is equiangular.
ANDIf a triangle is equiangular,
then it is equilateral.
A triangle is equilateral if and only if
it is equiangular.
Corollaries to Theorem 4.6/4.7
8. Find x.
A B
C
247x + 3 = 24
7x + 3x = 3
7x = 21 10x
– 6
10x – 6 = 7x + 3
3x = 910x - 6 = 24
10x = 30
ExperimentUsing a right angle,
a hypotenuse, and a leg, can you ONLY create 2 triangles
that are congruent?
Hypotenuse-Leg Congruence Theorem
X
Y
Z
A C
B
The triangles MUST be right triangles.
If Hyp BC YZ Leg AB XYthen ΔABC ΔXYZ
by HL
If the hypotenuse and a leg of two right triangles are congruent, then the two
triangles are congruent.
12. X is a midpoint. Does the diagram give enough info to use HL?
VWX _ _ _ by HL
V W
X
ZYYZX
Def. of midpoint
13. Does the diagram give enough info to use HL Congruence? W
X
Z
Y
YWX _ _ _ by HL YZX
Base Angles Converse
Reflexive Property