"To accomplish great things, we must dream as well as act." Anatole France
Bellwork – Write a theorem about the following
Justify your theorem
Leg-Leg Congruence
Leg Angle Congruence
Hypotenuse-Angle
Congruence
Hypotenuse-Angle
Congruence
"To accomplish great things, we must dream as well as act." Anatole France
Bellwork – Write a theorem about the following
Justify your theorem
Leg-Leg Congruence
If the legs of one right triangle are
congruent to the corresponding legs of
another right triangle, then the triangles
are congruent
Hypotenuseleg Congruence
If the hypotenuse and a leg of one right
triangle are congruent to the
hypotenuse and corresponding leg of
another right triangle, then the triangles
are congruent.
Leg Angle Congruence
If one leg and an acute angle of one
right triangle are congruent to the
corresponding leg and acute angle of
another right triangle, then the triangles
are congruent.
Hypotenuse-Angle
Congruence
If the hypotenuse and an acute angle of
one right triangle are congruent to the
hypotenuse and corresponding acute
angle of another right triangle then the
two triangles are congruent
Chapter 4.6 Isosceles Triangles
Objective: To understand and be able to
use the properties of isosceles and
equilateral triangles.
Check.4.10 Identify and apply properties and relationships of special
figures (e.g., isosceles and equilateral triangles, family of
quadrilaterals, polygons, and solids).
Spi.4.3 Identify, describe and/or apply the relationships and theorems
involving different types of triangles, quadrilaterals and other
polygons.
What are the 5 ways to prove Triangles
congruent
a. SSS, Side Side Side
b. SAS, Side Angle
Side
c. ASA, Angle Side
Angle
d. AAS, Angle Angle
Side
e. SSSAAA 3 sides, 3
angles
•
f. What does CPCTC
mean?
R
S
P
Given: PQR, PQRQ
Prove: P  R
Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
B
If AB CB then A  C
Q
Statement
1.
2.
3.
4.
5.
6.
7.
Reason
A
C
Let S be midpoint of PR
1. Every segment has one midpoint
Draw a segment SQ
2. Two points determine a line
If two angles of a triangle are congruent, then the sides opposite those angles
PScongruent.
RS
3. Midpoint
are
B Theorem
QS QS
4. Reflexive Property
If A  C then AB CB
PQRQ
5. Given
PQS RQS
6. SSS
P  R
7. CPCTC
A
C
Check.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of
quadrilaterals, polygons, and solids).
What about an equilateral triangle?
• If ABBCCA
• ABBC then A  C
B
• BCCA then A  B
Each Angle of an Equilateral Triangle = 60
C
A
B
A = 60
B
60
• =What
C = 60
•
•
•
•
does A, B and C equal?
A +B + C = 180
A
3 A = 180
A = 60,
B = 60, C = 60
C
K
Find the missing measure
G
H
If GHHK, HJJK and mGJK = 100
What is the measure of HGK?
KHJ + HKJ + KJH = 180
KHJ = HKJ, set equal to x
x+ x + 100 = 180
KHG + HGK + GKH = 180
2x = 180-100= 80
HGK = GKH, set to y
x = 40
140 + 2y = 180
KHJ = HKJ, = 40
2y = 40
KHG + KHJ = 180
y = 20
KHG = 140
HGK = GKH = 20
J
Find the missing measure
EFG is equilateral and EH bisects E
Find m1 and m2
E
m1 + m2 = 60
m1 = m 2 – bisector
m1 = 60/2 = 30
EFH + m1 + EHF = 180
60+ 30 + 15x = 180
15x = 90
x=6
1 2
15x
F
H
G
Find the missing measure
EFG is equilateral and EH bisects E, EJ bisects 2
Find HEJ and EJH and EJG
m1 + m2 = 60
m1 = m 2 – bisector
m1 = 60/2 = 30
EFH + m1 + EHF = 180
60+ 30 + 15x = 180
15x = 90
x=6
E
1 2
15x
F
H
J
G
HEJ = 1/2 m 2 – bisector
HEJ = ½ (30) =15
HEJ + EJH + JHE = 180
15 + EJH + 90= 180
EJH = 75
EJG = 105
Practice Assignment
• Page 287, 10 – 24 even
• Honors; page 288 16 – 42 even