Download 5 - Blue Valley Schools

Honors Geometry
Chapter 4
Section
Target
4.1
1
Triangle Sum Conjecture
4.2
1
Properties of Isosceles Triangles
Pg 208 1-10,18-21,24,25
4.3
1
Triangle Inequalities
Pg 218 1-17,19-21
4.4
2
Triangle Shortcuts
Pg 224 1-10,12-19
4.5
2
More Triangle Shortcuts
Pg 229 1-19,22
4.4/4.5
2/3
Triangle Proofs
4.4 & 4.5 WS
4.6
3
Corresponding Parts of Congruent
Triangles
Pg 233 1-9,12-15,18,22,23
4.7
3
Flow Chart Thinking
Pg 239 1-5,7-14
3
Congruence in Right Triangles
Congruence in Right Triangles WS
3
Proving Special Triangle
Conjectures
Pg245 1-7,12-14
4.8
Section Name
Notes
Date
Homework
Pg 203 2-9
Chapter 4 Learning Targets
1. The student will be able to know and apply side and angle properties of triangles.
2. The student will be able to recognize congruent triangles.
3. The student will be able to construct proofs for congruent triangles and their corresponding parts.
4.2 Properties of Isosceles Triangle
Honors Geometry
(C-17) Triangle Sum Conjecture – The sum of the measures of the angles of in every triangle is
______________.
Ex 1:
Ex 2:
Isosceles Triangle - An isosceles triangle has
at least two congruent sides.
(C-18) Isosceles Triangle Conjecture - If a triangle is isosceles, then base angles are _______ .
Ex 3:
Ex 4:
(C-19) Converse of the Isosceles Triangle Conjecture - If a triangle has two congruent angles,
then ________________ .
Ex 5:
Ex 6:
4.3 Triangle Inequalities
(C-20)
Honors Geometry
Triangle Inequality Conjecture - The sum of the lengths of any two sides of a
triangle is _____________ the length of the third side.
Ex 1: Are the following sides of a triangle possible?
a. 5 in, 7 in, 9in
b. 10 cm, 10 cm, 20 cm
c. 12 in, 24in, 2 ft
Ex 2: Two sides of a triangle are given. What is the range for the third side of the triangle?
a. 4 m & 7m
b. 10 mm & 14 mm
c. 20 yd & 15 yd
(C-21) Side-Angle Inequality Conjecture - In a triangle, if one side is longer than another
side, then the angle opposite the longer side is __________________ .
Put the unknown measures in order from least to greatest.
Ex 3:
Ex 5:
Ex 4:
If you extend one side of a triangle from one vertex, then you have constructed an exterior angle at
that vertex.
(C-22)
Triangle Exterior Angle Conjecture - The measure of an exterior angle of a
triangle is __________________________ .
Ex 6: Find the unknown angle
Ex 7: Find the unknown variables
x
x = ______
7y
y = ______
z = ______
z
31y
4.4 Triangle Shortcuts
Honors Geometry
What does it mean to have congruent triangles?
Is it necessary to prove all angles and all sides congruent in order to conclude the triangles are
congruent?
Today we will take a look at some possible shortcuts to prove triangles congruent.
C-23 If the three sides of one triangle are congruent to the three sides of another triangle, then
_____________________.
C-24 If two sides and the angle between them in one triangle are congruent to two sides and the angle
between them in another triangle, then _____________________.
C-25 If two sides and an angle that is not between the two sides in one triangle are congruent to two
sides and an angle that is not between the two sides in another triangle, then
_____________________.
Determine if the following triangles are congruent . If they are, name the congruent triangles and then
name why they are congruent.
D
1)
2)
A
3)
H
G
C
B
I
X
W
E
4)
Z
Y
5)
R
E
D
J
6)
L
O
F
S
T
U
H
M
G
N
4.5 More Triangle Shortcuts
Honors Geometry
C-25 If the two angles and the side between them in one triangle are congruent to
_____________________.
C-26 If two angles and a side that is not between them in one triangle are congruent to the
corresponding _____________________.
C-27 If the three angles of one triangle are congruent to _____________________.
Determine if the following triangles are congruent . If they are, name the congruent triangles and then
name why they are congruent.
1)
X
2)
R
3)
S
O
L
T
Y
N
M
4)
D
C
Q
Z
5)
P
W
FGHIJK is regular
X is the midpoint of KH
F
6)
DCEA is an isosceles
Trapezoid
C
D
K
B
G
A
B
J
X
E
A
H
I
E
4.4 and 4.5 Proofs
4.6 Corresponding Parts of Congruent Triangles
Honors Geometry
CPCTC – If two triangles are congruent, then _____________________________________
______________________________________________________________________ .
Determine whether the triangles are congruent, then state which conjecture (SSS, SAS, ASA, SAA) proves them
congruent.
1) m1 = m2
m3 = m4
2) ABDC
mA = mC
3) O is the midpoint of RQ
O is the midpoint of PS
R
A
Q
1
2
1 4
R
B
3
T
4
O
3 2
S
Is QR = QS?
Why?
S
C
D
Is AD = BC?
Why?
Given: mA = mB
AM = BM
Is mP = mS
Why?
B
C
2
1 M
Prove: AD = BC
A
P
D
Q
E
Given: AR = ER
EC = AC
Prove: mE = mA
R
C
A
A
Given: C is the midpoint of BD
AB  DE
Prove: AB = DE
B
C
D
E
4.7 Flow Chart Thinking
Honors Geometry
Prove: AD = BC
B
C
Given: mA = mB
AM = BM
2
1 M
D
A
E
Given: AR = ER
EC = AC
Prove: mE = mA
R
C
A
A
Given: C is the midpoint of BD
AB  DE
Prove: AB = DE
B
C
D
E
Congruence in Right Triangles
Honors Geometry
Hypotenuse-Leg (HL) Conjecture – If the hypotenuse and a leg of one right triangle are congruent to the
hypotenuse and a leg of another right triangle, then the triangles are congruent.
Ex 2: Using the HL
Conjecture
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
V
R
Ex 3: Using the HL Conjecture
Given: RV  RU and UT  RU ; S is midpoint of TV; RV = TU
S
Prove: mV  mT
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
7)
7)
T
U
Additional Ex:
For what values x and y are the triangles congruent by HL?
x+y
5x + 1
2x + y +3
2y - 4
What additional information you need to prove the triangles congruent with HL?
1.
2.
4.8 Proving Special Triangle Conjectures
Honors Geometry
(C-27) Vertex Angle Bisector Conjecture - In an isosceles triangle, the bisector of the vertex angle is also
_______________________ and _______________________________ .
Given:  ABC is isosceles AC  BC
CD is the bisector of C
Prove:  ADC   BDC
Given:  ABC is isosceles AC  BC
CD is the bisector of C
Prove: CD is an altitude
(C-28)Equilateral/Equiangular Triangle Conjecture - Every equilateral triangle is __________________,
and, conversely, every equiangular triangle is _________________________ .
This conjecture is a biconditional conjecture: Both the statement and its converse are true.
A triangle is equilateral if and only if it is equiangular.