7.
Conjecture: mB = 2(mA)
B
D
C
A
From HW # 6
Given:
B
1.
AB  BC
BD  BE
30
x = 15
E
Find the measure of the angle marked x.
x
A
D
C
From HW # 6
x = 15
x
2.
63
A
63
B
104 41
C
D
63
A
63
B
104 41
C
D
63
A

D
B
104
C
63
A
41

D
B
104
76
C
63
D
14
C
D
A
E
F
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1. A  B (isosceles triangle theorem)
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1. A  B (isosceles triangle theorem)
2. AE  EB (definition of midpoint)
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1. A  B (isosceles triangle theorem)
2. AE  EB (definition of midpoint)
3. ADE and BFE are congruent right angles
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1.
2.
3.
4.
A  B (isosceles triangle theorem)
AE  EB (definition of midpoint)
ADE and BFE are congruent right angles
ADE  BFE (AAS)
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1.
2.
3.
4.
5.
A  B (isosceles triangle theorem)
AE  EB (definition of midpoint)
ADE and BFE are congruent right angles
ADE  BFE (AAS)
DE  EF and AD  BF (CPCTC)
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1.
2.
3.
4.
5.
6.
A  B (isosceles triangle theorem)
AE  EB (definition of midpoint)
ADE and BFE are congruent right angles
ADE  BFE (AAS)
DE  EF and AD  BF (CPCTC)
DEF is isosceles (definition of isosceles )
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1.
2.
3.
4.
5.
6.
7.
A  B (isosceles triangle theorem)
AE  EB (definition of midpoint)
ADE and BFE are congruent right angles
ADE  BFE (AAS)
DE  EF and AD  BF (CPCTC)
DEF is isosceles (definition of isosceles )
DC  FC (subtraction post, AC – AD = BC – BF)
C
D
A
F
E
B
From HW # 6
5. In the diagram, triangle ABC is isosceles with base AB ,
E is the midpoint of AB, and ED  AC, EF  BC .
Prove that triangle DEF and triangle CDF are isosceles.
Outline of proof:
1.
2.
3.
4.
5.
6.
7.
8.
A  B (isosceles triangle theorem)
AE  EB (definition of midpoint)
ADE and BFE are congruent right angles
ADE  BFE (AAS)
DE  EF and AD  BF (CPCTC)
DEF is isosceles (definition of isosceles )
DC  FC (subtraction post, AC – AD = BC – BF)
CDF is isosceles.
D
A
C
F
E
B
From HW # 6
6. Think about how you would prove that the altitudes to the legs
of an isosceles triangle are congruent.
B
D
A
E
C
7.
Conjecture: mB = 2(mA)
B
D
C
A
Use Geometer’s Sketchpad to construct quadrilateral ABCD
in which AB is parallel to CD and BC is parallel to AD.
C
B
D
A
Quadrilaterals
Definitions
C
A parallelogram is a quadrilateral with
both pairs of opposite sides parallel.
D
B
A rhombus is a parallelogram with one
pair of adjacent sides congruent.
A
C
B
C
D
A
D
A rectangle is a parallelogram with one
right angle.
B
C
A
D
A square is a rhombus and a rectangle.
B
A
bases
C
D
A trapezoid is a quadrilateral with
exactly one pair of sides parallel.
B
A
legs
C
An isosceles trapezoid is a trapezoid
with the two non-parallel sides
congruent.
B
Base angles
(there are two pairs)
D
A
Parallelogram
Exactly 1 pair of
parallel sides
2 pairs of parallel
sides
Exactly 1 pair of
congruent sides
2 pairs of
congruent sides
All sides
congruent
Perpendicular
diagonals
Congruent
diagonals
Diagonals bisect
angles
Diagonals bisect
each other
Opposite angles
congruent
Four right angles
Base angles
congruent
Rhombus
Rectangle
Square
Trapezoid
Isosceles
Trapezoid
HW #7
Fill in the chart (question 5) on the HW 7 handout.
Begin work on the rest of the problems.
You will have all period on Monday to complete it.