Chapter 4 Notes

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Chapter 4 Notes
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4.1 Triangles and Angles
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A Triangle ? Three segments joining three
noncollinear points. Each point is a VERTEX of
the triangle. Segments are SIDES!
B
A
C
Equilateral All sides congruent
Isosceles At LEAST 2 congruent sides
Scalene No congruent sides
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Acute 3 acute angles
Obtuse One obtuse angle
Right One right angle
Equiangular all angles congruent
5
B
C
A
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TRIANGLE SUM THEOREM The sum of the measures of
the angles of a triangle is 180.
2
3
1
7
Exterior Angles Theorem The measure of an
exterior angle of a triangle equals the sum of
the two remote interior angles. (remote means
nonadjacent)
Statement Reason
3
4
2
1
8
Corollary to triangle sum theorem Acute angles
of a right triangle are complementary.
All angles 180, if one is 90, the other two add
up to 90, and are complementary
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4.2 Congruence and Triangles
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When TWO POLYGONS have the same size and shape,
they are called CONGRUENT! Their vertices and
sides must all match up to be congruent. When
two figures are congruent, their corresponding
sides and corresponding angles are congruent.
Identical twins!
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Name all the corresponding parts and sides, then
make a congruence statement.
If you notice, the way you name the triangle is
important, all the CORRESPONDING SIDES must line
up!
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3rd Angles Theorem If two angles of one
triangle are congruent to two angles of another
triangle, then the 3rd angles are congruent.
B
E
C
A
D
F
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E
A
G
M
H
T
Y
O
Find x
Find y
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• Note, triangles also have the following
  properties of congruent Reflexive, symmetric,
  and transitive.

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4.3 4.4 Proving Triangles are Congruent
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SSS Congruence Postulate If three sides of one
triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
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SAS Congruence Postulate If two sides and the
included angle of one triangle are congruent to
two sides and the included angle of a second
triangle, then the two triangles are congruent.
Included means IN BETWEEN
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A
D
B
E
C
F
ASA Congruence Postulate If two angles and the
included side of one triangles are congruent to
two angles and the included side of a second
triangle, then the two triangles are congruent.
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AAS Congruence Theorem If two angles and a
nonincluded side of one triangle are congruent to
two angles and the corresponding nonincluded side
of a second triangle, then the two trianges are
congruent.
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A
D
B
E
C
F
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A
B
C
E
F
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Helpful things for the future!
Reflexive sides
Reflexive angles
E
B
G
H
F
C
D
A
When you see shapes sharing a side, you state
that fact using the reflexive property of
congruence!
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A
B
C
D
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Draw and write down if the triangles are
congruent, and by what thrm\post
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Proofs! The way I like to think about it to look
at all the angles and sides, and dont be fooled
by the picture.
A
D
C
B
E
Tips, label the diagram as you go along.
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A
D
B
E
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E
B
C
G
What about the angle?
H
F
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Use SSS Congruence Postulate to show that
(5, 4)
D
(1, 3)
E
(-5, 1)
A
(2, 2)
F
C
(-3, -2)
B
(-4, -3)
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A
D
B
E
Tips, label the diagram as you go along.
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A
C
B
D
E
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B
A
C
E
D
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4.5 Using Congruent Triangles
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A
D
C
B
E
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Some Ideas that may help you. If they want you to
prove something, and you see triangles in the
picture, proving triangles to be congruent may be
helpful. If they want parallel lines, look to use
parallel line theorems (CAP, AIAT, AEAT,
CIAT) Know definitions (Definition of midpoint,
definition of angle bisectors, etc.) Sometimes
you prove one pair of triangles are congruent,
and then use that info to prove another pair of
triangles are congruent.
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A
D
B
E
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B
A
C
E
D
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N
A
E
L
G
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You try this classic proof!
D
1
5
3
A
C
4
E
6
2
B
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4.6 Isosceles, Equilateral, and Right Triangles.
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• Bring book Tuesday
• We will go over whats going to be on Wednesdays
  Quiz at end of Tuesday lesson

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Vertex Angle
Remember, definition of isosceles triangles is
that AT LEAST two congruent sides.
LEGS
BASE
Base Angles
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Base angles theorem If two sides of a triangle
are congruent, then the base angles are congruent.
Converse of Base angles theorem If base angles
are congruent, then the two opposite sides are
congruent.
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Corollary 1 An equilateral triangle is also
equiangular (Use isosceles triangle theorem
multiple times with transitive) Corollary 2 An
equilateral triangle has three 60 degree angles
(Use corollary 1 and angle of triangle equals 180)
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Hypotenuse Leg Theorem (HL) If the hypotenuse
and ONE of the legs of a RIGHT triangle are
congruent, then the triangles are congruent.
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A
B
C
D
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Draw and write down if the triangles are
congruent, and by what thrm\post
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Def of isosceles triangle
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A
C
B
D
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4.7 Triangles and Coordinate Proof
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Given a right triangle with one vertex (-20,
-10), and legs of 30 and 40, find two other
vertices, then find the length of the hypotenuse.
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Given a vertex of a rectangle at the origin, find
three other possible vertices if the base is 15
and the height is 10 for a rectangle. Then find
the area.
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Given the coordinates, prove that the AC is the
angle bisector of BCD
B
A
C
D
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(d, k)
(a, b)
(__,__)
(__,__)
(__,__)
(__, k)
(j,__)
(h,k)
(a,__)
(__,__)
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Picking convenient variable coordinates, prove
that the diagonals of a rectangle are congruent.