Title: Chapter 4 Notes
1Chapter 4 Notes
24.1 Triangles and Angles
3A 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
4Acute 3 acute angles
Obtuse One obtuse angle
Right One right angle
Equiangular all angles congruent
5B
C
A
6TRIANGLE SUM THEOREM The sum of the measures of
the angles of a triangle is 180.
2
3
1
7Exterior 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
8Corollary 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
94.2 Congruence and Triangles
10When 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!
11Name 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!
123rd 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
13E
A
G
M
H
T
Y
O
Find x
Find y
14- Note, triangles also have the following
properties of congruent Reflexive, symmetric,
and transitive.
154.3 4.4 Proving Triangles are Congruent
16SSS Congruence Postulate If three sides of one
triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
17SAS 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
18A
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.
19AAS 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.
20A
D
B
E
C
F
21A
B
C
E
F
22Helpful 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!
23A
B
C
D
24Draw and write down if the triangles are
congruent, and by what thrm\post
25Proofs! 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.
26A
D
B
E
27E
B
C
G
What about the angle?
H
F
28Use SSS Congruence Postulate to show that
(5, 4)
D
(1, 3)
E
(-5, 1)
A
(2, 2)
F
C
(-3, -2)
B
(-4, -3)
29A
D
B
E
Tips, label the diagram as you go along.
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31A
C
B
D
E
32B
A
C
E
D
334.5 Using Congruent Triangles
34A
D
C
B
E
35Some 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.
36A
D
B
E
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38B
A
C
E
D
39N
A
E
L
G
40You try this classic proof!
D
1
5
3
A
C
4
E
6
2
B
414.6 Isosceles, Equilateral, and Right Triangles.
42- Bring book Tuesday
- We will go over whats going to be on Wednesdays
Quiz at end of Tuesday lesson
43Vertex Angle
Remember, definition of isosceles triangles is
that AT LEAST two congruent sides.
LEGS
BASE
Base Angles
44Base 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.
45Corollary 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)
46Hypotenuse Leg Theorem (HL) If the hypotenuse
and ONE of the legs of a RIGHT triangle are
congruent, then the triangles are congruent.
47A
B
C
D
48Draw and write down if the triangles are
congruent, and by what thrm\post
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51Def of isosceles triangle
52A
C
B
D
534.7 Triangles and Coordinate Proof
54Given 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.
55Given 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.
56Given the coordinates, prove that the AC is the
angle bisector of BCD
B
A
C
D
57(d, k)
(a, b)
(__,__)
(__,__)
(__,__)
(__, k)
(j,__)
(h,k)
(a,__)
(__,__)
58Picking convenient variable coordinates, prove
that the diagonals of a rectangle are congruent.