Math for 800 10 lines, angles and triangles
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Transcript of Math for 800 10 lines, angles and triangles
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Two points determine a
straight line.
AB
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Is a segment that is extended
infinitely in one direction.
AB
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Is a section of a straight line, of
finite length with two endpoints.
AB
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Is the distance between its endpoints.
10 3
7
AB
103
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If point Q is between point P and
point R, then
PQ QR PR
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With a letter, a number, or
a group of three letters.
a
A
B C
Angle a Angle ABC
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It is extremely unlikely that you’ll
see any fractional angles (23 ½o).
There are no negative angles (−30o).
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Angles
greater than
zero but less
than 90o.
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Angles with
measure
equal to 90o.
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Angles
greater than
90o but less
than 180o.
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An angle
that
measures
exactly 180o.
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Two angles
that sum up
to 90o.
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Two
angles
that sum
up to
180o.
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Edwin Lapuerta, May 2014
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LINES AND
ANGLES
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Angles opposite each other
when two lines cross.
ac
bd
a b
c d
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Angles that have a common vertex
and share a side.
ac
bd
,
180
180
180
180
a b c d
a c
a d
b c
b d
360a b c d
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ABC CBD
A
B
C
D
If BC is the angle bisector
of angle ABD, then
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L1
L2
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Two lines which not meet.
L1
L2
L1 || L2
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L1
L2
14
23
8 57 6
L1 || L2
1 3 5 7
2 4 6 8
small angles
big angles
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L1
L2
14
23
8 57 6
L1 || L2
180small angle big angle
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An angle is a right angle
only if:
You’re expressly told, “this
is a right angle”.
You see the
perpendicular symbol ().
You see the box in the
angle.
L1
L2
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TRIANGLES
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A triangle with
three equal sides
and three equal
angles (60o).
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A triangle with
two equal sides
and two equal
angles.
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A triangle with no
equal sides and no
equal angles.
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If two sides of
a triangle are
unequal, the
angles
opposite these
sides are
unequal.
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If two angles
of a triangle
are unequal,
the sides
opposite these
angles are
unequal.
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The largest
angle is
opposite to the
largest side.
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A triangle has 3
possible
midsegments. d
mED
B
CA
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The midsegment is
always parallel to
the third side of
the triangle. d
mED
B
CA
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The midsegment is
always half of the
length of the third
side. d
mED
B
CA
1
2m d
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d
mED
B
CADBE ABC
DB EB DE
AB CB AC
The triangle formed
by the midsegment
is similar to the
original triangle.
DBE ABC
DB EB DE
AB CB AC
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2a 2b
2c
a b
c
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CongurentTriangles
have exactly the same three sides and exactly the same three angles.
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Corresponding
sides and angles
are equal .
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SAS
(Side-Angle-Side)
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SSS
(Side-Side-Side)
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ASA
(Angle-Side-Angle)
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AAS
(Angle-Angle-Side)
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The sum of the lengths of the sides.
a b
c
P a b c
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The sum of the length of two sides
must be greater than the length of
the third side.
a b
c
a b c
b a c
c a b
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The difference of two sides must be
less than the third side.
a b
c
b c a
a c b
a b c
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ab
c
b c a b c
a c b a c
a b c a b
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a b
c AB
C
The sum of the length of two sides must be
greater than the length of the third side.
+
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In any triangle, the sum of the interior
angles is 180o.
a b
c AB
C
180A B C
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The largest side is the hypotenuse and the
sides adjacent to the right angle (C = 90o) are
the legs.
C A
B
ac
b
180
90
A B C
A B
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The measure of exterior angles of a triangle
is equal to the sum of the two remote
interior angles.
a
b
c
y
xz
360
x b c
y a c
z a b
x y z
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The square of the length of the hypotenuse
is equal to the sum of the squares of the
lengths of the legs of the triangle.
a
b
c2 2 2a b c
It works only in right triangles.
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3:4:5 (2 sides)
5
4
3
6:8:10 2 3: 4:5
15: 20: 25 5 3: 4:5
30: 40:50 10 3: 4:5
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5:12:13 (2 sides)
13
12
5
10: 24: 26 2 5:12:13
25:60:65 5 5:12:13
50:120:130 10 5:12:13
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Isosceles right triangle (1 side) 1:1:1 2
x
x 2x
45o
45o
: : 2 1:1:1 2
2: 2 : 2 2 2 1:1:1 2
5:5:5 2 5 1:1:1 2
x x x x
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(1 side) 1: 3 : 2
2xx
3x
30o
60o
: 3 : 2 1: 3 : 2
2: 2 3 : 4 2 1: 3 : 2
5:5 3 :10 5 1: 3 : 2
x x x x
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1
2A base height
The term "base" denotes any side, and
"height" denotes the length of a
perpendicular from the vertex opposite
the side onto the line containing the
side itself.
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1
2A base height
h
b
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1
2A base height
h
b
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1
2A base height
h
b
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