summarizing angles. angles in a triangle for any triangle, a b c a + b + c = 180°
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Summarizing Angles
Angles in a triangle
For any triangle,For any triangle,
ab
c
a + b + c = 180°a + b + c = 180°
Angles in a triangle
We can prove that the sum of the angles in a triangle is 180° by drawing a line parallel to one of the sides through the opposite vertex.
These angles are equal because they are alternate angles.
a
a
b
b
Call this angle c.
c
a + b + c = 180° because they lie on a straight line.
The angles a, b and c in the triangle also add up to 180°.
Sum of the interior angles in a pentagon
What is the sum of the interior angles in a pentagon?
We can work this out by using lines from one vertex to divide the pentagon into three triangles .
a + b + c = 180° and d + e + f = 180°
So, (a + b + c) + (d + e + f ) + (g + h + i) = 540°
The sum of the interior angles in a pentagon is 540°.The sum of the interior angles in a pentagon is 540°.
c
a
b
and g + h + i = 180°
d
f
eg
ih
Sum of the interior angles in a polygon
We’ve seen that a quadrilateral can be divided into two triangles …
… and a pentagon can be divided into three triangles.
How many triangles can a hexagon be divided into?A hexagon can be divided into
four triangles.
To find the interior angles of a convex polygon:(n-2) x 180
Angles made with parallel lines
Types of angle
Acute angle0º < a < 90º
Acute angle0º < a < 90º a
Right anglea = 90º
Right anglea = 90º
a
Obtuse angle90º < a < 180º
Obtuse angle90º < a < 180º
a
Reflex angle 180º < a < 360º
Reflex angle 180º < a < 360º
a
Angles at a point add up to 360
ab
cd
a
b
Angles on a line add up to 180
c
Angles on a straight line and at a point
a + b + c + d = 360
because there are 360 in a full turn.
a + b + c = 180°
because there are 180° in a half turn.
Complementary and supplementary angles
a
b
a + b = 90°
Two complementary angles add up to 90°.
Two complementary angles add up to 90°.
Two supplementary angles add up to 180°.
Two supplementary angles add up to 180°.
a b
a + b = 180°
Intersecting lines
Vertically opposite angles
When two lines intersect, two pairs of vertically opposite angles are formed.
a
b
c
d
a = c and b = d
Vertically opposite angles are equal.Vertically opposite angles are equal.
Angles made with parallel lines
When a straight line crosses two parallel lines eight angles are formed.
Which angles are equal to each other?
a
b
c
d
ef
g
hThis line is called a
traversal.
Alternate angles are
equal
Alternate angles are
equal
a
b
a = ba = b
Corresponding, alternate and interior angles
Look for an F-shape
Look for a Z-shape
Corresponding angles are
equal
Corresponding angles are
equal
a
b
a = ba = b
Look for a C- or U-shape
Interior angles add up to 180°
Interior angles add up to 180°
a
b
a + b = 180°a + b = 180°
dd
hh
a
b
ce
f
g
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
ce
f
g
d = h because
Corresponding angles are equalCorresponding angles are equal
ee
aa
b
c
d
f
g
h
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b
c
d
f
g
h
a = e because
Corresponding angles are equalCorresponding angles are equal
gg
cc
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
c = g because
a
bd
e
fh
Corresponding angles are equalCorresponding angles are equal
ff
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b = f because
a
b
c
d
e
g
h
b
Corresponding angles are equalCorresponding angles are equal
ff
dd
Alternate angles
There are two pairs of alternate angles, or Z-angles.
d = f because
Alternate angles are equalAlternate angles are equal
a
b
ce
g
h
ccee
Alternate angles
There are two pairs of alternate angles, or Z-angles.
c = e because
a
b
g
h
d
f
Alternate angles are equalAlternate angles are equal