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Angles
•
An angle is: the union of two rays having a common vertex.
•
Angles can be measured in both degrees and radians. A circle of 360° in radian measure
is equal to 2π radians.
•
If you draw a circle with two lines stemming from the centre of the circle, the two lines
are called rays, the meeting point of the two lines is called the vertex, and the arc of the
circle between the rays is called the angle.
A
B
•
C
From this drawing, there are two angles one can determine, the smaller angle between the
two rays, and the larger one on the outside of the two rays.
•
When labeling the angle, it is labeled from the way it is measured. For example, in the
above angle, you would label the angle <ABC, or <B. The rays would then be labeled
RBA and RBC. Since <ABC represents the smaller angle, <CBA would represent the
larger angle.
•
If A, B, and C are in a straight line, then the angle is called a straight angle. Straight
angles always measures 180° since they are always half of a circle, whose measurement is
360°.
A
B
C
1
•
If the angle you are measuring is exactly half the size of the straight angle it is called a
right angle, and measures exactly 90°.
A
•
If you are asked to find the measurement of <ABC, it will be abbreviated as m(<ABC).
•
If you have two angles whose sum is equal to 180°, the angles are said to be
supplementary.
Z
•
Two angles that have a ray in common are called adjacent angles. The ray that divides
the angle in half is called the angle bisector.
•
To draw the bisector of an angle, first draw two connecting rays. Then, using your
compass, draw an arc that crosses those rays. Next, place the protractor on each
intersection point of the arc and the ray. Draw another arc from each of these two points.
At the intersection of these two new arcs, you can then draw the bisector, which will go
through the vertex and the intersecting arcs.
2
•
To figure out the length of the arc on a circle, you would divide 360° by the degree of the
angle, and then multiply that number by the length of the circumference of the circle. In
the following diagram, the angle is 40°, and the circumference is 6cm. Find the length of
the arc.
40°
•
When two lines intersect, the angles across from each other are said to be vertically
opposite angles. These angles will be the same measurement, and are therefore
congruent.
•
Two lines are considered perpendicular if they intersect each other at right angles, and
two rays are perpendicular if the lines on which they lie are perpendicular.
•
To draw a perpendicular line when given one line, find a point where the measurement of
the angle is 90°. From this point, draw a line that intersects the first line. The result will
be a line that is perpendicular to the first line.
3
Angles Review
¾
The basic definition of an angle is: the union of two rays having a common vertex.
¾
The lines of the angle are called rays, and the meeting point is the vertex.
¾
A straight angle measures 180°, and is half of the full circle whose angle is always 360°.
¾
A right angle has a measurement of 90°, and is exactly half of a straight angle.
¾
Supplementary angles always add up to 180°.
¾
Adjacent angles have one ray in common.
¾
The line that cuts an angle in half is called an angle bisector.
¾
Opposite angles are vertically or horizontally across from each other and are congruent.
¾
Perpendicular lines intersect each other at right angles.
4
Angles Assignment:
1. Name the following indicated angles:
a)A
C
P
b)
X
K
c)
G
R
I
B
2. What is the value of “x” in the following examples:
a)
b)
40°
x
x
65°
3. If the circumference of the circle is 36 cm, how long is the arc if m<A is:
a) 45°
b) 180°
c) 60°
4. Draw the angle bisector for the following:
5. In the following diagram, if m<A is 30°, find m<B, m<C, and m<D in degrees.
C
B
D
30°
5
Assignment Answers
1. a) <ABC b) <RPX c) <KGI
2. a) 140° b) 295°
3. a) 288 b) 72 c) 216
4.
5. a) 150° b) 30° c) 150°
6
Triangles
Labeling Triangles: The corners of a triangle are called vertices. To label a triangle, we assign
capital letters to each vertex (ex: A, B, and C). We can then label the triangle ∆ABC.
Notation:
*To indicate that two sides of a triangle are of equal length, we use:
*To indicate that two angles of a triangle are of equal measure we use:
* |PQ| implies “the length of PQ.”
Angles of a Triangle: The sum of the angles in any triangle is 180°.
That is, if A, B, and C are the angles of a triangle, then: m(<A) + m(<B) + m(<C) = 180°
A Right Triangle is a triangle in which one angle is 90°. The sides which determine the right
angle are called the legs. The side diagonal from the right angle is called the hypotenuse.
Angles of a Right Triangle: The sum of the angles of a right triangle other than the right angle
will be 90°. That is, if <A and <B are the other angles of a right triangle, then m(<A) + m(<B)=
90°
Pythagoras Theorem: Let ∆ABC be a right triangle with legs of lengths a and b and a
hypotenuse of length c. Then a2+b2= c2
Example: Find the missing length in each of the following diagrams:
An Isosceles Triangle is a triangle where two sides have the same lengths.
Isosceles Triangle Theorem: Given an isosceles triangle with |PQ|=|QR|, the angles opposite to
the sides PQ and QR have the same measure. That is, m(<P)=m(<R)
Example:
a) ∆ABC is isosceles, with |AB|=|AC|. If m(<A)=40°, find the value of m(<B):
An Equilateral Triangle is a triangle where all three sides have the same length.
*Note: Any equilateral triangle is also an isosceles triangle.
Corollary Theorem (Angles of an Equilateral Triangle): all the angles of an equilateral
triangle have the same measure—60°.
Area of Triangles: The area of a triangle is ½ the product of the lengths of the base and its
corresponding height, regardless of which side is chosen as a base. If b is the length of the base
and h is the height then the area of a triangle is area= ½bh.
7
Any side can be chosen as a base. Once the base is chosen, the perpendicular segment from the
vertex opposite to the line of the base is the height.
Example: Find the base of a triangle if its area is 5 cm2 and its height is 3cm.
Area of a Right Triangle: The area of a right triangle is one-half the product of the lengths of
the two legs. That is, Area= ½ |BC| x |AB|.
a= ½ (|BC|)(|AB|)
Example: Find the area of a right triangle whose legs have the measures of 6cm and 5 cm.
Congruent Triangles: Two figures are congruent if we can lay one on top of the other without
changing its shape. The sides and angles are the same. The symbol for congruence is ≅ .
How to Determine if Triangles are Congruent:
1. Side-Angle-Side: if two sides and the included angle of one triangle are congruent to the
corresponding parts of another triangle, the two triangles are congruent.
*Note: the included angle is the angle created by the two sides.
Ex: Is ∆PQR congruent to ∆STV by SAS?
2. Angle-Side-Angle: if two angles and the included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent.
*Note: the included side is the side that lies between the two angles.
Ex: Show that ∆ABP is congruent to ∆DCP.
3. Angle-Angle-Side: if two angles and a non-included side of one triangle are congruent the
corresponding parts of another triangle, the two triangles are congruent.
*Note: a non-included side is any side that does not lie between the two angles.
Ex. Prove that ∆CAB is congruent to ∆ZXY.
8
4. Side-Side-Side: if three sides of one triangle are congruent to three sides of another triangle,
the two triangles are congruent.
Similar Triangles: triangles are similar if their corresponding angles are congruent and their
corresponding sides are proportional.
∆UVW is similar to ∆XYZ, if and only if
The symbol ~ indicates similarity.
Example: ∆UVW and ∆XYZ are two similar triangles. The two corresponding sides of the two
triangles have lengths of 3cm and 5 cm respectively. If another side of ∆UVW has a length of
7cm, what is the length of the corresponding side of ∆XYZ?
Using
Special Triangles: Special triangles are right triangles whose sides are in a particular ratio. The
two types we will look at are the 45°-45°-90° triangles and the 30°-60°-90° triangles.
45°-45°-90° Triangles: a special right triangle whose angles are 45°, 45°and 90°. The lengths of
the sides of a 45°- 45°- 90° triangle are in the ratio of
A right triangle with two sides of equal lengths is a 45°- 45°- 90° triangle. A triangle with a 45°
angle will also be a 45°- 45°- 90° triangle.
Example: Find the length of the hypotenuse of a right triangle if the lengths of the other two
sides are both 3 inches.
9
30º-60º-90º Triangles: a special right triangle whose angles are 30°, 60°and 90°. The lengths of
the sides of a 30°- 60°- 90° triangle are in the ratio of
A right triangle with a 30° or 60° angle must be a
30°- 60°- 90° triangle.
Example: Find the length of the hypotenuse of a right triangle if the lengths of the other two
sides are 4 inches and
inches.
10
Definitions of Shapes:
Polygon – A closed figure made by joining line segments, where each line segment intersects
exactly two others.
Example)
Parallelogram – A four-sided polygon with two pairs of parallel sides. Opposite angles are
congruent (the same). Special parallelograms are rectangles, squares, and rhombuses.
Example)
p
q
pq parallel to nm
pn parallel to qm
n
m
Quadrilateral – A four-sided polygon. The sum of the interior angles of the quadrilateral adds to
360°.
4
Example)
1
3
2
Rectangle – A four-sided polygon that has four right angles. A rectangle has two pairs of sides
that are parallel to each other (this means that it is also a parallelogram). The sum of the
interior angles adds to 360°.
Example)
Square – A four-sided polygon that has four right angles, and four sides that are all equal to each
other. The sum of its interior angles is 360°. The sides that are across from each other are
also parallel (this means that a square is also a parallelogram).
Example)
11
Rhombus – A parallelogram with four equal sides. A rhombus DOES NOT have to have four
right angles. The sum of its interior angles is 360°. The sides that are across from each
other are parallel.
Example)
Note: If two parallel lines are
crossed by another, same side
interior angles add up to 180°.
Also, any two angles that are
Next to each other, add to 180°.
Quadrilateral
D
A
C
B
Remember:
A Quadrilateral is any shape that has four sides. The sum of the interior angles of a
quadrilateral is 360°.
<B & <C are examples of consecutive angles
AB & DC are examples of opposite sides
<B & <D are examples of opposite angles
Formula for the Sum of Angles:
S = (n-2)(180°)
S = the sum of the measure of its angles (360°)
n = the number of sides of the polygon
12
Try:
Find the measure of angle x.
-the symbol x on the diagram represents the measure
-we know that <Z is supplementary to 60°, so z =
-we also know that <Y is congruent to 80°, so y =
-and we are given 75°
-we learned in the definition of a quadrilateral, that the sum of its interior angles is 360°.
To solve for x we must first find w
-from this answer, we can find the value for x, since <X is supplementary to <W:
13
Parallelogram
height
b
D
C
a
a
A
B
b
Remember:
A Parallelogram is a convex (a polygon with no interior angle greater than 180°) quadrilateral
with two pairs of parallel sides. The sum of its interior angles is 360°.
AB // DC line AB is parallel to line DC
AD // BC line AD is parallel to line BC
*A line over the letters, indicates the length of the line between point A and point B for example.
Formula to Calculate the Perimeter:
P = 2(Length of Side a + Length of Side b)
Formula to Calculate the Area:
A = Base × Height (always correct for any parallelogram)
*A = Length × Width can only be used when finding the area of a rectangle or square.
Try:
Calculate the Perimeter and the Area of the given Parallelogram.
5
6
7
-by applying the formula given to calculate the perimeter:
Perimeter = 2a + 2b
14
-we know that the parallelogram has two pairs of parallel sides. From this we can figure
out that the base of the parallelogram.
-by applying the formula to calculate the area :
Area = bh
=
Rectangle
D
A
C
B
Remember:
A Rectangle is a parallelogram that has four right angles (90°), and two pairs of congruent
opposite sides.
AD // BC
AB // DC
m(<A) = m(<B) = m(<C) =m(<D)
*All angles are equal and all angles equal 90°.
Note: To calculate the Area of a Rectangle is the same as a parallelogram, square, and rhombus
(a parallelogram with four equal sides, and two pairs of congruent angles).
Base × Height
15
Square
D
C
A
B
Remember:
A Square is a parallelogram with four right angles (90°), four congruent sides, and two
congruent diagonal lines.
AB // DC
AD // BC
mAB = mBC = mCD = mDA
m<A = m<B = m<C = m<D = 90°
Formula to Calculate the Diagonal Line:
Diagonal Line2 = base2 × height2
Try:
Calculate the length of the diagonal line.
4
4
4
4
16
Diagonal Line2 = base2 × height2
Rhombus
D
a
b
A
C
b
B
a
Remember:
A Rhombus is a parallelogram with four equal sides, two pairs of congruent opposite angles,
and two perpendicular diagonals that bisect the angles of the rhombus. The sum of its angles
equals 360°.
AB // DC
AD // BC
m(<A) = m(<C)
m(<B) = m(<D)
m(AB) = m(BC) = m(CD) = m(DA) *the measure of line from A to B is equal to the measure
of line from B to C, which is equal to the measure of line from C to D, which is equal to
the measure of line from D to A.
AC
BC
*The diagonal line from A to C is perpendicular to the line from B to C.
17
Formula to Calculate the Area:
Base × Height
*The Area is equal to ½ the product of the lengths of the first diagonal and the second diagonal
line.
d1
d2
d1 = 2
d2 = 2
A = ½ (2)(2)
A = ½ (4)
A=2
Try:
Calculate the measure of the angles.
x
4x
-we know that two angles are supplementary of their sum is 180 degrees.
18
Assignment:
1.Calculate the Area and Perimeter of the Parallelogram:
b
13
a
14
16
2. What is the value of angle x?
120°
60°
60°
x
3. Determine the value of x.
x
45°
19
4.Solve for z.
v z
60
145
x
y
60
w
20
Circles
Circle- The set of points that are the same distance from the point; O. O is the center of the
circle.
Radius (r) - Measure of a circle from the centerpoint, O, to the outermost point of the circle, this
measure should be the same from the center, to any of the outer points.
Diameter (2r)- A chord that goes through O, the center of the circle. The length of the
diameter is 2r, r being the radius.
Circumference- The outer line of a circle, the measure around a circle. The circumference of a
circle of radius r is 2 Π r, and is the distance around the closed curve representing the circle.
Chord- A chord is a line segment that connects two points on a circle. The longest chord on a
circle is the diameter of the circle.
Degree- A degree is a measure of angle, there are 360 degrees in a circle.
Radian- A supplementary unit of the international system used in angular measure. One radian
is equal to the angle subtended, to be opposite to, the center of a circle by an arc equal in length
to the radius of the circle, approximately 57.2957° (1 degree = 0.0174532 radians). There are 2Π
radians in 360°.
Arc- An arc is a segment along the circumference of a circle, or the part of any curve between
two points.
Central Angle- Given a circle with Center, O, and two points P and Q on the circle the angle
created by QOP is called a central angle because the vertex, O, is at the center of the circle. The
measure of an Arc is equal to it’s central angle.
Inscribed Angle- The angle created by three points along the circumference of a circle, it differs
from a central angle because the vertex is NOT at the center.
21
Theorem 5.4
The Measure of an inscribed angle is one-half that of its intercepted arc.
Area of a circle
The equation for the Area of a circle is A= Π r2
If the Radius of a given circle is 10cm, what is the area of the circle?
Apollonius Theorem
If A, and B, are two points lying on a circle and M1 and M2 are two points which lie on the arc
between A and B.
Then Angle <A M1 B = <A M2 B
22
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