Geometry Tutorial

8/2/2019 Geometry Tutorial

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What is an Angle?

Two rays that share the same endpoint form an angle. The point where the rays intersect is called

the vertex of the angle. The two rays are called the sides of the angle or arms of the angle.

Example: Here are some examples of angles.

We can specify an angle by using a point on each ray and the vertex. The angle below may be

specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA.Note how the vertex point is always given in the middle.


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Degrees: Measuring Angles

We measure the size of an angle using degrees.

Example: Here are some examples of angles and their degree measurements.

As the Angle Increases, the Name Changes

Type of Angle Description

Acute Angle an angle that is less than 90

Right Angle an angle that is 90 exactly

Obtuse Anglean angle that is greater than 90 but less than

180

Straight Angle an angle that is 180 exactly

Reflex Angle an angle that is greater than 180
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Acute Angles

An acute angle is an angle measuring between 0 and 90 degrees.

Example:

The following angles are all acute angles.

Obtuse Angles

An obtuse angle is an angle measuring between 90 and 180 degrees.

Example:

The following angles are all obtuse.


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Right Angles

A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right

angle are said to be perpendicular. Note that any two right angles are supplementary angles (a

right angle is its own angle supplement).

Example:

The following angles are both right angles.


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Complementary Angles

Two angles are called complementary angles if the sum of their degree measurements equals 90

degrees (right angle). One of the complementary angles is said to be the complement of the

other.

The two angles do not need to be together or adjacent.

They just need to add up to 90 degrees. If the two complementary angles are adjacent then they

will form a right angle.

ABCis the complement ofCBD

These two angles (40 and 50) are Complementary Angles,

because they add up to 90.

These two are complementary because 27 + 63 = 90


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Example: What is the complementary angle of 43o?

Solution: 90o - 43o = 47o

OR

x+ 43o= 90

x = 90 43o

x = 47o

Example:

x andy are complementary angles. Givenx= 35, find the valuey.

Solution:x +y= 90

35 +y= 90

y= 90 35 = 55

Q If two angles are complementary and one of them is 77, what is the size of the other

angle?

(A) 13 (B) 23 (C) 77 (D) 103

Ans (A) 13


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Supplementary Angles

Two angles are called supplementary angles if the sum of their degree measurements equals 180

degrees. One of the supplementary angles is said to be the supplement of the other.

The two angles do not need to be together or adjacent.

They just need to add up to 180 degrees. If the two supplementary angles are adjacent then they will

form a straight line.

ABCis the supplement ofCBD

These two are supplementary because 60 + 120 = 180

Example: What is the supplementary angle of 43o?

Solution: 180o

- 43o

= 137o

OR

x + 43o= 180

x = 180 43o

x = 137o


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Example:x andy are supplementary angles. Givenx= 72, find the valuey.

Solution: x +y= 180

72 +y= 180

y= 180 72 = 108

Q Two angles are supplementary and one of them is 31 ,What is the size of the other

angle?

(A) 31 (B) 59 (C) 121 (D) 149

Ans (D) 149


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Adjacent Angles

Twoanglesare Adjacent if they have a

(a)common side or arm(b)a common vertex (corner point)(c)don't overlap or no common interior points

Angle ABC is adjacent to angle CBD

Because:

they have a common side (line CB) they have a common vertex (point B)


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What Is and Isn't an Adjacent Angle

These ARE Adjacent Angles

they share a vertex anda side

NOT Adjacent Angles

they only share a vertex, notaside

NOT Adjacent Angles

they only share a side, notavertex

Don't Overlap! Or No common interior points

ALSO the angles must not overlap.

NOT Adjacent Angles

angles a and b overlap


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Straight Angle

A straight angle is 180 degrees

This is a straight angle

A straight angle changes the direction to point the opposite way.

Sometimes people say "You did a complete 180 on that!" ... meaning you completely changed

your mind, idea or direction.

All the angles below are straight angles:

Angles On One Side of A Straight Line

Angles on one side of a straight line will always add to 180 degrees.

If a line is split into 2 and you know one angle you can always find the other one.


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30 + 150 = 180

Example: If we know one angle is 45 what is angle "a" ?

Angle a is 180 45 = 135

This method can be used for several angles on one side of a straight line.

Example: What is angle "b" ?

Angle b is simply 180 less the sum of the otherangles.

Sum of known angles = 45 + 39 + 24Sum of known angles = 108

Angle b= 180 108Angle b = 72


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Interior Angle

An Interior Angle is an angle inside a shape.

Note: When you add up the Interior Angle and Exterior Angle you get a straight line, 180.

The Interior Angles of a Triangle add up to 180

90 + 60 + 30 = 180 80 + 70 + 30 = 180

It works for this triangle! Let's tilt a line by 10 ...

It still works, because one angle went up by

10, but the other went down by 10

Alternate Interior Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the

diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles

have the same degree measurement. Angle B and angle C are also alternate interior angles.


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Exterior Angle

The Exterior Angle is the angle between any side of a shape, and a line extendedfrom the next side.

Note: When you add up the Interior Angle and Exterior Angle you get a straight line, 180.

Alternate Exterior Angles


diagram below, angle A and angle D are called alternate exterior angles. Alternate exteriorangles have the same degree measurement. Angle B and angle C are also alternate exterior

angles.


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Corresponding Angles


diagram below, angle A and angle C are called corresponding angles. Corresponding angles have

the same degree measurement. Angle B and angle D are also corresponding angles.

A Linear Pair is 2 adjacent angles whose non-common sides

form opposite rays. The angles MUST be adjacent.


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Angle Bisector

An angle bisector is a ray that divides an angle into two equal angles.

Example:

The blue ray on the right is the angle bisector of the angle on the left.

The red ray on the right is the angle bisector of the angle on the left.

Perpendicular Lines

Two lines that meet at a right angle are perpendicular.

These lines intersect and form four right

angles. They are perpendicular lines.


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PAIR OF LINES

Two lines can be related to each other in four different ways.
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TRANSVERSAL

A line that intersects two or more lines at a distinct points is called Transversal.

Transversal crossing This Transversal crosses This Transversal

two lines two parallel lines cuts across three lines
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These two lines are parallel, and are cut by a transversal. Eight angles appear, in four

corresponding pairs that have the same measure, so therefore are congruent.

These four corresponding pairs are:

angles a and e angles candg angles b and f angles dand h

The angles that lie in the interior area, or the area between the two lines that are cut by thetransversal, are called interior angles.

Interior angles are c, d, e and f

The angles that lie in the exterior area, that are cut by the transversal, are called interior

angles.

Exterior angles are a, b, g, and h

Angles on opposite sides of the transversal are called alternate angles.

Angles cand f, and dand e, are alternate interior angles.

Angles a and h, and b and g, are alternate exterior angles.

Note that these alternate pairs are also congruent.


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