Download Sum of Angles

Definitions
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Line - A series of connected points that extends out infinitely.
Line Segment - A portion of a line.
Parallel Lines - Two or more lines that never intersect.
Perpendicular Lines - Two lines whose intersection forms a 90° angle.
RayPoint-
Line Basics
In the world of geometry, a line refers to a straight line that extends out infinitely in both
directions. For any two points, there is one and only one line that connects them. A line segment
is a portion of a line from one point to another. For example:
Line l can be called the line nm, sometimes denoted nm. The portion of the line from n to m is a
line segment. nm can also denote the line segment as well as the length of the line segment.
Context is the best means to determine the use of the notation.
Parallel Lines
Two lines in the same plane are said to be parallel if they never cross or intersect. In coordinate
geometry, parallel lines have the same slope but different x-intercepts. For example:
Lines D and E are parallel (denoted D || E), while lines F and G are not parallel since they
intersect. Since the slope of parallel lines must be the same, the slope of lines D and E is the
same. Note that parallel lines do not have to be horizontal.
Perpendicular Lines
Two lines are perpendicular when they intersect and form 90° angles. For example:
Lines K and H are perpendicular while lines P and Q are not perpendicular. A right angle symbol
indicates that K and H are perpendicular (the symbol is composed of the two lines that form a
square at the intersection of lines K and H).
Definitions
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Angle - A figure formed by two lines sharing a common endpoint.
Ray - A straight line extending from a point.
Acute Angle - An angle less than 90° and greater than 0°
Obtuse Angle - An angle between 90° and 180°
Right Angle - An angle that is exactly 90°
Complementary Angles - Two angles whose sum is 90°
Supplementary Angles - Two angles whose sum is 180°
Adjacent Angles - Two angles that share a side (i.e., are adjacent).
Angle Bisector - A ray that divides an angle into two equal parts.
Vertical Angles - The two non-adjacent angles that share a vertex when two lines intersect.
Alternate Interior Angles - When two parallel lines are cut by a transversal, the two angles on
opposite sides of the transversal and between the parallel lines (an example below).
Alternate Exterior Angles - When two parallel lines are cut by a transversal, the two angles on
opposite sides of the transversal and outside the parallel lines (an example below).
Types of Angles
Acute Angle
An acute angle is one whose measurement is less than 90°. Angle A, shown below, is an acute
angle.
Obtuse Angle
An obtuse angle is one whose measurement is greater than 90° and less than 180°. Angle B,
shown below, is an obtuse angle.
Right Angle
A right angle is one whose measurement is exactly 90°. Angle C, shown below, is a right angle.
Note: The lines that surround the letter C are the formal way of denoting a right angle. If you see
this symbol, you know that you are dealing with a right angle. You cannot assume an angle is
right simply because it appears to be right.
Intersecting Lines
When two lines intersect, they form four angles. Angles that are not adjacent (i.e., do not share a
common side) are called vertical angles and these angles are congruent (or equal in
measurement).
Due to the properties of intersecting lines, vertical angles are congruent. Consequently, angle D
= angle F and angle G = angle E.
Parallel Lines Cut by a Transversal
When two parallel lines are cut by a transversal (i.e., a third line intersects the two parallel lines),
a number of relationships exist between the resulting angles.
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Alternate Interior Angles Are Equal: K = L; O = J
Alternate Exterior Angles Are Equal: H = M; N = I
Corresponding Angles Are Equal: K = N; J = M; H = O; I = L
Non-Alternate Interior Angles Are Supplementary: L + J = 180; K + O = 180
It is important to note that if two lines cut by a transversal have any of the above properties, then
the two lines must be parallel. For example, if alternate interior angles are equal, then the two
lines cut by a transversal must be parallel.
Sum of Angles
The sum of the interior angles of a polygon is 180*(n-2) where n is the number of sides. For
example, the sum of the interior angles of a triangle is 180 = (180)(3-2) while the sum of the
interior angles of a square is 360 = (180)(4-2).
Types of GMAT Problems
1. Problems Involving Various Aspects of Angles.
Many times, a problem will require the use of several properties of angles. If stuck on a
problem, always write out as much of the information as can be deduced from the given
information.
A) 20°
B) 55°
C) 35°
D) 125°
E) 45°
 Determining if Lines are Parallel
At times, diagrams in GMAT problems make lines appear parallel. However, you cannot assume
that two lines are parallel or an angle is a right angle simply because it looks that way. In order to
determine whether lines are parallel, assess the relationship between the angles.
Which of the following depicts line l parallel to line m?
A) A
B) B
C) C
D) A and C
E) B and C
Properties of All Triangles
A triangle is a three-sided shape whose three inner angles must sum to 180°. The largest angle will be
across from the longest side while the smallest angle will be across from the shortest side of the
triangle. If and only if two sides of a triangle are equal, the angles opposite them will be equal as well.
Sum of Angles is 180
The sum of the interior angles in any triangle is 180°
Triangle Inequality Theorem
The sum of any two sides of a triangle must be greater than the third side of a triangle.
For example, the figure below is not physically possible since the sum of two sides is smaller
than the third side.
Longest Side Opposite Largest Angle
The longest side of a triangle is opposite the largest angle of a triangle. Conversely, the
smallest side of a triangle is opposite the smallest angle of a triangle. Consider the following
example:
Rank the size of the angles of triangle ABC from largest to smallest.
Since side AC is the longest side, angle B is the largest angle.
Since side AB is the next longest side, angle C is the next largest angle.
Since side BC is the shortest side, angle A is the smallest angle.
Exterior Angle Theorem
The exterior angle theorem states:
Angle EFG + Angle EGF = Angle DEG
Consider the following example:
In the figure above, if angle EFG is 60° and angle DEG is 120°, what is the measure of angle FGE?
Angle EFG + Angle EGF = Angle DEG
60 + Angle EGF = 120
Angle EGF = 60
Types Triangles
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Right Triangle
o One angle is 90°
Scalene
o Each side has a different length
o Each angle has a different measurement
Obtuse
o One angle of the triangle is greater than 90°
Acute
o Every angle of the triangle is less than 90°
Equilateral
o Every angle of the triangle is equal (i.e., 60°)
o Every side of the triangle is equal in length
Isosceles
o Two angles of the triangle are equal
o Two sides of the triangle are equal in length
Similar Triangles
When Are Triangles Similar?
Two triangles are similar if any one of the following three possible scenarios is met:
1. AAA [Angle Angle Angle] - The corresponding angles of each triangle have the same
measurement.
In other words, the above triangles are similar if:
Angle L = Angle O; Angle N = Angle Q; Angle M = Angle P
2. SAS [Side Angle Side] - An angle in one triangle is the same measurement as an angle in the
other triangle and the two sides containing these angles have the same ratio.
In other words, the above triangles are similar if:
Angle L = Angle O; Side LM/Side OP = Side LN/Side OQ
Note: Any other combination of side, angle, side also proves similarity.
3. SSS [Side Side Side] - Each pair of corresponding sides have the same ratio.
In other words, the above triangles are similar if:
Side LM
/Side OP = Side LN/Side OQ = Side MN/Side PQ
Properties of Similar Triangles
1. Corresponding angles are the same measurement.
2. The perimeter of each triangle is in the same ratio as the sides.
3. Corresponding sides are all in the same proportion.
Congruent Triangles
When Are Triangles Congruent?
Two triangles are congruent if any one of the following three possible scenarios is met:
1. SAS [Side Angle Side] - Two pairs of corresponding sides are equal and the corresponding angle
between the sides is equal.
In other words, the above triangles are congruent if:
Side SW = Side UV; Angle W = Angle V; Side WR = Side VT
Note: Any other combination of side, angle, side also proves congruence.
2. ASA [Angle Side Angle] - Two pairs of corresponding angles are equal and the corresponding side
between them is equal.
In other words, the above triangles are congruent if:
Angle R = Angle T; Side RW = Side TV; Angle W = Angle V
Note: Any other combination of angle, side, angle also proves congruence.
3. SSS [Side Side Side] - All three pairs of corresponding sides are equal.
In other words, the above triangles are congruent if:
Side RS = Side TU; = Side RW = Side TV; Side SW = Side UV
Properties of Congruent Triangles
1. Corresponding angles have the same measurement.
2. Corresponding sides have the same measurement.
Area of a Triangle
The area of a triangle is given by the following formula:
Area = 1/2(base)(height)
The height of the triangle is the length of the line which is perpendicular to the base and goes through
the opposite vertex (i.e., line KH in the triangle below).
To reiterate, the area of a triangle can be found using the equation: A = 1/2bh. In this case, b
stands for the base of the triangle and h stands for the height. Any side can be chosen to be the
base, but the height is the line that is perpendicular to the base and goes through the opposing
vertex. The perimeter of a triangle is the sum of the three sides.
Consider the following example:
Referring to the triangle immediately above, what is the area of triangle KIJ if KJ = 13, KH = 5, IH = 3, and
JI = 9?
Area = 1/2(JI)(HK)
Area = 1/2(9)(5) = 22.5
1. Similar Triangles
Two triangles are similar if: (a) 3 angles of one triangle are the same as 3 angles of the
other triangle (b) 3 pairs of corresponding sides are in the same ratio (c) an angle of 1
triangle is the same as the angle of the other triangle and the sides containing these angles
are in the same ratio. For example:
What is the value of x?
A) 6
B) 7
C) 8
D) 9
E) It Cannot Be Determined