Plane geometry angles
59 flashcards covering Plane geometry angles for the GMAT Quantitative section.
Plane geometry angles are the measures of the space between two lines that meet at a point on a flat surface. They form the building blocks of shapes like triangles, circles, and polygons, helping us understand how these figures relate to one another. For instance, angles can be acute (less than 90 degrees), right (exactly 90 degrees), obtuse (between 90 and 180 degrees), or straight (180 degrees). Mastering angles is essential because they reveal key properties of geometric figures, making it easier to solve problems involving space and measurement.
On the GMAT Quantitative section, angles often appear in questions about triangles, parallel lines, and polygons, where you might calculate measures or identify relationships like vertical or supplementary angles. Common traps include misinterpreting angle pairs or forgetting rules such as the sum of angles in a triangle equaling 180 degrees. Focus on practicing visualization techniques and memorizing theorems to handle these efficiently and avoid careless mistakes. Always draw a diagram if one isn't provided.
Terms (59)
- 01
Acute Angle
An acute angle is an angle that measures greater than 0 degrees but less than 90 degrees.
- 02
Obtuse Angle
An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees.
- 03
Right Angle
A right angle is an angle that measures exactly 90 degrees.
- 04
Straight Angle
A straight angle is an angle that measures exactly 180 degrees and forms a straight line.
- 05
Reflex Angle
A reflex angle is an angle that measures greater than 180 degrees but less than 360 degrees.
- 06
Full Angle
A full angle is an angle that measures exactly 360 degrees, representing a complete rotation.
- 07
Complementary Angles
Complementary angles are two angles whose measures add up to exactly 90 degrees.
- 08
Supplementary Angles
Supplementary angles are two angles whose measures add up to exactly 180 degrees.
- 09
Vertical Angles
Vertical angles are the angles opposite each other when two lines intersect, and they are always equal in measure.
- 10
Adjacent Angles
Adjacent angles are two angles that share a common vertex and a common side, but do not overlap.
- 11
Linear Pair
A linear pair consists of two adjacent angles whose non-common sides form a straight line, making them supplementary.
- 12
Corresponding Angles
Corresponding angles are angles in the same position at each intersection when a transversal crosses two lines, and they are equal if the lines are parallel.
- 13
Alternate Interior Angles
Alternate interior angles are angles on opposite sides of a transversal and inside two lines, and they are equal if the lines are parallel.
- 14
Alternate Exterior Angles
Alternate exterior angles are angles on opposite sides of a transversal and outside two lines, and they are equal if the lines are parallel.
- 15
Same-Side Interior Angles
Same-side interior angles are angles on the same side of a transversal and inside two lines, and they are supplementary if the lines are parallel.
- 16
Sum of Angles in a Triangle
The sum of the interior angles in any triangle is always 180 degrees.
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Exterior Angle of a Triangle
An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
- 18
Base Angles of Isosceles Triangle
In an isosceles triangle, the two base angles are equal in measure.
- 19
Angles in Equilateral Triangle
In an equilateral triangle, all three interior angles are equal to 60 degrees.
- 20
Angles in Right Triangle
In a right triangle, one angle is exactly 90 degrees, and the other two are acute and sum to 90 degrees.
- 21
Sum of Interior Angles in Quadrilateral
The sum of the interior angles in any quadrilateral is 360 degrees.
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Sum of Interior Angles in Polygon
The sum of the interior angles in a polygon with n sides is (n-2) multiplied by 180 degrees.
- 23
Exterior Angle of Polygon
An exterior angle of a polygon is equal to the sum of the interior angles that are not adjacent to it.
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Central Angle in Circle
A central angle in a circle is an angle whose vertex is at the center and whose sides pass through two points on the circumference.
- 25
Inscribed Angle in Circle
An inscribed angle in a circle is an angle whose vertex is on the circumference and whose sides pass through two other points on the circumference.
- 26
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
- 27
Angle Subtended by Arc
The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circumference.
- 28
Angle Bisector
An angle bisector is a line that divides an angle into two equal parts.
- 29
Perpendicular Lines
Perpendicular lines are lines that intersect at a 90-degree angle.
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Parallel Lines
Parallel lines are lines in a plane that do not intersect and are always the same distance apart.
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Transversal Line
A transversal is a line that intersects two or more other lines at distinct points.
- 32
Vertical Angles Theorem
Vertical angles formed by intersecting lines are always equal in measure.
- 33
Angles Around a Point
The sum of all angles around a single point is 360 degrees.
- 34
Angles on a Straight Line
The sum of angles on a straight line is 180 degrees.
- 35
Common Trap: Confusing Acute and Obtuse
A common error is mistaking an acute angle (less than 90 degrees) for an obtuse angle (greater than 90 degrees) in diagrams.
- 36
Strategy: Using Vertical Angles
To solve for unknown angles, identify vertical angles and use their equality to set up equations.
- 37
Example: Supplementary Angles Calculation
If two angles are supplementary and one measures 70 degrees, the other measures 110 degrees.
In a linear pair, 70 degrees + unknown = 180 degrees, so unknown is 110 degrees.
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Trap: Assuming Equal Angles
Do not assume angles are equal unless proven, such as in isosceles triangles or with parallel lines.
- 39
Congruent Angles
Congruent angles are angles that have the same measure.
- 40
Similar Triangles and Angles
In similar triangles, corresponding angles are equal.
- 41
Regular Polygon Angles
In a regular polygon with n sides, each interior angle measures [(n-2) 180 / n] degrees.
- 42
Example: Pentagon Interior Angle
Each interior angle of a regular pentagon is 108 degrees.
For a pentagon (n=5), [(5-2)180]/5 = 108 degrees.
- 43
Trap: Forgetting Angle Sum
A frequent mistake is overlooking that angles in a triangle sum to 180 degrees, leading to incorrect calculations.
- 44
Strategy: Angle Chasing
Angle chasing involves systematically using known angle relationships to find unknown angles in a diagram.
- 45
Example: Triangle Angle Sum
In a triangle with angles of 30 degrees and 60 degrees, the third angle is 90 degrees.
+ 60 + unknown = 180, so unknown is 90 degrees.
- 46
Co-interior Angles
Co-interior angles are angles on the same side of a transversal and inside two lines, summing to 180 degrees if the lines are parallel.
- 47
Opposite Angles in Parallelogram
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary.
- 48
Angles in Trapezoid
In a trapezoid with one pair of parallel sides, the angles adjacent to the same leg are supplementary.
- 49
Right Angle in Semicircle
An angle inscribed in a semicircle is a right angle.
- 50
Angle Between Tangent and Radius
The angle between a tangent to a circle and the radius at the point of tangency is 90 degrees.
- 51
Angle Between Tangent and Chord
The angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord.
- 52
Opposite Angles in Cyclic Quadrilateral
In a cyclic quadrilateral, opposite angles sum to 180 degrees.
- 53
Example: Inscribed Angle Calculation
If a central angle is 80 degrees, the inscribed angle subtending the same arc is 40 degrees.
- 54
Trap: Misidentifying Arcs
A common error is using the wrong arc when calculating inscribed angles, such as confusing major and minor arcs.
- 55
Strategy: Proving Lines Parallel
To prove lines are parallel, show that corresponding angles are equal or alternate interior angles are equal.
- 56
Formula: Interior Angle of Regular Polygon
For a regular polygon with n sides, each interior angle is [(n-2) 180 / n] degrees.
- 57
Example: Exterior Angle of Triangle
In a triangle with angles 40 degrees and 60 degrees, an exterior angle at the third vertex is 100 degrees.
Exterior angle = 40 + 60 = 100 degrees.
- 58
Angle of Elevation
The angle of elevation is the angle formed by the line of sight above the horizontal line.
- 59
Angle of Depression
The angle of depression is the angle formed by the line of sight below the horizontal line.