Angle Measure Units
Degrees and radians are the most common units used to represent angles.
In order to convert one unit to the other, remember that one turn measures:
- 360º (Degrees).
- $2\pi$ (in Radians) (angles expressed in radians are dimensionless)
- 400gr (Grads)
Radians
1 radian is defined in a circle as the angle whose arc is equal to one circle's radius (following figure).
One turn is 2π radians. The number π is a constant, that equals approximately 3.14159265359 (in tests it is usually enough to use only three digits: 3.14).
Types of Angles
- Zero Angle: an angle that measures 0°
- Acute Angle: an angle that measures less than 90°
- Right Angle: an angle that measures 90° or π/2
- Obtuse Angle: an angle that measures between 90° and 180°
- Straight Angle: an angle that measures 180°
- Reflex Angle: an angle that measures between 180° and 360°
- Full (Complete, Round or Perigon) Angle: an angle that measures 360°
- Oblique Angle: not a right angle or a multiple of a right angle
- Complementary Angles: two angles whose measures sum 90°
- Supplementary Angles: two angles whose measures sum 180°
- Explementary (or Conjugate) Angles: two angles whose measures sum 360°
Internal Angle Bisector
Internal Angle Bisector is a line segment that splits an angle into two equal parts.
Vertical Angle Theorem
In the following figure, lines "x" and "z" intersect forming four angles (A, B, C e D):
Angles A and C, and angles B and D are called vertical angles or opposite angles or vertically opposite angles.
According to the Vertical Angle Theorem, vertically opposite angles are equal (or congruent).
This is the demonstration of this theorem:
A and B are supplementary angles, therefore A+B=180º (equation 1).
B and C are supplementary angles, therefore B+C=180º (equation 2).
A and D are supplementary angles, therefore A+D=180º (equation 3).
Subtracting equation 2 from equation 1:
$A+B-B-C=180-180$
$A-C=0$
$A=C$
Subtracting equation 3 from equation 1:
$A+B-A-D=180-180$
$B-D=0$
$B=D$
$A+B-B-C=180-180$
$A-C=0$
$A=C$
Subtracting equation 3 from equation 1:
$A+B-A-D=180-180$
$B-D=0$
$B=D$
Angles of a Transversal
In the following figure, "r" and "s" are parallel lines, and "t" is a transversal.
A transversal forms eight angles with the two parallel lines. These angles can be alternate, corresponding, or consecutive interior angles.
Alternate angles are located in distinct vertices, lie on opposite sides of the transversal, and
are both interior or both exterior angles. Alternate angles are congruent.
In the figure above these are the pairs of alternate angles: 1 and 7; 2 and 8; 3 and 5; 4 and 6.
Corresponding angles are located in distinct vertices, lie on the same side of the transversal, and
if one angle is interior the other is exterior. Corresponding angles are congruent.
In the figure above these are the pairs of corresponding angles: 1 and 5; 2 and 6; 3 and 7; 4 and 8.
Consecutive interior angles are located in distinct vertices, lie on the same side of the transversal, and (as the name says) are both interior. Consecutive interior angles are supplementary.
In the figure above these are the pairs of consecutive interior angles: 4 and 5; 3 and 6.
Solved SAT Practice Tests
Find Practice Tests in the following link:
SAT Practice Tests - Angles and
Additional Practice Tests - Angles
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