Angles
56 flashcards covering Angles for the SAT Math section.
Angles are basic geometric figures formed by two rays sharing a common endpoint, called the vertex. They are measured in degrees, with a full circle equaling 360 degrees. Types include acute angles (less than 90 degrees), right angles (exactly 90 degrees), and obtuse angles (greater than 90 degrees but less than 180 degrees). Mastering angles is essential for understanding shapes, lines, and spatial relationships, which form the foundation of geometry problems on standardized tests.
On the SAT Math section, angles often appear in questions involving triangles, parallel lines, or polygons, where you might need to calculate measures or identify relationships like vertical, supplementary, or complementary angles. Common traps include mistaking angle pairs or forgetting that the sum of angles in a triangle is always 180 degrees. Focus on practicing angle chasing, applying theorems such as those for parallel lines cut by a transversal, and double-checking diagrams to avoid errors. Always draw a clear diagram if one isn't provided.
Terms (56)
- 01
Acute Angle
An acute angle is an angle that measures less than 90 degrees.
- 02
Right Angle
A right angle is an angle that measures exactly 90 degrees.
- 03
Obtuse Angle
An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees.
- 04
Straight Angle
A straight angle is an angle that measures exactly 180 degrees, forming a straight line.
- 05
Reflex Angle
A reflex angle is an angle that measures more than 180 degrees but less than 360 degrees.
- 06
Vertical Angles
Vertical angles are the angles opposite each other when two lines intersect, and they are always equal in measure.
- 07
Adjacent Angles
Adjacent angles are two angles that share a common vertex and a common side, but do not overlap.
- 08
Complementary Angles
Complementary angles are two angles whose measures add up to exactly 90 degrees.
- 09
Supplementary Angles
Supplementary angles are two angles whose measures add up to exactly 180 degrees.
- 10
Angle Sum in a Triangle
The sum of the interior angles in any triangle is always 180 degrees.
- 11
Exterior Angle of a Triangle
An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
- 12
Parallel Lines
Parallel lines are lines in a plane that never intersect and are always the same distance apart.
- 13
Transversal Line
A transversal is a line that intersects two or more other lines at distinct points.
- 14
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.
- 15
Alternate Interior Angles
Alternate interior angles are angles on opposite sides of the transversal and inside the two lines, and they are equal if the lines are parallel.
- 16
Alternate Exterior Angles
Alternate exterior angles are angles on opposite sides of the transversal and outside the two lines, and they are equal if the lines are parallel.
- 17
Same-Side Interior Angles
Same-side interior angles are angles on the same side of the transversal and inside the two lines, and they are supplementary if the lines are parallel.
- 18
Angle on a Straight Line
Angles on a straight line add up to 180 degrees.
- 19
Angles Around a Point
The sum of angles around a single point is 360 degrees.
- 20
Central Angle
A central angle is an angle at the center of a circle, with its sides passing through two points on the circle.
- 21
Inscribed Angle
An inscribed angle is an angle formed by two chords in a circle that meet at a point on the circle.
- 22
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
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Interior Angle of a Polygon
An interior angle of a polygon is an angle inside the polygon at each vertex.
- 24
Interior Angle Sum of a Polygon
The sum of the interior angles of a polygon with n sides is (n-2) times 180 degrees.
- 25
Exterior Angle of a Polygon
An exterior angle of a polygon is formed by extending one side, and it equals the sum of the two non-adjacent interior angles.
- 26
Exterior Angle Sum of a Polygon
The sum of the exterior angles of any polygon, one at each vertex, is always 360 degrees.
- 27
Regular Polygon Angles
In a regular polygon, all interior angles are equal, and each measures [(n-2) times 180 degrees] divided by n, where n is the number of sides.
- 28
Sine of an Angle
In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
- 29
Cosine of an Angle
In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
- 30
Tangent of an Angle
In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
- 31
Pythagorean Theorem
In a right triangle, the square of the hypotenuse's length equals the sum of the squares of the other two sides.
- 32
Angle Bisector
An angle bisector is a line that divides an angle into two equal parts.
- 33
Perpendicular Lines
Perpendicular lines intersect at a 90-degree angle.
- 34
Common Trap: Confusing Complementary and Supplementary
Students often mix up complementary angles, which sum to 90 degrees, with supplementary angles, which sum to 180 degrees.
- 35
Strategy for Angle Chasing
To solve angle problems, start by identifying known angles and use properties like vertical angles or supplementary pairs to find unknowns step by step.
- 36
Example: Angles in a Triangle
If a triangle has angles of 40 degrees and 60 degrees, the third angle is 80 degrees since the total must be 180 degrees.
In triangle ABC, angle A is 40°, angle B is 60°, so angle C is 80°.
- 37
Vertical Angles Example
When two lines intersect, the vertical angles are equal; for instance, if one is 70 degrees, the opposite is also 70 degrees.
Lines intersect forming 70° and 110° angles; the vertical angle to 70° is 70°.
- 38
Parallel Lines with Transversal Example
When a transversal crosses parallel lines, corresponding angles are equal, such as both being 50 degrees if one is.
Transversal crosses parallel lines; if one corresponding angle is 50°, the matching one is also 50°.
- 39
Inscribed Angle Example
An inscribed angle subtending a 100-degree arc measures 50 degrees, as it is half the central angle.
In a circle, an arc of 100° has an inscribed angle of 50° at the circumference.
- 40
Polygon Interior Angle Example
For a quadrilateral with four sides, the sum of interior angles is 360 degrees.
A square has four 90° angles, summing to 360°.
- 41
Trigonometry in Right Triangles
In right triangles, trigonometric ratios relate angles to side lengths, helping solve for unknowns.
- 42
Common Trap: Assuming All Angles Equal
In irregular polygons, angles are not equal, so avoid assuming symmetry unless specified.
- 43
Angle Measure in Degrees
Angles are measured in degrees, where a full circle is 360 degrees.
- 44
Example: Supplementary Angles
If one angle is 120 degrees, its supplementary angle is 60 degrees because they add to 180 degrees.
Angle A is 120°, so its supplement is 60°.
- 45
Example: Complementary Angles
If one angle is 30 degrees, its complementary angle is 60 degrees because they add to 90 degrees.
Angle X is 30°, so its complement is 60°.
- 46
Reflex Angle Example
A reflex angle greater than 180 degrees, like 270 degrees, is the larger angle formed by two rays.
Two rays form a 270° reflex angle and a 90° acute angle.
- 47
Central Angle Example
A central angle of 120 degrees subtends an arc that is one-third of the circle.
In a circle, a 120° central angle covers a 120° arc.
- 48
Exterior Angle Example in Triangle
In a triangle with angles 50° and 60°, the exterior angle at the third vertex is 110°.
Triangle with 50° and 60° angles; exterior at 70° vertex is 110°.
- 49
Strategy for Multiple Angles
When dealing with multiple intersecting lines, label all angles and use equality and supplementary properties to solve.
- 50
Common Trap: Forgetting Angle Sum
In polygons, forgetting to subtract 2 from the number of sides when calculating interior angle sum leads to errors.
- 51
Perpendicular Bisector
A perpendicular bisector cuts a line segment into two equal parts at a 90-degree angle.
- 52
Example: Tangent Ratio
For a right triangle with opposite side 3 and adjacent side 4, the tangent of the angle is 3/4 or 0.75.
Angle with opposite 3, adjacent 4; tan = 3/4.
- 53
Isosceles Triangle Angles
In an isosceles triangle, the base angles are equal.
- 54
Equilateral Triangle Angles
In an equilateral triangle, all three angles are equal to 60 degrees.
- 55
Example: Sine Ratio
In a right triangle with opposite side 5 and hypotenuse 13, the sine of the angle is 5/13.
Angle with opposite 5, hypotenuse 13; sin = 5/13.
- 56
Obtuse Triangle Angles
An obtuse triangle has one angle greater than 90 degrees and two less than 90 degrees.