Plane geometry
61 flashcards covering Plane geometry for the ACT Math section.
Plane geometry is the study of flat, two-dimensional shapes and their properties, focusing on things like points, lines, angles, triangles, circles, and polygons. It explores how these shapes relate to one another through concepts such as congruence, similarity, and parallelism, and involves calculating measurements like lengths, areas, and angles. This foundational topic helps build problem-solving skills that apply to real-world scenarios, from architecture to navigation.
On the ACT Math section, plane geometry appears in questions that test your ability to work with shapes, such as finding areas, perimeters, or angles in triangles and circles, or solving problems with coordinate planes. Common traps include misapplying theorems like the Pythagorean theorem or overlooking scale in similar figures, so watch for tricky wording that might lead to calculation errors. Focus on key formulas, angle relationships, and visualizing problems to handle these efficiently.
Practice sketching diagrams for every geometry question.
Terms (61)
- 01
Point
A point is a location in space that has no size or dimension and is represented by a dot.
- 02
Line
A line is a straight path that extends infinitely in both directions and is defined by two points.
- 03
Plane
A plane is a flat, two-dimensional surface that extends infinitely in all directions and contains points, lines, and shapes.
- 04
Angle
An angle is formed by two rays sharing a common endpoint, measured in degrees, and can be acute, right, obtuse, or straight.
- 05
Parallel lines
Parallel lines are lines in a plane that never intersect and maintain the same distance apart from each other.
- 06
Perpendicular lines
Perpendicular lines are lines in a plane that intersect at a right angle, forming 90-degree angles.
- 07
Vertical angles
Vertical angles are the pairs of opposite angles formed by two intersecting lines, and they are always equal in measure.
- 08
Complementary angles
Complementary angles are two angles whose measures add up to 90 degrees.
- 09
Supplementary angles
Supplementary angles are two angles whose measures add up to 180 degrees.
- 10
Triangle
A triangle is a polygon with three sides and three angles, and the sum of its interior angles is always 180 degrees.
- 11
Equilateral triangle
An equilateral triangle is a triangle with all three sides equal in length and all three angles equal to 60 degrees.
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Isosceles triangle
An isosceles triangle is a triangle with at least two sides equal in length, resulting in at least two equal angles.
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Scalene triangle
A scalene triangle is a triangle with all three sides of different lengths and all three angles of different measures.
- 14
Right triangle
A right triangle is a triangle with one angle measuring exactly 90 degrees, and it follows the Pythagorean theorem for its sides.
- 15
Acute triangle
An acute triangle is a triangle where all three interior angles are less than 90 degrees.
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Obtuse triangle
An obtuse triangle is a triangle with one interior angle greater than 90 degrees and the other two less than 90 degrees.
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Pythagorean theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides.
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Congruent triangles
Congruent triangles are triangles that have the same size and shape, meaning all corresponding sides and angles are equal.
- 19
Similar triangles
Similar triangles are triangles that have the same shape but not necessarily the same size, with corresponding angles equal and sides proportional.
- 20
Area of a triangle
The area of a triangle is calculated using the formula one-half times base times height, where the base and height form a right angle.
- 21
Perimeter of a triangle
The perimeter of a triangle is the total length around it, found by adding the lengths of all three sides.
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Quadrilateral
A quadrilateral is a polygon with four sides and four angles, and the sum of its interior angles is 360 degrees.
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Square
A square is a quadrilateral with all four sides equal and all four angles equal to 90 degrees.
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Rectangle
A rectangle is a quadrilateral with opposite sides equal and all four angles equal to 90 degrees.
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Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel and equal in length, and opposite angles equal.
- 26
Rhombus
A rhombus is a quadrilateral with all four sides equal in length, though its angles are not necessarily 90 degrees.
- 27
Trapezoid
A trapezoid is a quadrilateral with at least one pair of parallel sides, called the bases.
- 28
Kite
A kite is a quadrilateral with two pairs of adjacent sides equal in length, forming two pairs of equal angles.
- 29
Polygon
A polygon is a closed shape with straight sides, and the sum of its interior angles is (n-2) times 180 degrees, where n is the number of sides.
- 30
Regular polygon
A regular polygon is a polygon with all sides equal and all interior angles equal, such as an equilateral triangle or square.
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Interior angle sum of a polygon
The sum of the interior angles of a polygon with n sides is (n-2) multiplied by 180 degrees.
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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.
- 33
Circle
A circle is a set of points in a plane equidistant from a fixed center point, with that distance being the radius.
- 34
Radius
The radius of a circle is the distance from the center to any point on the circle.
- 35
Diameter
The diameter of a circle is a line segment passing through the center and connecting two points on the circle, equal to twice the radius.
- 36
Circumference
The circumference of a circle is the distance around it, calculated using the formula 2 times pi times the radius.
- 37
Area of a circle
The area of a circle is the space inside it, found using the formula pi times the radius squared.
- 38
Arc
An arc is a portion of the circumference of a circle, and its length is a fraction of the full circumference based on the central angle.
- 39
Sector
A sector is a region bounded by two radii and an arc of a circle, like a slice of pie.
- 40
Tangent to a circle
A tangent to a circle is a straight line that touches the circle at exactly one point and is perpendicular to the radius at that point.
- 41
Chord
A chord is a straight line segment with endpoints on the circle, and the longest chord is the diameter.
- 42
Coordinate plane
The coordinate plane is a two-dimensional surface formed by two perpendicular number lines, called the x-axis and y-axis, intersecting at the origin.
- 43
Distance formula
The distance formula calculates the length between two points (x1, y1) and (x2, y2) as the square root of [(x2 - x1)^2 + (y2 - y1)^2].
- 44
Midpoint formula
The midpoint formula finds the midpoint of a line segment between two points (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2).
- 45
Slope of a line
The slope of a line is a measure of its steepness, calculated as the change in y-coordinates divided by the change in x-coordinates between two points.
- 46
Equation of a line
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
- 47
Parallel lines in coordinate plane
Parallel lines in the coordinate plane have the same slope but different y-intercepts.
- 48
Perpendicular lines in coordinate plane
Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.
- 49
Reflection over a line
Reflection over a line is a transformation that flips a point or shape across the line, resulting in a mirror image.
- 50
Rotation
Rotation is a transformation that turns a point or shape around a fixed point by a certain angle, such as 90 or 180 degrees.
- 51
Translation
Translation is a transformation that slides a point or shape in a specified direction without rotating or flipping it.
- 52
Area of a rectangle
The area of a rectangle is calculated by multiplying its length by its width.
- 53
Perimeter of a rectangle
The perimeter of a rectangle is the sum of all its sides, calculated as 2 times (length plus width).
- 54
Special right triangles
Special right triangles include 30-60-90 and 45-45-90 triangles, which have specific side ratios that simplify calculations.
- 55
Sine in right triangles
Sine is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
- 56
Cosine in right triangles
Cosine is the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
- 57
Tangent in right triangles
Tangent is the ratio of the length of the opposite side to the adjacent side in a right triangle.
- 58
Common trap: Confusing area and perimeter
A common error is mixing up area, which is a two-dimensional measure, with perimeter, which is the boundary length, leading to incorrect calculations.
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Strategy for similar triangles problems
To solve similar triangles problems, set up proportions using corresponding sides and ensure angles are equal, then cross-multiply to find unknown lengths.
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Example: Finding the area of a triangle
For a triangle with base 6 units and height 4 units, the area is calculated as one-half times base times height, resulting in 12 square units.
If base is 6 and height is 4, area = 0.5 6 4 = 12.
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Example: Using Pythagorean theorem
To find the hypotenuse of a right triangle with legs 3 and 4, apply the Pythagorean theorem: hypotenuse squared equals 3 squared plus 4 squared, so hypotenuse is 5.
For legs 3 and 4, hypotenuse = sqrt(9 + 16) = sqrt(25) = 5.