Download Math 131 (Lecture 2) - Fall 2012, Facts and Definitions

Math 131 (Lecture 2) - Fall 2012, Facts and
Definitions
Facts
Lines and Points
Through any two different points, there is one and only one line.
If two different lines intersect, their intersection is a single point.
Collinear Measurement: Lengths along a line add.
Non-collinear Measurement: In any triangle, the length of the longest side is less than the
total length of the other two sides.
Angles
There is an angle with any specified measure.
Adjacent angles add. (∠s add)
The total measure of adjacent angles around a point is 360◦ . (∠s at a pt)
The total measure of adjacent angles forming a straight line is 180◦ . (∠s on a line)
The sum of adjacent angles in a right angle is 90◦ . (∠s in rt. ∠)
Vertical angles have equal measure. (vert. ∠s)
Triangles
In any triangle, the sum of the measures of the interior angles is 180◦ . (∠ sum of ∆)
When one angle of a triangle is a right angle, the measures of the other two angles add up
to 90◦ . (∠ sum of rt. ∆)
The measure of each exterior angle of a triangle is the sum of the measures of the opposite
interior angles. (ext. ∠ of ∆)
Two triangles are congruent if corresponding sides are equal under some correspondence.
(SSS)
Two triangles are congruent if two pairs of corresponding angles and the sides between them
are equal. (ASA)
Two triangles are congruent if two pairs of corresponding sides and their shared angle are
equal. (SAS)
Two right triangles are congruent if they have hypotenuse of equal length, and a pair of legs
with equal length. (RHL)
Quadrilaterals
Opposite angles in a parallelogram are equal. (opp. ∠s k-ogram)
Interior angles between two parallel sides in a trapezoid (or parallelogram) are supplementary. (int. ∠s, AB k CD)
Polygons
The sum of the interior angles of an n-gon is 180(n − 2) degrees. (∠ sum of n-gon)
The sum of the exterior angles, one at each vertex, of a convex polygon is 360◦ . (ext. ∠s
1
of polygon)
Parallel Lines
If a transversal intersects two parallel lines, then corresponding angles are equal. (corr. ∠s,
AB k CD)
If a transversal intersects two lines and corresponding angles are equal, then the two lines
are parallel. (corr. ∠s converse)
If a transversal intersects two parallel lines, then the interior angles on the same side of the
transversal are supplementary. (int. ∠s, AB k CD)
If a transversal intersects two lines and the interior angles on the same side of the transversal
are supplementary, then the two lines are parallel. (int. ∠s converse)
If a transversal intersects two parallel lines, then alternating angles are equal. (alt. ∠s,
AB k CD)
If a transversal intersects two lines and alternating angles are equal, then the two lines are
parallel. (alt. ∠s converse)
Areas/Perimeters
For any region R, Area(R) ≥ 0. (area ≥ 0)
Congruent figures have equal area. (∼
= figs)
If two figures R and S intersect only in vertices and sides (or not at all), then their areas
add: Area(R ∪ S) = Area(R) + Area(S). (areas add)
The area of a rectangular region R is the product of the lengths of two adjacent sides:
Area(R) = Length × Width. (area of rect.)
For any two regions R and S, Area(R ∪ S) = Area(R) + Area(S) − Area(R ∩ S).
Among rectangles with a fixed perimeter, squares have the largest area.
Among rectangles with a fixed area, squares have the smallest perimeter.
Area of triangle = 12 Base × Height.
Area of parallelogram = Base × Height.
Pythagorean Theorem
If a right triangle has legs of lengths a and b and hypotenuse of length c, then a2 + b2 = c2 .
(Converse) If the lengths a, b, c of the sides of a triangle are related by a2 + b2 = c2 , then the
triangle is a right triangle (with the right angle√
opposite the side of length c).
In an isosceles right triangle, the hypotenuse is 2 times as long as each leg. (45-45-90 4)
In
√ a 30-60-90 triangle, the hypotenuse is twice as long as the short leg, and the other leg is
3 times the short leg. (30-60-90 4)
Similarity
Scaling a triangle yields a similar triangle.
If two triangles have three pairs of equal angles, then the triangles are similar. (equiangular/AAA)
If all 3 sides have the same scale factor, then the triangles are similar. (3 sides proportional)
If two pairs of sides have the scale factor and the included angles are equal, then the triangles
are similar. (ratio of 2 sides, incl. ∠)
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In a right triangle, the altitude to the hypotenuse separates the triangle into two triangles,
each similar to the original. (Mother-Daughter Theorem)
When parallel lines intersect two crossed lines, the triangles formed are similar. (k cuts)
If two figures are similar and are related by a scale factor of k, then their areas are related
by a factor of k 2 .
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Definitions
Lines
Line: Infinite on both ends and determined by two points.
Line segment: The part of a line between two points.
Ray: The part of a line on one side of a point.
Transversal: Given a pair of lines in a plane, a third line is a transversal if it intersects
them both in distinct a single point.
Angles
Acute angle: Less than 90◦
Right angle: 90◦
Obtuse angle: Between 90◦ and 180◦
Straight angle: 180◦
Reflex angle: Between 180◦ and 360◦
Full rotation (complete turn): 360◦
Adjacent angles: Two angles with the same vertex that share a side.
Bisector: A ray that separates the angle into two adjacent angles with equal measure.
Supplementary: Two angles whose measure sums to 180◦ .
Complementary: Two angles whose measure sums to 90◦ .
Perpendicular: Two segments, rays, or lines are perpendicular if the lines containing them
←→
←→
←→ ←→
intersect to form a 90◦ angle. If AB is perpendicular to CD, we write AB ⊥ CD.
Parallel: Two lines, segments, or rays are parallel if they lie in the same plane and are both
←→
←→
←→ ←→
perpendicular to a third line. If AB is parallel to CD, we write AB k CD.
Congruent: Two angles are congruent if they have the same measure.
Circles
Circle: The circle with center P and radius R is the set of all points in the plane that are
distance R from the point P .
Triangles
Triangle:
• Grade 2 - A figure enclosed by 3 straight segments.
• Grade 4-7 - A triangle consists of 3 non-collinear points and the line segments joining
them. (The points are called vertices, and the segments are called the sides of the
triangle.)
Exterior angle: An angle formed by one side of a triangle and the straight extension of
another side is called and exterior angle of the triangle.
Classified by angles:
• Acute: An acute triangle has all angles with measure less than 90◦ .
• Right: A right triangle has one angle with measure equal to 90◦ .
• Obtuse: An obtuse triangle has one angle with measure more than 90◦ .
Classified by side lengths:
• Equilateral: An equilateral triangle has all sides the same length.
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• Isosceles: An isosceles triangle has two sides the same length.
• Scalene: A scalene triangle has all sides of different length.
Quadrilaterals
Quadrilateral:
• Grade 2 - A figure enclosed by 4 straight segments.
• Grade 4-7 - A quadrilateral consists of 4 distinct points A, B, C, D in the plane, no
three of which are collinear, and four segments AB, BC, CD, DA that intersect only
at their endpoints. (The endpoints are called vertices, and the segments are called the
sides of the quadrilateral.)
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Equivalent criteria:
• Pairs of opposite sides are congruent.
• Pairs of opposite angles are congruent.
• The diagonals bisect each other.
• One pair of opposite sides are parallel and congruent.
Rectangle: A parallelogram all of whose angles are right angles.
Other rectangle facts:
• If a figure is a rectangle, then it has all the properties of a parallelogram.
• If a figure is a rectangle, then its diagonals are equal length.
• If a figure is a parallelogram with diagonals equal length, then it’s a rectangle.
Rhombus: A parallelogram with all sides of equal length.
Other rhombus facts:
• If a figure is a rhombus, then it has all the properties of a parallelogram and a kite.
• If a figure is a rhombus, then it has that the diagonals bisect the interior angles.
• If a figure is a rhombus, then it has that the diagonals are perpendicular.
• If a figure is a parallelogram with diagonals that bisect the interior angles, then it’s a
rhombus.
Square: A rectangle with all sides of equal length.
Other square facts:
• If a figure is a square, then it has all the properties of parallelograms, rhombuses, kites,
and rectangles.
• If a figure has diagonals that are equal length and perpendicular bisectors of each other,
then it is a square.
Trapezoid: A quadrilateral in which at least one pair of opposite sides are parallel. The
parallel sides are called bases of the trapezoid and the other two sides are called transversals.
Kite: A quadrilateral in which two consecutive sides have equal length and the remaining
two sides also have equal length.
Other kite facts:
• If a figure is a kite, then at least one pair of opposite angles is equal, but the converse
is not necessarily true.
• If a figure is a kite, then the diagonals are perpendicular to each other, but the converse
is not necessarily true.
• If a figure is a kite, then there is a diagonal that bisects a pair of opposite angles, and
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the converse is also true.
Area and Shapes
Unit Square: For each unit of length a unit square is any square whose sides are 1 unit
long. The corresponding unit of area, called a square unit is the area of a unit square.
Perimeter: The perimeter of a polygon is the sum of the lengths of the sides of the polygon.
Altitude of a triangle: Any side of a triangle can be called the base. The corresponding
altitude is the segment from the opposite vertex to the baseline that intersects the baseline
perpendicularly.
Altitude of a parallelogram: Any side of a parallelogram can be called the base. An
altitude corresponding to this base is a segment perpendicular to the base from any point
on the opposite side.
Pythagorean TheoremIrrational Numbers: A number is irrational if it cannot be
written as a fraction pq where p, q are integers.
Square Root: Every non-negative real number
√ x has a unique non-negative square root,
called the principal square root and denoted x. Pythagorean Triples: Triples of whole
numbers (a, b, c) satisfying a2 + b2 = c2 are called pythagorean triples. Multiples of these
numbers also form triples.
Common triples:
• (3,4,5)
• (5,12,13)
• (6,8,10)
• (7,24,25)
• (8,15,17)
SimilarityProportionality: We say that corresponding sides are proportional if there is
a number k such that each side of one triangle is k times the length of the corresponding side
of the other triangle. The number k is called the scale factor or proportionality constant.
Similar Triangles: We say that two triangles are similar if, under some correspondence,
corresponding angles are equal and corresponding sides are proportional.
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