Arithmetic Reasoning and Mathematics Knowledge Review: Geometry

Lines and Planes
A point is a fixed location in space; has no size or dimensions; commonly represented by a dot. A line is a set of points that extends infinitely in two opposite directions. It has length, but no width or depth. A line can be defined by any two distinct points that it contains. A line segment is a portion of a line that has definite endpoints. A ray is a portion of a line that extends from a single point on that line in one direction along the line. It has a definite beginning, but no ending.

Point - Line - Segment - Ray



Line



Segment



Ray



Intersecting lines are lines that have exactly one point in common. Concurrent lines are multiple lines that intersect at a single point. Perpendicular lines are lines that intersect at right angles. They are represented by the symbol

.

The shortest distance from a line to a point not on the line is a perpendicular segment from the point to the line. Parallel lines are lines in the same plane that have no points in common and never meet. It is possible for lines to be in different planes, have no points in common, and never meet, but they are not parallel because they are in different planes.

Intersecting - Concurrent - Perpendicular - Parallel




A transversal is a line that intersects at least two other lines, which may or may not be parallel to one another. A transversal that intersects parallel lines is a common occurrence in geometry. A bisector is a line or line segment that divides another line segment into two equal lengths. A perpendicular bisector of a line segment is composed of points that are equidistant from the endpoints of the segment it is dividing.

Transversal Bisector
Perpendicular bisector






The projection of a point on a line is the point at which a perpendicular line drawn from the given point to the given line intersects the line.  This is also the shortest distance from the given point to the line. The projection of a segment on a line is a segment whose endpoints are the points formed when perpendicular lines are drawn from the endpoints of the given segment to the given line. 

This is similar to the length a diagonal line appears to be when viewed from above.

Projection of a point on a line
Projection of a segment on a line




A plane is a two-dimensional flat surface defined by three non-collinear points. A plane extends an infinite distance in all directions in those two dimensions. It contains an infinite number of points, parallel lines and segments, intersecting lines and segments, as well as parallel or intersecting rays. A plane will never contain a three-dimensional figure or skew lines, lines that don't intersect and are not parallel. Two given planes are either parallel or they intersect at a line. A plane may intersect a circular conic surface to form conic sections, such as a parabola, hyperbola, circle or ellipse.



Angles
An angle is formed when two lines or line segments meet at a common point. It may be a common starting point for a pair of segments or rays, or it may be the intersection of lines. Angles are represented by the symbol ∠.

The vertex is the point at which two segments or rays meet to form an angle. If the angle is formed by intersecting rays, lines, and/or line segments, the vertex is the point at which four angles are formed. The pairs of angles opposite one another are called vertical angles, and their measures are equal. A. acute angle is an angle with a degree measure less than 90°.

A right angle is an angle with a degree measure of exactly A. obtuse angle is an angle with a degree measure greater than 90° but less than 180°.
A straight angle is an angle with a degree measure of exactly 180°. This is also a semicircle.
A reflex angle is an angle with a degree measure greater than 180° but less than 360°.
A full angle is an angle with a degree measure of exactly 360°.
Two angles whose sum is exactly 90° are said to be complementary.
The two angles may or may not be adjacent. In a right triangle, the two acute angles are complementary.
Two angles whose sum is exactly 180° are said to be supplementary.
The two angles may or may not be adjacent. Two intersecting lines always form two pairs of supplementary angles. Adjacent supplementary angles will always form a straight line.

Two angles that have the same vertex and share a side are said to be adjacent. Vertical angles are not adjacent because they share a vertex but no common side.





When two parallel lines are cut by a transversal, the angles that are between the two parallel lines are interior angles. In the diagram below, angles 3, 4, 5, and 6 are interior angles. that are outside the parallel lines are exterior angles. In the diagram below, angles 1, 2, 7, and 8 are exterior angles. that are in the same position relative to the transversal and a parallel line are corresponding angles. The diagram below has four pairs of corresponding angles: angles 1 and 5; angles 2 and 6; angles 3 and 7; and angles 4 and 8. Corresponding angles formed by parallel lines are congruent.
When two parallel lines are cut by a transversal, the two interior angles that are on opposite sides of the transversal are called alternate interior angles. In the diagram below, there are two pairs of alternate interior angles: angles 3 and 6, and angles 4 and 5. Alternate interior angles formed by parallel lines are congruent.
When two parallel lines are cut by a transversal, the two exterior angles that are on opposite sides of the transversal are called alternate exterior angles.

In the diagram below, there are two pairs of alternate exterior angles: angles 1 and 8, and angles 2 and 7.
Alternate exterior angles formed by parallel lines are congruent.


When two lines intersect, four angles are formed. The non-adjacent angles at this vertex are called vertical angles. Vertical angles are congruent. In the diagram,

 and

.


P1. Find the measure of angles (a), (b), and (c) based on the figure with two parallel lines, two perpendicular lines and one transversal:


P1. (a) The vertical angle paired with (a) is part of a right triangle with the 40° angle. Thus the measure can be found:





The triangle formed by the supplementary angle to (b) is part of a triangle with the vertical angle paired with (a) and the given angle of 27°. Since

:











(c) As they are part of a transversal crossing parallel lines, angles (b) and (c) are supplementary. Thus





Triangles
A triangle is a three-sided figure with the sum of its interior angles being

. For an equilateral triangle, this is the same as

, where a is any side length, since all three sides are the same length.


The area of any triangle can be found by taking half the product of one side length referred to as the base, often given the variable b and the perpendicular distance from that side to the opposite vertex called the altitude or height and given the variable h. In equation form that is
a, b, and c are the lengths of the three sides. Special cases include isosceles triangles:
a is the length of a side.


Quadrilaterals
A quadrilateral is a closed two-dimensional geometric figure that has four straight sides. The sum of the interior angles of any quadrilateral is

.



Trapezoid: A trapezoid is defined as a quadrilateral that has at least one pair of parallel sides.
There are no rules for the second pair of sides. So, there are no rules for the diagonals and no lines of symmetry for a trapezoid.






The area of a trapezoid is found by the formula
a, b1, c, and b2 are the four sides of the trapezoid.



Parallelogram: A quadrilateral that has two pairs of opposite parallel sides. As such it is a special type of trapezoid. The sides that are parallel are also congruent. The opposite interior angles are always congruent, and the consecutive interior angles are supplementary. The diagonals of a parallelogram divide each other. Each diagonal divides the parallelogram into two congruent triangles. A parallelogram has no line of symmetry, but does have 180-degree rotational symmetry about the midpoint.


The area of a parallelogram is found by the formula
a and b are the lengths of the two sides.

Isosceles trapezoid: A trapezoid with equal base angles. This gives rise to other properties including: the two nonparallel sides have the same length, the two non-base angles are also equal, and there is one line of symmetry through the midpoints of the parallel sides.



Rectangle: A quadrilateral with four right angles. All rectangles are parallelograms and trapezoids, but not all parallelograms or trapezoids are rectangles. The diagonals of a rectangle are congruent. Rectangles have 2 lines of symmetry
(through each pair of opposing midpoints) and 180-degree rotational symmetry about the midpoint.



The area of a rectangle is found by the formula
l is the length, and w is the width. It may be easier to add the length and width first and then double the result, as in the second formula.

Square: A quadrilateral with four right angles and four congruent sides. Squares satisfy the criteria of all other types of quadrilaterals. The diagonals of a square are congruent and perpendicular to each other. Squares have 4 lines of symmetry
(through each pair of opposing midpoints and along each of the diagonals) as well as 90-degree rotational symmetry about the midpoint.



The area of a square is found by using the formula
s is the length of one side. Because all four sides are equal in a square, it is faster to multiply the length of one side by 4 than to add the same number four times. You could use the formulas for rectangles and get the same answer.

Circles
The center of a circle is the single point from which every point on the circle is equidistant.

The radius is a line segment that joins the center of the circle and any one point on the circle. All radii of a circle are equal. Circles that have the same center, but not the same length of radii are concentric. The diameter is a line segment that passes through the center of the circle and has both endpoints on the circle. The length of the diameter is exactly twice the length of the radius. Point O in the diagram below is the center of the circle, segments

,

, and

 are radii, and segment

is a diameter.




Diameter, Radius, and Circumference of Circles

The area of a circle is found by the formula
r is the radius. Again, remember to convert the diameter if you are given that measure rather than the radius.

P1. Find the area and perimeter of the following quadrilaterals:
(a) A square with side length 2.5 cm.
A parallelogram with height 3 m, base 4 m, and other side 6 m.
P2. Calculate the area of a triangle with side lengths of 7 ft, 8 ft, and 9 ft.
P1. (a)

;


(b)


P2. Given only side lengths, we can use the semi perimeter to the find the area based on the formula,

:









Triangle Classification and Properties
A scalene triangle is a triangle with no congruent sides. A scalene triangle will also have three angles of different measures. The angle with the largest measure is opposite the longest side, and the angle with the smallest measure is opposite the shortest side. An acute triangle is a triangle whose three angles are all less than 90°. If two of the angles are equal, the acute triangle is also an isosceles triangle. An isosceles triangle will also have two congruent angles opposite the two congruent sides. If the three angles are all equal, the acute triangle is also an equilateral triangle. An equilateral triangle will also have three congruent angles, each 60°. All equilateral triangles are also acute triangles. An obtuse triangle is a triangle with exactly one angle greater than 90°. The other two angles may or may not be equal. If the two remaining angles are equal, the obtuse triangle is also an isosceles triangle. A right triangle is a triangle with exactly one angle equal to 90°. All right triangles follow the Pythagorean theorem. A right triangle can never be acute or obtuse.

The table below illustrates how each descriptor places a different restriction on the triangle:



Isosceles:
Two equal side



















Equilateral:
Three equal side








Similarity and Congruence Rules
Similar triangles are triangles whose corresponding angles are equal and whose corresponding sides are proportional. Represented by AAA. Similar triangles whose corresponding sides are congruent are also congruent triangles.
The triangles can be shown to be congruent in 5 ways:
SSS: Three sides of one triangle are congruent to the three corresponding sides of the second triangle.
SAS: Two sides and the included angle (the angle formed by those two sides) of one triangle are congruent to the corresponding two sides and included angle of the second triangle.
ASA: Two angles and the included side (the side that joins the two angles) of one triangle are congruent to the corresponding two angles and included side of the second triangle.
AAS: Two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of the second triangle.
HL: The hypotenuse and leg of one right triangle are congruent to the corresponding hypotenuse and leg of the second right triangle.

General Rules for Triangles
The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is always greater than the measure of the third side. If the sum of the measures of two sides were equal to the third side, a triangle would be impossible because the two sides would lie flat across the third side and there would be no vertex. If the sum of the measures of two of the sides was less than the third side, a closed figure would be impossible because the two shortest sides would never meet. In other words, for a triangle with sides lengths A, B, and C:

,

, and


The sum of the measures of the interior angles of a triangle is always 180°. Therefore, a triangle can never have more than one angle greater than or equal to 90°.
In any triangle, the angles opposite congruent sides are congruent, and the sides opposite congruent angles are congruent. The largest angle is always opposite the longest side, and the smallest angle is always opposite the shortest side.
The line segment that joins the midpoints of any two sides of a triangle is always parallel to the third side and exactly half the length of the third side.

P1. Given the following pairs of triangles, determine whether they are similar, congruent, or neither (note that the figures are not drawn to scale):
(a).


(b).


(c).


P1. (a). Neither: We are given that two sides lengths and an angle are equal, however, the angle given is not between the given side lengths. That means there are two possible triangles that could satisfy the given measurements. Thus, we cannot be certain of congruence:


(b) Similar: Since we are given a side-angle-side of each triangle and the side lengths given are scaled evenly

 and the angles are equal. Thus,

. If the side lengths were equal, then they would be congruent.
(c) Congruent: Even though we aren't given a measurement for the shared side of the figure, since it is shared it is equal. So, this is a case of SAS. Thus,



Three-Dimensional Shapes
The surface area of a solid object is the area of all sides or exterior surfaces. For objects such as prisms and pyramids, a further distinction is made between base surface area (B) and lateral surface area (LA). For a prism, the total surface area
(SA) is

. For a pyramid or cone, the total surface area is

.

The surface area of a sphere can be found by the formula
r is the radius. Both quantities are generally given in terms of π.



The volume of any prism is found by the formula

, where P is the perimeter of the base.



For a rectangular prism, the volume can be found by the formula

 or

.



The volume of a cube can be found by the formula
SA is the total surface area and s is the length of a side. These formulas are the same as the ones used for the volume and surface area of a rectangular prism, but simplified since all three quantities (length, width, and height) are the same.

The volume of a cylinder can be calculated by the formula

. The first term is the base area multiplied by two, and the second term is the perimeter of the base multiplied by the height.



The volume of a pyramid is found by the formula

 times the volume of a prism. Like a prism, the base of a pyramid can be any shape.
Finding the surface area of a pyramid is not as simple as the other shapes we've looked at thus far.

If the pyramid is a right pyramid, meaning the base is a regular polygon and the vertex is directly over the center of that polygon, the surface area can be calculated as



The volume of a cone is found by the formula

 times the volume of a cylinder. The surface area can be calculated as

, where s is the slant height. The slant height can be calculated using the Pythagorean theorem to be

, so the surface area formula can also be written as

.



P1. Find the surface area and volume of the following solids:
A cylinder with radius 5 m and height 0.5 m.

A trapezoidal prism with base area of 254

, base perimeter 74 mm, and height 10 mm.
(c) A half sphere (radius 5 yds) on the base of an inverted cone with the same radius and a height of 7 yds.
P1. (a)

;



(b)

;



(c) We can find s, the slant height using Pythagoras' theorem, and since this solid is made of parts of simple solids, we can combine the formulas to find surface area and volume: