Geometry: Praxis I Exam (page 6)

This section will familiarize you with the properties of angles, lines, polygons, triangles, and circles, as well as the formulas for area, volume, and perimeter.

Geometry is the study of shapes and the relationships among them. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and concepts. Therefore, a list of definitions and important formulas is provided.

Geometry Terms

Coordinate Geometry

Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal x-axis and a vertical y-axis. These two axes intersect at one coordinate point, (0,0), the origin. A coordinate point, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and the second number representing the vertical placement. Coordinate points are given in the form of (x,y).

Graphing Ordered Pairs (Points)

• The x-coordinate is listed first in the ordered pair and tells you how many units to move to either the left or the right. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left.
• The y-coordinate is listed second and tells you how many units to move up or down. If the y-coordinate is positive, move up. If the y-coordinate is negative, move down.

Example

Graph the following points: (2,3), (3,–2), (–2,3), and (–3,–2).



• Notice that the graph is broken up into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs based on their signs:

Lines, Line Segments, and Rays

A line is a straight geometric object that goes on forever in both directions. It is infinite in length, and is represented by a straight line with an arrow at both ends. Lines can be labeled with one letter (usually in italics) or with two capital letters near the arrows. Line segments are portions of lines. They have two endpoints and a definitive length. Line segments are named by their endpoints. Rays have an endpoint and continue straight in one direction. Rays are named by their endpoint and one point on the ray.

Parallel and Perpendicular Lines

Parallel lines (or line segments) are a pair of lines that, if extended, would never intersect or meet. The symbol || is used to denote that two lines are parallel. Perpendicular lines (or line segments) are lines that intersect to form right angles, and are denoted with the symbol .

Lengths of Horizontal and Vertical Segments

Two points with the same y-coordinate lie on the same horizontal line and two points with the same x-coordinate lie on the same vertical line. The distance between a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points.

Example

Find the lengths of line segments and

Distance of Coordinate Points

The distance between any two points is given by the formula d = , where (x₁,y₁) represents the coordinates of one point and (x₂,y₂) is the other. The subscripts are used to differentiate between the two different coordinate pairs.

Example

Find the distance between points A(–3,5) and B(1,–4).

Let (x₁,y₁) represent point A and let (x₂,y₂) represent point B. This means that x₁ = –3, y₁ = 5, x₂ = 1, and y₂ = –4. Substituting these values into the formula gives us:

Midpoint

The midpoint of a line segment is a point located at an equal distance from each endpoint. This point is in the exact center of the line segment, and is said to be equidistant from the segment's endpoints.

In coordinate geometry, the formula for finding the coordinates of the midpoint of a line segment whose endpoints are (x₁,y₁) and (x₂,y₂) is given by M =

Example

Determine the midpoint of the line segment with A(–3,5) and B(1,–4).

Let (x₁,y₁) represent point A and let (x₂,y₂) represent point B. This means that x₁ = –3, y₁ = 5, x₂ = 1, and y₂ = –4. Substituting these values into the formula gives us:

Note: There is no such thing as the midpoint of a line, as lines are infinite in length.

Slope

The slope of a line (or line segment) is a numerical value given to show how steep a line is. A line or segment can have one of four types of slope:

1. A line with a positive slope increases from the bottom left to the upper right on a graph.
2. A line with a negative slope decreases from the upper left to the bottom right on a graph.
3. A horizontal line is said to have a zero slope.
4. A vertical line is said to have no slope (undefined).

Parallel lines have equal slopes.

Perpendicular lines have slopes that are negative reciprocals of each other.

The slope of a line can be found if you know the coordinates of any two points that lie on the line. It does not matter which two points you use. It is found by writing the change in the y-coordinates of any two points on the line, over the change in the corresponding x-coordinates. (This is also known as the rise over the run.)

The formula for the slope of a line (or line segment) containing points (x₁, y₁) and (x₂, y₂):

Example

Determine the slope of the line joining points A(–3,5) and B(1,–4).

Let (x₁,y₁) represent point A and let (x₂,y₂) represent point B. This means that x₁ = –3, y₁ = 5, x₂ = 1, and y₂ = –4. Substituting these values into the formula gives us:

 

Example

Determine the slope of the line graphed below.

Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x₁, y₁), and let (0,–1) = (x₂, y₂). This means that x₁ = 3, y₁ = 1, x₂ = 0, and y₂ = –1. Substituting these values into the formula gives us:

Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope , move up seven units and to the right five units. Another point on the line, thus, is (13,16).

Determining the Equation of a Line

The equation of a line is given by y = mx + b where:

• y and x are variables such that every coordinate pair (x,y) is on the line
• m is the slope of the line
• b is the y-intercept, the y-value at which the line intersects (or intercepts) the y-axis

In order to determine the equation of a line from a graph, determine the slope and y-intercept and substitute it in the appropriate place in the general form of the equation.

Example

Determine the equation of the line in the graph below.

In order to determine the slope of the line, choose two points that can be easily determined on the graph. Two easy points are (–1,4) and (1,–4). Let (–1,4) = (x₁, y₁), and let (1,–4) = (x₂, y₂). This means that x₁ = –1, y₁ = 4, x₂ = 1, and y₂ = –4. Substituting these values into the formula gives us:

Looking at the graph, we can see that the line crosses the y-axis at the point (0,0). The y-coordinate of this point is 0. This is the y-intercept.

Substituting these values into the general formula gives us y = –4x + 0, or just y = –4x.

Example

Determine the equation of the line in the following graph.

Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x₁,y₁), and let (3,6) = (x₂,y₂). Substituting these values into the formula gives us:

We can see from the graph that the line crosses the y-axis at the point (0,4). This means the y-intercept is 4.

Substituting these values into the general formula gives us y = x + 4.

Angles

Naming Angles

An angle is a figure composed of two rays or line segments joined at their endpoints. The point at which the rays or line segments meet is called the vertex of the angle. Angles are usually named by three capital letters, where the first and last letter are points on the end of the rays, and the middle letter is the vertex.

This angle can be named either ABC or CBA, but because the vertex of the angle is point B, letter B must be in the middle.

We can sometimes name an angle by its vertex, as long as there is no ambiguity in the diagram. For example, in the angle above, we may call the angle B, because there is only one angle in the diagram that has B as its vertex.

But, in the following diagram, there are a number of angles that have point B as their vertex, so we must name each angle in the diagram with three letters.

Angles may also be numbered (not measured) with numbers written between the sides of the angles, in the interior of the angle, near the vertex.

Classifying Angles

The unit of measure for angles is the degree.

Angles can be classified into the following categories: acute, right, obtuse, and straight.

• An acute angle is an angle that measures between 0 and 90 degrees.



• A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex.



• An obtuse angle is an angle that measures more than 90°, but less than 180



• A straight angle is an angle that measures 180°. Thus, both of its sides form a line.

Special Angle Pairs

• Adjacent angles are two angles that share a common vertex and a common side. There is no numerical relationship between the measures of the angles.



• A linear pair is a pair of adjacent angles whose measures add to 180°.
• Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplementary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear pair.



• Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or may not be adjacent.

Example

Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles?

Because the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x:

2x + 3x + 20 = 90

5x + 20 = 90

5x = 70

x = 14

Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary.

Example

One angle is 40 more than 6 times its supplement. What are the measures of the angles?

Let x = one angle.

Let 6x + 40 = its supplement.

Because the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x:

x + 6x + 40 = 180

7x + 40 = 180

7x = 140

x = 20

Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, proving that the angles are supplementary.

Note: A good way to remember the difference between supplementary and complementary angles is that the letter c comes before s in the alphabet; likewise 90 comes before 180 numerically.

Angles of Intersecting Lines

Important mathematical relationships between angles are formed when lines intersect. When two lines intersect, four smaller angles are formed.

Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplementary. In this diagram, 1 and 2, 2 and 3, 3 and 4, and 4 and 1 are all examples of linear pairs.

Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles that share a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, 1 and 3 are vertical angles, so 1 3; 2 and 4 are congruent vertical angles as well.

Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angles or perpendicular angles, which often have vertical components.

Example

Determine the value of y in the following diagram:

The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are equal. We can set up and solve the following equation for y:

3y + 5 = 5y

5 = 2y

2.5 = y

Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the two vertical angles are congruent, with each measuring 12.5°.

Parallel Lines and Transversals

Important mathematical relationships are formed when two parallel lines are intersected by a third, non parallel line called a transversal.

In the preceding diagram, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and vertical angle pairs are formed in this diagram, too.

Other congruent angle pairs are formed:

• Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transversal: 3 and 6; 4 and 5.
• Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of the transversal: 1 and 5; 2 and 6; 3 and 7; 4 and 8.

Example

In the following diagram, line l is parallel to line m. Determine the value of x.

The two angles labeled are corresponding angle pairs, because they are located on top of the parallel lines and on the same side of the transversal (same relative location). This means that they are congruent, and we can determine the value of x by solving the equation:

4x + 10 = 8x– 25

10 = 4x – 25

35 = 4x

8.75 = x

We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25:

4(8.75) + 10 = 8(8.75) – 25

45 = 45

Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the problem would be solved in the same way.

Area, Circumference, and Volume Formulas

Here are the basic formulas for finding area, circumference, and volume. They are discussed in detail in the sections that follow.

Triangles

The sum of the measures of the three angles in a triangle always equals 180°.

Exterior Angles

An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles:

• An exterior angle and an interior angle that share the same vertex are supplementary. In other words, exterior angles and interior angles form straight lines with each other.
• An exterior angle is equal to the sum of the nonadjacent interior angles.
• The sum of the exterior angles of a triangle equals 360°.

Example

Classifying Triangles

It is possible to classify triangles into three categories based on the number of congruent (indicated by the symbol ) sides. Sides are congruent when they have equal lengths.

Angle-Side Relationships

Knowing the angle-side relationships in isosceles, equilateral, and right triangles is helpful.

• In isosceles triangles, congruent angles are opposite congruent sides.



• In equilateral triangles, all sides are congruent and all angles are congruent. The measure of each angle in an equilateral triangle is always 60°.



• In a right triangle, the side opposite the right angle is called the hypotenuse and the other sides are called legs. The box in the angle of the 90-degree angle symbolizes that the triangle is, in fact, a right triangle

Pythagorean Theorem

The Pythagorean theorem is an important tool for working with right triangles. It states: a² + b² = c², where a and b represent the legs and c represents the hypotenuse.

This theorem makes it easy to find the length of any side as long as the measures of two sides are known. So, if leg a = 1 and leg b = 2 in the triangle below, it is possible to find the measure of the hypotenuse, c.

Pythagorean Triples

Sometimes, the measures of all three sides of a right triangle are integers. If three integers are the lengths of the sides of a right triangle; we call them Pythagorean triples. Some popular Pythagorean triples are:

3, 4, 5

5, 12, 13

8, 15, 17

9, 40, 41

The smaller two numbers in each triple represent the lengths of the legs, and the largest number represents the length of the hypotenuse.

Multiples of Pythagorean Triples

Whole-number multiples of each triple are also triples. For example, if we multiply each of the lengths of the triple 3, 4, 5 by 2, we get 6, 8, 10. This is also a triple

Example

If given a right triangle with sides measuring 6, x, and a hypotenuse 10, what is the value of x?

3, 4, 5 is a Pythagorean triple, and a multiple of that is 6, 8, 10. Therefore, the missing side length is 8.

Comparing Triangles

Triangles are said to be congruent (indicated by the symbol ) when they have exactly the same size and shape. Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is easy to tell if two triangles are congruent by looking at them. However, in geometry, we must be able to prove that the triangles are congruent.

There are a number of ways to prove that two triangles are congruent:

Used less often but also valid:

Example

Determine if these two triangles are congruent.

Although the triangles are not aligned the same, there are two congruent corresponding sides, and the angle between them (150°) is congruent. Therefore, the triangles are congruent by the SAS postulate.

Example

Determine if these two triangles are congruent.

Although the triangles have two congruent corresponding sides and a corresponding congruent angle, the 150° angle is not included between them. This would be SSA, but SSA is not a way to prove that two triangles are congruent.

Area of a Triangle

Area is the amount of space inside a two-dimensional object. Area is measured in square units, often written as unit². So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet².

A triangle has three sides, each of which can be considered a base of the triangle. A perpendicular line segment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height. It measures how tall the triangle stands.

It is important to note that the height of a triangle is not necessarily one of the sides of the triangle. The correct height for the following triangle is 8, not 10. The height will always be associated with a line segment (called an altitude) that comes from one vertex (angle) to the opposite side and forms a right angle (signified by the box). In other words, the height must always be perpendicular to (form a right angle with) the base. Note that in an obtuse triangle, the height can be outside the triangle, and in a right triangle the height is usually one of the sides.

The formula for the area of a triangle is given by , where b is the base of the triangle, and h is the height.

Example

Determine the area of the following triangle.

Volume Formulas

A prism is a three-dimensional object that has matching polygons as its top and bottom. The matching top and bottom are called the bases of the prism. The prism is named for the shape of the prism's base, so a triangular prism has congruent triangles as its bases.

Note: This can be confusing. The base of the prism is the shape of the polygon that forms it; the base of a triangle is one of its sides.

Volume is the amount of space inside a three-dimensional object. Volume is measured in cubic units, often written as unit³. So, if the edge of a triangular prism is measured in feet, the volume of it is measured in feet³.

The volume of ANY prism is given by the formula V = A_bh, where A_b is the area of the prism's base, and h is the height of the prism.

Example

Determine the volume of the following triangular prism:

The area of the triangular base can be found by using the formula bh, so the area of the base is (15)(20) = 150. The volume of the prism can be found by using the formula V = A_bh, so the volume is V= (150)(40) = 6,000 cubic feet.

A pyramid is a three-dimensional object that has a polygon as one base, and instead of a matching polygon as the other, there is a point. Each of the sides of a pyramid is a triangle. Pyramids are also named for the shape of their (non-point) base.

The volume of a pyramid is determined by the formula A_bh.

Example

Determine the volume of a pyramid whose base has an area of 20 square feet and stands 50 feet tall.

Because the area of the base is given to us, we only need to replace the appropriate values into the formula.

The pyramid has a volume of 33 cubic feet.

Polygons

A polygon is a closed figure with three or more sides—for example, triangles, rectangles, and pentagons.

Terms Related to Polygons

• Vertices are corner points, also called endpoints, of a polygon. The vertices in the above polygon are A, B, C, D, E, and F, and they are always labeled with capital letters.
• A regular polygon has congruent sides and congruent angles
• An equiangular polygon has congruent angles.

Interior Angles

To find the sum of the interior angles of any polygon, use this formula:

S = (x – 2)180°, where x = the number of sides of the polygon.

Example

Find the sum of the interior angles in this polygon:

The polygon is a pentagon that has five sides, so substitute 5 for x in the formula:

S = (5 – 2) × 180°

S = 3 × 180°

S = 540°

Exterior Angles

Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.

Similar Polygons

If two polygons are similar, their corresponding angles are congruent and the ratios of the corresponding sides are in proportion.

Example

These two polygons are similar because their angles are congruent and the ratios of the corresponding sides are in proportion

Quadrilaterals

A quadrilateral is a four-sided polygon. Because a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360°.

Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides.

In the figure, . Parallel lines are symbolized with matching numbers of triangles or arrows

A parallelogram has:

• opposite sides that are congruent
• opposite angles that are congruent (A = C and B = D)
• consecutive angles that are supplementary (A + B = 180°, B + C = 180°, C + D = 180°, D + A = 180° )
• diagonals (line segments joining opposite vertices) that bisect each other (divide each other in half)

Special Types of Parallelograms

• A rectangle is a parallelogram that has four right angles



• A rhombus is a parallelogram that has four equal sides.



• A square is a parallelogram in which all angles are equal to 90 degrees and all sides are congruent. A square is a special case of a rectangle where all the sides are congruent. A square is also a special type of rhombus where all the angles are congruent

Diagonals of Parallelograms

In this diagram, parallelogram ABCD has diagonals and that intersect at point E. The diagonals of a parallelogram bisect each other, which means that .

In addition, the following properties hold true:

• The diagonals of a rhombus are perpendicular.
• The diagonals of a rectangle are congruent.
• The diagonals of a square are both perpendicular and congruent.

Example

In parallelogram ABCD, the diagonal = 5x + 10 and the diagonal = 9x. Determine the value of x.

Because the diagonals of a parallelogram are congruent, the lengths are equal. We can then set up and solve the equation 5x + 10 = 9x to determine the value of x.

5x + 10 = 9x	Subtract x from both sides of the equation.
10 = 4x	Divide 4 from both sides of the equation.
2.5 = x

Area and Volume Formulas

The area of any parallelogram can be found with the formula A = bh, where b is the base of the parallelogram, and h is the height. The base and height of a parallelogram is defined the same as in a triangle.

Note: Sometimes b is called the length (l) and h is called the width (w) instead. If this is the case, the area formula is A = lw.

A rectangular prism (or rectangular solid) is a prism that has rectangles as bases. Recall that the formula for any prism is V = A_bh. Because the area of the rectangular base is A = lw, we can replace lw for A_b in the formula giving us the more common, easier to remember formula, V = lwh. If a prism has a different quadrilateralshaped base, use the general prisms formula for volume.

Note: A cube is a special rectangular prism with six congruent squares as sides. This means that you can use the V = lwh formula for it, too.