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Geometry: Praxis I Exam (page 2)

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Updated on Jul 5, 2011

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 .

Parallel and Perpendicular Lines

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

Lengths of Horizontal and Vertical Segments

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.

Midpoint

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.

Slope

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.

Slope

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.

Determining the Equation of a Line

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.

Determining the Equation of a Line

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.

Angles

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

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.

Angles

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.
  • Acute Angle

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

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

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

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.
  • Special Angle Pairs

  • 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.
  • Special Angle Pairs

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

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.

Angles of Intersecting Lines

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:

Angles of Intersecting Lines

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.

Parallel Lines and Transversals

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.

Parallel Lines and Transversals

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.

Area, Circumference, and Volume Formulas

Triangles

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

Triangles

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

Exterior Angles

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.

Classifying Triangles

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.
  • Isosceles Triangle

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

  • 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
  •  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 Theorem

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:

Side-Side-Side (SSS) If the three sides of one triangle are congruent to the three corresponding sides of another triangle, the triangles are congruent.
Side-Angle-Side (SAS) If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, the triangles are congruent.

Used less often but also valid:

Angle-Angle-Side (AAS) If two angles and the non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, the triangles are congruent.
Hypotenuse-Leg (Hy-Leg) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, the triangles are congruent.

Comparing Triangles

Example

Determine if these two triangles are congruent.

Comparing Triangles

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.

Comparing Triangles

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.

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