Thea practice test_6 pps

Vertical angles are angles who share a vertex and whose sides are two pairs of opposite rays.. Determine the value of y in the diagram below: The angles marked 3y + 5 and 5y are vertical

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): m =y x2

2

– –

y x

1 1

.

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:

–5

–1–2–3–4

x

5

m = –01––31

m = ––23= 2

3 

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 7

5 ,

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

substi-tute it in the appropriate place in the general form of the equation

–2–4

x

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:

m = 1––4(––41)= –28= – 4

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 graph below

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:

–2–4

x

6–6

–66

N AMING A NGLES

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 either 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 ple, in the angle above, we may call the angle ∠B, because there is only one angle in the diagram that has B as itsvertex

exam-But, in the following diagram, there are a number of angles which 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, on theinterior of the angle, near the vertex

C LASSIFYING A NGLES

The unit of measure for angles is the degree

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

B

C A

■ 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.

Straight Angle

180 °

Obtuse Angle

Right Angle

Symbol

Acute Angle

S PECIAL A NGLE P AIRS

■ 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

supplemen-tary angles A linear pair is always supplemensupplemen-tary, but supplemensupplemen-tary angles do not have to form a linearpair

■ 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?

Since 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

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.

Since 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, ing that the angles are supplementary

prov-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.

A NGLES OF I NTERSECTING L INES

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 tary In this diagram,∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs

supplemen-Also, the angles that are opposite each other are called vertical angles Vertical angles are angles who share

a vertex and whose sides are two pairs of opposite rays Vertical angles are congruent In this diagram,∠1 and ∠3are 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 angle or

per-pendicular angles, which often have vertical components

2

4

Determine the value of y in the diagram below:

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°

P ARALLEL L INES AND T RANSVERSALS

Important mathematical relationships are formed when two parallel lines are intersected by a third, non-parallel

line called a transversal.

In the diagram above, parallel lines l and m are intersected by transversal n Supplementary angle pairs and

vertical angle pairs are formed in this diagram, too

Supplementary Angle Pairs Vertical Angle Pairs

21

65

Other congruent angle pairs are formed:

■ Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the

transver-sal:∠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 diagram below, 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

con-gruent, and we can determine the value of x by solving the equation:

Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the

prob-lem would be solved in the same way

m

8x – 25

n

Area, Circumference, and Volume Formulas

Here are the basic formulas for finding area, circumference, and volume They will be discussed in detail in thefollowing sections

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 non-adjacent interior angles

■ The sum of the exterior angles of a triangle equals 360 degrees

Example

m∠1 + m∠2 = 180° m∠1 = m∠3 + m∠5m∠3 + m∠4 = 180° m∠4 = m∠2 + m∠5m∠5 + m∠6 = 180° m∠6 = m∠3 + m∠2

m∠1 + m∠4 + m∠6 = 360°

C LASSIFYING T RIANGLES

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

sym-Scalene Triangle Isosceles Triangle Equilateral Triangle

no sides congruent more than 2 congruent sides all sides congruent

It is also possible to classify triangles into three categories based on the measure of the greatest angle:

Acute Triangle Right Triangle Obtuse Triangle

greatest angle is acute greatest angle is 90° greatest angle is obtuse

1 2

3 5

6

4

A NGLE -S IDE R ELATIONSHIPS

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.

Hypotenuse Leg

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 measure of two sides is 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.

Sometimes, the measures of all three sides of a right triangle are integers If three integers are the lengths of a right

triangle, we call them Pythagorean triples Some popular Pythagorean triples are:

M ULTIPLES OF P YTHAGOREAN T RIPLES

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

C OMPARING T RIANGLES

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 iseasy to tell if two triangles are congruent by looking at them However, in geometry, it must be able to be proventhat 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

cor-responding two sides and included angle of another triangle, the trianglesare congruent

Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to the

cor-responding two angles and included side of another triangle, the trianglesare 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

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 tulate

pos-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 ment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height It measures how

seg-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 rect height for the following triangle is 8, not 10 The height will always be associated with a line segment (called

cor-Obtuse Triangle

b h

Right Triangle

b

h

b h

The formula for the area of a triangle is given by A = 1

2 bh, where b is the base of the triangle, and h is the

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

Height of prismBase of prism

10"

5"

12 8

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 b h, 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 A = 12bh, so the area of the base is

A = 12(15)(20) = 150 The volume of the prism can be found by using the formula V = A b h, 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 13A b h.

Example

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

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

15'