Week 7 Geometry Notes Unit 3 Angles and Parallels

Unit 3 Lesson 1 Angle Definitions

Angles are present everywhere and are very important in the study of geometry. Lines, line segments, and rays can be placed in such a way as to form angles.

angle

The union of two rays that have a common endpoint.

acute angle

An angle whose measure is greater than zero and less than 90°.

right angle

An angle whose measure equals 90°.

obtuse angle

An angle with a measure greater than 90° but less than 180°.

bisector of an angle

A set of points which divides the angle into two angles of equal measure.

perpendicular lines

Lines that intersect to form right angles.

An angle is the intersection of two noncollinear rays that have a common end point.Remember that noncollinear means not on the same line and a ray begins at a point and extends infinitely in one direction.

The two rays that form the angle are called its sides, and the common endpoint is called the vertex of the angle. The symbol for angle is .

Example:

1.

Two lines form angles if there is an intersection point. Point O is a vertex.

2.

Ray OA and ray OB form the sides of  AOB.  Point O is a vertex.

Models:

The angles above are formed by the union of the two rays.     In the angle formed by and ; the sides are  and .     Its vertex is B. We can name the angle  ABC,  CBA,  B, or 1.     The angle to the right of  ABC is the union of the two rays  and . The vertex is point S.     This angle can be called  RST,  TSR,  S, or 2.

Notice that when three letters are used to name an angle, the vertex letter is always placed between the other two.

If no confusion will result, the vertex letter can be used alone; or a numeral can be used. You should never use a single letter when several angles have the same vertex as in the figure below.

 A could mean:

      BAC, the angle formed by  and ; or      CAD, the angle formed by  and ; or      DAE, the angle formed by  and ; or      BAD, the angle formed by  and ; or      CAE, the angle formed by  and ; or      BAE, the angle formed by  and 

     Now let's look at some types of angles.

An acute angle is an angle whose measure is less than 90°.

These angles are all acute angles.

A right angle is an angle whose measure equals exactly 90°.

These angles are all right angles. A little box at the vertex indicates a right angle.

An obtuse angle is an angle with a measure greater than 90° but less than 180°.

These angles are all obtuse angles.

The bisector of an angle is a set of points which divides that angle into two angles of equal measure. The bisector of an angle could be a ray, line, or line segment.

Model:

     

If the measure of  ABD(m ABD) equals the measure of  DBC(m DBC), then  bisects  ABC.

MODEL

 bisects ∠CAD. If m∠BAD = x + 15 and m∠BAC = 2x - 20, find x.

∠BAD = ∠BACx + 15 = 2x - 2015 = x - 2035 = x

Angle Definitions

Perpendicular lines, right angles, and ninety degree measurements all go together. Perpendicular lines are two lines that intersect to form right angles. The upside down capital T means “is perpendicular to.”

In this diagram, line AB is perpendicular to line CD. Notice the small red square at the point of intersection. This tells you that there are four ninety degree angles at this location.

Segments and rays can also be perpendicular.

[Segment XY is perpendicular to segment YZ. A red box at Y indicates the intersection of the segments creates a 90-degree angle.]In this figure, the red mark inside the angle indicates that angle Y is a right angle. It is also true that the segment XY is perpendicular to segment YZ, and that the measure of angle Y is 90 degrees.

Do not assume something is perpendicular simply because it appears to be so in a diagram. In triangle DEF, it appears the segment DE is perpendicular to segment EF, but without the correct markings we cannot assume that they are perpendicular.

Let’s look at the concept of betweeness in reference to rays and angles.

Betweeness of rays means that for any angle AOC, the ray OB is between rays OA and OC when all three rays have the same endpoint, and when ray OB lies in the interior of angle AOC. Both of these conditions must be true for ray OB to be between rays OA and OC.

In Figure 1, using the concept of betweeness, what is the relationship of the three rays? 

Ray MO is between rays MN and MP as all three rays share a common endpoint and ray MO lies in the interior of angle NMP.

How about figure 2?

Is ray CD between rays CA and CB? It is true that they share a common endpoint, but ray CD is not in the interior of angle ACB. Therefore, ray CD is not between rays CA and CB.

Unit 3 Lesson 2 Angle Measurement

Angle Measurement

The concept of angle is one of the most important concepts in geometry.

The concepts of equality, sums, and differences of angles are important and will be used throughout this course.

· Measuring angles is pretty simple: the size of an angle is based on how wide the angle is open.

· To help you remember approximate sizes of angles, you will want to remember that a circle, once around, measures 360 degrees. If you removed a quarter of the circle, the angle created at the center of the circle would be 90 degrees. This you already know is a right angle. If you cut one of the quarters in half, you would have half of 90 degrees which is 45 degrees. It is beneficial to have a good mental picture of the approximate size of any angle.

· A protractor is the most common tool for measuring angles. For the purpose of this lesson, you will be given a protractor with different angle measurements and angles drawn on the protractor. Let's look at the following example.

· If angle TOK was 22 degrees, what is the measure of angle TOS?From the symbols in the illustration, you can see angle SOL is 90 degrees. Since this angle is supplementary with angle SOK that means that angle SOK must also measure 90 degrees. Therefore, angle TOS is 90 minus 22, which is 68 degrees.

If you know angle QOL is 49 degrees, what is the measure of angle TOQ? From the previous exercise, you determined that angle TOS was 68 degrees. By repeating the same process, you can determine that angle SOQ is 90 minus 49, which is equal to 41 degrees.Now by the angle addition postulate, you can add angles TOS and SOQ to get angle TOQ. By substituting known values for angles TOS and SOQ, you get the angle measure for angle TOQ is 109 degrees.

Being able to break large angles into smaller angles is a key strategy that you will often be able to use in solving problems.

Angle Measurement

POSTULATE 6

P6: Every angle corresponds with a unique real number greater than 0° and less than 180°.     (This is called the Angle measurement postulate)     P6 tells us that every angle has a measure that is a real number between 0° and 180°.

The next postulate tells us how to find the measurement number.

POSTULATE 7

     P7: The set of rays on the same side of a line with a common endpoint in the line     can be put in one-to-one correspondence with the real numbers from 0° to 180° inclusive in such a way:

          1. That one of the two opposite rays lying in the line is paired with zero and the other is paired with 180°.

          2. That the measure of any angle whose sides are rays of that given set is          equal to the absolute value of the difference between the numbers          corresponding to its sides. (Protractor postulate)

     P7 tells us how to build and use a protractor to measure an angle.

Take a line and a point O on that line.

Place a set of rays all on the same side of the line with common endpoint O.

From a point (o) use two opposite rays as a base. Pair one ray with 0° and the other with 180°. Additional rays can pass through point o creating angles with measures between 0° and 180°.

From this information we can build a protractor.

· To find the measure of an angle with a protractor, line up the vertex of the angle to be measured with point O on the protractor. Line up one of the sides of the angle with the rays forming the base of the protractor. The other side of the angle should pass through the measure of the angle.

Given the measure of the ray forming sides of an angle, subtraction is used to find the measure of the angle.

mAOB = | 20° - 0° | = | 20° | = 20°mBOC = | 40° - 20° | = | 20° | = 20°mCOD = | 90° - 40° | = | 50° | = 50°

Because absolute values are used, the order in which you subtract does not affect the measure.

THEOREM 3-1ANGLE ADDITION THEOREM

 

Theorems are part of the building blocks of geometry. Once axioms and postulates are in place, theorems can be proven.

This section will prove the angle addition theorem using the Angle Measurement Postulate and the Protractor Postulate.

Proofs are limited to definitions and axioms. Note that what is given relates to "betweenness," an undefined postulate of incidence.

Prove the angle addition theorem:

THEOREM 3-1

 If  lies between  and , then m  BOA + m  AOC = m BOC.

The proof of Theorem 3-1follows:

Given:

      between  and .

To prove:

     m BOA + mAOC = mBOC

Plan:

Use reflexive property, addition property of equality, the Angle Measurement Postulate, and the Protractor Postulate.

 

In practical problems, angle measures that are not integers are sometimes expressed in degrees (°), minutes ('), and seconds (").

Angle Measurement 2

In practical angle measurement problems, the measures that cannot be expressed as an integer are sometimes expressed in degrees, minutes, and seconds.

· In algebra, we learned that a degree is made up of sixty minutes and a minute is made up of 60 seconds.

· This is the same as if we were measuring time, so we don’t have to learn a new system of measurement.

· We can further our understanding of the degree, minute, and second relationship by creating some conversions.One minute is one-sixtieth of a degree, one second is one-sixtieth of a minute, and one second is also one-three thousand six hundredth of a degree.

·

· To find the decimal equivalent to a degree, minute and second measure, you must express the minutes and seconds in degree equivalents, divide and combine terms.In this figure, angle RST is measured as 35 degrees, 45 minutes and 36 seconds.We usually don’t include the minutes and seconds, and instead change this to a decimal.This is done by placing the 45 over 60, the 36 over 3600 and dividing.

· Notice that the seconds are divided by 3600 which is 60 times 60.Once the fractions have been converted to decimals, we can add all the parts together to get the measure of angle RST as thirty-five point eight one degrees.

· When you add several measures of angles that have minutes and seconds, line up like units, and add.In this example, there are three angle measures recorded in degrees, minutes, and seconds.

· Line up the degrees, the minutes and the seconds and add each column.In the seconds column, there are seventy-three seconds.We know that sixty seconds equals one minute, so we can carry the one minute to the next column and leave the remaining thirteen seconds.The same process is used for the minute column for a final answer of fifty degrees, eight minutes, and thirteen seconds.

·

Unit 3 Lesson 4 Angle Relationship Definitions

Angles are related to each other in many ways. They can be related by position or by measure.

adjacent angles

Two angles in the same plane that have a common vertex and a common side, but no interior points in common.

complementary angles

Two angles with measures that, when added together, equal 90°. Each angle is called the complement of the other.

supplementary angles

Two angles with measures that, when added together, equal 180°. Each angle is called the supplement of the other.

vertical angles

Angles with sides that form two pairs of opposite rays.

Angle Relationships

Earlier in this course, you were introduced to angles and their various relationships. In this video, you will spend a little more time identifying the different relationships.

Adjacent angles are angles that share a common ray and vertex. Size doesn't matter as long as they are next to each other.

Vertical angles are created when two lines intersect. It creates two sets of congruent angles where the opposite angles are congruent. We will leave the proof to you in a future lesson.

Complementary angles are angles whose sum is 90 degrees. Complementary angles do not need to be adjacent to be complementary. However, when they are they are easy to identify because together they make a right angle.

Supplementary angles are angles whose sum is 180 degrees.

Supplementary angles do not need to be adjacent to be supplementary. When they are though they form a line. This is why two of such angles are at times called linear pairs.

Let's look at an example that has all four types of angles.

[Line Z, N has midpoint O. Line B, K also has midpoint O, as it intersects line Z, N. Line segment O, V is perpendicular to line Z, N. Angle N, O, K is 45 degrees and angle V, K, O is 45 degrees.]

Can you name two sets of supplementary angles? Angles B, O, Z and B, O, N are supplementary as they would add up to 180 degrees. Also, angles B, O, N and N, O, K are supplementary. Now look for a pair of complementary angles. Because of the right angle markings, you know that angles V, O, K and N, O, K are complementary.

Can you name two sets of vertical angles? As lines B, K and N, Z are intersecting, you have two sets of vertical angles. They are angles B, O, Z and N, O, K and angles B, O, N and Z, O, K. Lastly, can you name a set of adjacent angles? Angles Z, O, V and V, O, K are adjacent. There are actually five sets of adjacent angles in this diagram.

Adjacent Angles

· Recall that adjacent angles are two coplanar angles with a common side, a common vertex, and no common interior points. According to this definition, adjacent angles must meet four requirements.

·

The first requirement for adjacent angles is that the two angles be coplanar.

[Two planes. Line A O is on the first plane. C O is on the second plane. The planes intersect on line B O.]

In this figure, the angles A O B and C O B share a common side and a common vertex, but they are on two different planes. Remember that two planes intersect at a line. The common side for the adjacent angles is ray OB, which lays on the line where the two planes intersect, but ray O A and ray O C are on different planes. Although angles A O B and C O B look as if they are adjacent angles, they are not, because they lay on different planes.

The second requirement for adjacent angles is that they share a common vertex.

[Line B O is intersected by lines A O and line B C]

In this figure, angles A O B and O B C share a common side of O B, but they do not share a common vertex. Angle A O B has a vertex point of O and angle O B C has a vertex point of B. Although these angles are next to each other, they are not adjacent angles.

The third requirement for adjacent angles is that they share a common side.

[Lines O B. O A. O D. O C]

There are several angles in this figure that can be labeled as adjacent angles and some angle pairs that are not adjacent. Angles B O A and A O D are adjacent angles, while angles B O A and D O C are not adjacent angles, because they do share a common vertex but not a common side.

The fourth requirement is that they have no interior points in common.

Lines O A. O B. O C. Point x is in the interior of angle A O B as it is between lines O A and O B.

Point x is in the interior of angle A O B and is in the interior of angle A O C, so angles A O B and A O C are not adjacent angles. Even though angles A O B and A O C are not adjacent angles, two angles in the figure do meet all four requirements. The adjacent angles are angles A O B and B O C. They are in the same plane, they have point O as a common vertex, they share a common side in ray O B, and no point in the interior of angle A O B is also in the interior of angle B O C.

Unit 3 Lesson Angle Relationship Theorems (1)

OBJECTIVES

· Use theorems about adjacent, complementary, supplementary and vertical angles to answer questions and complete proofs.

Adjacent Angle Theorems

THEOREM 3-2If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.

Given:

∠ADC, ∠CDB are adjacent angles.,  are opposite rays.

To Prove:

∠ADC, ∠angles.

One way to prove a theorem or to analyze certain mathematical situations is by breaking it into cases and determining what happens in each one. One way to disprove a conjecture is by coming up with a single counterexample. We look at both of these in the following Modeling Mathematical Practices.

THEOREM 3-3If two lines are perpendicular, then they form right angles.

Given:

l  m

Prove:

∠1 is a right angle, ∠3 is a right angle, ∠2 is a right angle, ∠4 is a right angle.

Plan:

Show that m ∠1 = 90° by using Theorem 3-2 and properties of algebra.

By a similar method, we can show ∠2, ∠3 and ∠4 are also right ∠'s.

THEOREM 3-4If two adjacent angles have their exterior sides in perpendicular lines, then the angles are complementary.

Given:l  m

Given the following figure, prove that 1, 2 are adjacent angles.

Plan:Use angle addition theorem and Theorem 3-3 to show m ∠1 + m∠2 = 90°

Complementary and Supplementary Angle Theorems

Angle Relationship Theorems: Theorems 3-5 and 3-6

· Many of the theorems dealing with angle measures and angles relationships are used extensively in triangle proofs and parallel line proofs.

· Let’s look at a proof involving complementary angles.

· If we were given the conditional statement “If two angles are complements of the same angle or equal angles, then the two angles are congruent.” Our first step could be to draw a figure and label it so that our “given” and “prove” statements will be in terms of the diagram and not general statements.

Angle Relationship Theorems: Theorem 3-7

We’ve talked about deductive reasoning on several occasions, and the process of proving that a conjecture is true. A statement that we prove to be true is called a theorem. Let’s take a look at some theorems related to angles and work through the proof.

Consider the conditional, “If two lines intersect, the opposite or vertical angles are congruent.” As this is already written in conditional form, we can identify the “given” and “prove” portion of the proof easily. The “given” is contained in the hypothesis which is two lines intersect. At this point, we should create a figure that will illustrate the problem.

Start by drawing two intersecting lines and labeling the angles that are created. As a result of the two intersecting lines, angles 1 through 4 have been created.

The “prove” statement is found in the conclusion of the conditional, which is “opposite or vertical angles are congruent.”

MODEL

Using the diagram above, use the given information to solve the following problem.

If m∠1 = 5x - 30 and m∠2 = 3x + 10, find m∠1.

Because ∠1 and ∠2 are vertical angles, their measures are equal.m∠1 = m∠25x - 30 = 3x + 102x = 40x = 20m∠1 = 5(20) - 30 = 70