Unit9: Angles and Lines

Indice Lines Bisectors Sexagesimal system and angles Sexagesimal system and time Adding and Subtracting Exercises

Angles and Lines

Matematicas 1o E.S.O.Alberto Pardo Milanes

1 Lines

2 Perpendicular bisector of a segment and bisector of an angle

3 Sexagesimal system and angles

4 Sexagesimal system and time

5 Adding and Subtracting in the sexagesimal system

6 Exercises

Alberto Pardo Milanes Angles and Lines

Plane Geometry

Plane Geometry deals with figures in a plane (circles, lines,polygons. . . )A line is a set of points extending in two opposite directionswithout end, It’s a straight one-dimensional figure with nothickness and extending infinitely in both directions. A line issometimes called a straight line. Points are said to be collinearpoints if they lie on a single straight line. A plane is a flat surfacethat has no thickness and extends without ending in ALLdirections.

Points are said to be coplanar points ifthey lie on a common plane.A line segment is a finite portion of aninfinite line. A line segment always hasa beginning and an end.

Plane Geometry

Plane Geometry deals with figures in a plane (circles, lines,polygons. . . )A line is a set of points extending in two opposite directionswithout end, It’s a straight one-dimensional figure with nothickness and extending infinitely in both directions. A line issometimes called a straight line. Points are said to be collinearpoints if they lie on a single straight line. A plane is a flat surfacethat has no thickness and extends without ending in ALLdirections.

Points are said to be coplanar points ifthey lie on a common plane.A line segment is a finite portion of aninfinite line. A line segment always hasa beginning and an end.

Perpendicular bisector of a segment and bisector of anangle

Lines that intersect in a point are called intersecting lines. Linesthat do not intersect are called parallel lines (in the plane). Twolines that lie on top of one another are called coincident lines.Two lines or line segments which are perpendicular are said to beorthogonal (two lines are said to be perpendicular if they meet ata 90oangle).

Given two intersecting lines, the pointof intersection is called the vertexand the amount of rotation about thevertex required to bring one line intocorrespondence with the other is calledthe angle between them.

One full rotation corresponds to 360 degrees.Half a full rotation is called a straight angle, and a quarter of afull rotation is called a right angle.An angle less than a right angle is called an acute angle, an anglegreater than a right angle (but less than a straight angle) is calledan obtuse angle, and an angle greater than a straight angle (butless than a full angle) is called a reflex angle.

Perpendicular bisector of asegment and bisector of an

Perpendicular bisector of a segment and bisector of an angle

Bisector of a line segment

A perpendicular bisector of a line segment AB is a lineperpendicular to AB and passing through the midpoint M of AB.

Perpendicular bisector of a segment and bisector of an angle

Bisector of an angle

The bisector of an angle is the line that divides the angle intotwo equal parts.

Sexagesimal system and angles

The size of an angle

There are several ways to measure the size of an angle, for exampleusing units of degrees: In a complete circle there are threehundred and sixty degrees. You can express angles to any precisionusing decimal numbers.

Example: 30 degrees and a half is 30.5 degrees.

There is another way to state the size of an angle, subdividing adegree. The degree is divided into sixty parts called minutes.These minutes are further divided into sixty parts called seconds.

The size of an angle

There are several ways to measure the size of an angle, for exampleusing units of degrees: In a complete circle there are threehundred and sixty degrees. You can express angles to any precisionusing decimal numbers.

Example: 30 degrees and a half is 30.5 degrees.

There is another way to state the size of an angle, subdividing adegree. The degree is divided into sixty parts called minutes.These minutes are further divided into sixty parts called seconds.

Angles: Degrees, minutes and seconds

In a full circle there are 360 degrees, each degree is split up into 60parts called minutes (each part being 1/60 of a degree), eachminute is split up into 60 parts called seconds (each part being1/60 of a minute).

Example: 40 degrees, 20 minutes and 50 seconds is usually writtenthis way: 40o 20′ 50′′.

Example: 30.5 degrees is 30o and 30′, because 0.5 degrees are0.5 · 60 = 30 minutes.

Angles: Degrees, minutes and seconds

In a full circle there are 360 degrees, each degree is split up into 60parts called minutes (each part being 1/60 of a degree), eachminute is split up into 60 parts called seconds (each part being1/60 of a minute).

Example: 40 degrees, 20 minutes and 50 seconds is usually writtenthis way: 40o 20′ 50′′.

Example: 30.5 degrees is 30o and 30′, because 0.5 degrees are0.5 · 60 = 30 minutes.

Sexagesimal system and time

Time: Hours, minutes and seconds

To measure time we can use hours: In a day there are twenty-fourhours. You can express time to any precision using decimalnumbers.Example: 3 hours and a quarter is 3.25 hours.There is another way to state the amount of time, subdividing anhour. The hour is divided into sixty parts called minutes. Theseminutes are further divided into sixty parts called seconds.In a day there are 24 hours, each hour is split up into 60 partscalled minutes (each part being 1/60 of an hour), each minute issplit up into 60 parts called seconds (each part being 1/60 of aminute).Example: 14 hours, 56 minutes and 12 seconds is usually writtenthis way: 14 hr 56 min 12 sec.Example: 3.25 hours is 3 hr and 15 min, because 0.25 hours are0.25 · 60 = 15 minutes.

Adding and Subtracting in thesexagesimal system

Adding and Subtracting in the sexagesimal system

Add in the sexagesimal system

You must add or subtract the units separately. But you may need todo some adjusting if the hours end up 24 or more, the minutes endup 60 or more, the seconds end up 60 or more,or less than zero.

AddingAdd the hours. If the hours are 24 or more, subtract 24 from thehours and add 1 to the days. Add the minutes. If the minutes are60 or more, subtract 60 from the hours and add 1 to the hours.And so on.

Example: 40o 20′ 50′′ + 10o 33′ 6′′ = 50o 55′ 56′′.

Example: 14 hr 20 min 50 sec + 15 hr 43 min 26 sec=29 hr 63 min 76 sec =1 day 6 hr 4 min 16 sec.

Example: 40o 20′ 50′′ + 10o 33′ 6′′ = 50o 55′ 56′′.

Subtract in the sexagesimal system

SubtractingSubtract the hours. If the hours are negative, add 24 to the hoursand subtract 1 from days. Subtract the minutes. If the minutes arenegative, add 60 to the minutes and subtract 1 from hours. And soon.

Example: 23 hr 40 min 51 sec − 15 hr 33 min 21 sec =8 hr 7 min 30 sec.

Example:45o 33′ 12′′ −11o 42′ 20′′ =44o 92′ 72′′ −11o 42′ 20′′ =33o 50′ 52′′.

Exercises

Exercise 1

Find five objects in your classroom that have right angles:

Exercises

Exercise 1

Find five objects in your classroom that have right angles:

Exercises

Exercise 2

Using the set square, draw an angle of a)45o b)30o c)105o d)15o

Exercises

Exercise 2

45o 30o

105o 15o

Exercises

Exercise 3

For each angle below, state whether the angle is acute, obtuse orreflex:

Exercises

Exercise 3

For each angle below, state whether the angle is acute, obtuse orreflex:

Exercises

Exercise 4

Measure the following angles:

Exercises

Exercise 4

Measure the following angles:

Exercises

Exercise 6

Write down the following amounts:

1 43o 34′ 25′′:

2 5o 17′ 30′′:

3 3 hr 14 min 18 sec:

4 15 hr 56 min 24 sec:

Exercises

Exercise 6

1 43o 34′ 25′′: forty-three degrees, thirty-four minutes andtwenty-five seconds.

2 5o 17′ 30′′: five degrees, seventeen minutes and thirtyseconds.

3 3 hr 14 min 18 sec: three hours, fourteen minutes andeighteen seconds.

4 15 hr 56 min 24 sec: fifteen hours, fifty-six minutes andtwenty-four seconds.

Exercises

Exercise 6

Exercises

Exercise 6

Exercises

Exercise 6

Exercises

Exercise 6

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =

2 55o 44′ 52′′ + 24o 53′ 34′′ =

3 3 hr 34 min 17 sec + 4 hr 11 min 34 sec =

4 18 hr 34 min 37 sec + 12 hr 26 min 34 sec =

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =45o 27′ 30′′

2 55o 44′ 52′′ + 24o 53′ 34′′ =80o 38′ 26′′

3 3 hr 34 min 17 sec +4 hr 11 min 34 sec =7 hr 45 min 51 sec

4 18 hr 34 min 37 sec +12 hr 26 min 34 sec =1 day 7 hr 1 min 11 sec

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =45o 27′ 30′′

2 55o 44′ 52′′ + 24o 53′ 34′′ =80o 38′ 26′′

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =45o 27′ 30′′

2 55o 44′ 52′′ + 24o 53′ 34′′ =80o 38′ 26′′

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =45o 27′ 30′′

2 55o 44′ 52′′ + 24o 53′ 34′′ =80o 38′ 26′′

Exercises

Exercise 7

1 34o 14′ 15′′ + 11o 13′ 15′′ =45o 27′ 30′′

2 55o 44′ 52′′ + 24o 53′ 34′′ =80o 38′ 26′′

Exercises

Exercise 8

1 45o 21′ 45′′ − 33o 12′ 15′′ =

2 53o 24′ 12′′ − 32o 43′ 44′′ =

3 6 hr 24 min 15 sec − 4 hr 1 min 5 sec =

4 8 hr 24 min 13 sec − 5 hr 25 min 36 sec =

Exercises

Exercise 8