units 11+12 - theory and word problems (basic geometry 2d)

Mathematics 1 ESO. IES Don Bosco (Albacete). European Section 1

Unit 10: LINES AND ANGLES

10.1.- BASIC TERMS IN GEOMETRY

Geometry is the branch of Mathematics that deals with the properties,

measurement, and relationships of points, lines, angles, surfaces and solids.

Point

A point is a dot on a piece of paper. We identify this point with a number or

letter. A point has no length or width, it just specifies an exact location.

Example: The following is a diagram of points A, B and C.

Line

A line as a straight line that we might draw with a ruler on a piece of paper. A line extends forever in both directions. We write the name of a line passing

through two different points A and B as line AB.

Example: The following is a diagram of two lines: line AB and line HG.

Ray

A ray is a straight line that begins at a certain point and extends forever in

one direction. The point where the ray begins is known as its endpoint. We write the name of a ray with endpoint A and passing through a point B as ray AB.

The arrows signify that the lines drawn

extend indefinitely in each direction.

2 Mathematics 1 ESO. IES Don Bosco (Albacete). European Section

Example: The following is a diagram of two rays: ray HG and ray AB.

Line segment

A line segment is a portion of a straight line. A line segment does not extend forever, but has two distinct endpoints. We write the name of a line segment

with endpoints A and B as line segment AB or as AB .

Example: The following is a diagram of two line segments: line segment CD and line segment PN, or simply segment CD and segment PN.

Intersection

The term intersect is used when lines, rays, line segments of figures meet, that

is, they share a common point. The point they share is called the point of

intersection. We say that these figures intersect.

Example 1: In the diagram below, line AB and line GH intersect at point D.


Example 2: In the diagram below, line 1 intersects the square in points M and N.

Example 3: In the diagram below, line 2 intersects the circle at point P.

Parallel lines

Two lines in the same plane which never intersect are called parallel lines. We say that two segments are parallel is the lines that they lie on are parallel.

Example 1: Lines 1 and 2 below are parallel.

Example 2: The opposite sides of the rectangle below are parallel. The lines passing through them never meet.


10.2.- ANGLES

An angle measures the amount of turn.

The corner point of an angle is called the vertex.

And the two straight sides are called arms.

The angle is the amount of turn between each arm.

There are two main ways to label angles:

1. by giving the angle a name, usually a lower-

case letter like a or b, or sometimes a Greek

letter like (alpha) or (theta),

2. or by the three letters on the shape that

define the angle, with the middle letter being

where the angle actually is (its vertex).

Example: angle a is BAC, and angle is BCD.

Measuring angles

We can measure angles in degrees (). There are 360 in one full rotation (one

complete circle around).

You measure and draw angles in degrees using a protractor.

A full circle is 360

Half a circle is 180 (called a straight angle)

Quarter of a circle is 90 (called a right angle)

Why 360 degrees? Probably because old calendars

(such as the Persian Calendar) used 360 days for a

year - when they watched the stars they saw them

revolve around the North Star one degree per day.

This angle is 74


Types of angles

As the angle increases, the name changes:

Acute angle: an angle that is less than 90.

Right angle: an angle that is 90 exactly.

Obtuse angle: an angle that is greater than 90 but less than 180.

Straight angle: an angle that is 180 exactly.

Reflex angle: an angle that is greater than 180.

Full rotation: an angle that is 360 exactly.

Angles that add to 180 are called supplementary angles.

These two angles (140 and 40) are complementary

angles, because they add up to 180).

Angles that add to 90 are called complementary angles.

These two angles (40 and 50) are complementary angles,

because they add up to 90).

Exercise 1

Find the missing angles in these diagrams by calculating.

i)

ii)

iii)

iv)

v)

vi)


Exercise 2

Measure these angles:

Exercise 3

Construct a triangle ABC with AC 10= cm, A 40= and C 60= .

10.3.- BISECTING ANGLES AND SEGMENTS

If you bisect a segment or angle you cut it exactly in half:

A perpendicular bisector is the perpendicular line to the segment that cuts it in two equal parts.

The points of the perpendicular bisector are

equidistant from the endpoints of the segment.

PA PB= QA QB=

An angle bisector is a line passing through the vertex of the angle that cuts it into two equal smaller angles.

The points of the angle bisector are

equidistant from the sides of the angle.

PR PS= QR' QS'=


Vertically opposite angles are equal: 1 3= 5 7= 2 4= 6 8=

Corresponding angles are equal: 1 5= 2 6= 3 7= 4 8=

Alternate interior angles are equal: 3 5= 2 8=

Alternate exterior angles are equal: 1 7= 4 6=

Exercise 6

Find the missing angles in each diagram. Write down which angle fact you are

using each time.

a)

b)

10.5.- ANGLES IN POLYGONS

A polygon is a plane closed shape with three or more straight sides.

Polygon

(straight sides)

Not a Polygon

(has a curve)

Not a Polygon

(open, not closed)

Polygon comes from Greek. Poly- means many and gon means angle.

A regular polygon has all sides the same length and all interior angles equal.

The interior angles are inside the polygon.

The exterior angles are made by extending each side in the same direction.

Exterior angles are outside the polygon.


Exercise 7

Prove that the sum of angles in a triangle is 180.

(Hint: Add a line parallel to AB.)

Interior angle sum of any polygon

Look at this pentagon.

A pentagon divides into 3 triangles. Angles in a triangle

add up to 180.

So interior angle sum of a pentagon = = 3 180 540

You can divide any polygon into triangles by drawing

diagonals from a vertex (corner).

The number of triangles is always two less than the

number of sides.

Exercise 8

Complete the table.

Regular polygons

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Number of sides Number of triangles the shape

splits into

Sum of the interior angles in

the shape

Size of one interior angle

For a polygon with n sides the interior angle sum ( )= n 2 180


Exercise 10

Find the missing angles.

a)

b)

c)

d)

e)

f)

10.7.- SYMMETRIC SHAPES

The simplest symmetry is reflection symmetry (sometimes called line symmetry of mirror symmetry). It is easy to recognise, because one half is the reflection of the other half.

You can find if a shape has a line of symmetry by folding it.

When the folded part sits perfectly on top (all edges matching), then the fold

line is a line of symmetry.

Examples:

This is not a line of symmetry.

The white line down the centre is the

line of symmetry.


This is a line of symmetry.

Not all shapes have lines of symmetry, or they may have several lines of

symmetry. For example, a triangle can have 3, or 1 or no lines of symmetry.

Equilateral triangle

(all sides equal,

all angles equal)

Isosceles triangle

(two sides equal,

two angles equal)

Scalene triangle

(no sides equal,

no angles equal)

Exercise 11

a) How many lines of symmetry does a regular polygon of n sides have?

b) How many lines of symmetry does a circle have?

Exercise 12

Find lines of symmetry in the following capital letters of the alphabet.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Exercise 13

Complete the figure in order that it has the two indicated lines of symmetry.


Unit 11: PLANE AND 3-D SHAPES

11.1.- TRIANGLES

A triangle is a polygon that has three sides and three

angles. The three angles always add to 180.

It is usual to name each vertex of a triangle with a

single capital letter. The sides can be named with a single small letter, and named after the opposite

angle. So in the figure on the right, you can see that

side b is opposite vertex B, side c is opposite vertex

and so on.

Equilateral, Isosceles and Scalene

There are three special names given to triangles that tell how many sides (or

angles) are equal.

There can be 3, 2 or no equal sides/angles:

Equilateral Triangle Three equal sides

Three equal angles, always 60

Isosceles Triangle Two equal sides

Two equal angles

Scalene Triangle No equal sides

No equal angles

What type of angle?

Triangles can also have names that tell you what type of angle is inside:

Acute Triangle All angles are less than 90

Right Triangle Has a right angle (90)

Obtuse Triangle Has an angle more than 90


Constructing triangles

You can construct a triangle given 3 sides (SSS) using a ruler and a compass, but no protractor as you dont know any angles.

Example:

There are two types of triangle that you can construct using a ruler and a

protractor. The first is when you are given two sides and the included angle (SAS). The second is where you are given two angles and the included side

(ASA).

If you want to see all these constructions, click on the following link:

http://www.bbc.co.uk/schools/ks3bitesize/maths/shape_space/triangles_bearin

gs/revise1.shtml

Exercise 1

Classify the following triangles:

Exercise 2

Construct each of these triangles. Measure the angles in each triangle and write

them on your page.

i. =a 8 cm , =b 7 cm , =c 4 cm

ii. = A 30 , = B 50 , =c 6 cm

iii. =a 5 cm , =b 5 cm , = C 120

What did the three angles in each triangle add up to?

I.E.S. Andrs de Vandelvira Seccin Europea Mathematics

11-2

Exercise 1 Classify the following triangles by their sides and by their angles. Find the asked angles of the triangles.

Triangle By the sides By the angles Calculate ABC

DEF

HGI

=

H JKL

MNO

=

N YAZ

=

Y =

Z BCD

=

C =

D QPR

=

P SUT

XVW

=

X MKL

=

L NPO

=

O GEF

=

F QSR

=

R UTV

=== VTU HI

=

I


Exercise 3

Some triangles can not be constructed. Think about a triangle with sides of

length 12 cm, 5 cm and 4 cm. If you draw the base of 12 cm, the other two sides

are not long enough to join up with each other. Try it!

Which of these triangles is it possible to draw?

i. =a 7 cm , =b 6 cm , =c 4 cm ii. =a 15 cm , =b 12 cm , =c 8 cm

iii. =a 8 cm , =b 3 cm, =c 3 cm iv. =a 6 cm , =b 6 cm , =c 6 cm

Medians of a triangle. Barycentre

A median of a triangle is a line segment joining a vertex to the midpoint of the

opposite side.

The three medians of a triangle always intersect at a single point. This point is

called the barycentre of the triangle (the centre of gravity of the triangle).

Altitudes of a triangle. Ortocentre

An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex.

Sometimes the opposite side is not quite long enough to draw an altitude, so we

are allowed to extend it to make an altitude possible.

The three altitudes of a triangle intersect in a single point. This point is called

the ortocentre of the triangle.

Median

Barycentre

Ortocentre

ycentreoid

Perpendicular bisectors of a triangle. Circumcentre

A perpendicular bisector of a triangle is a straight lines passing through the midpoint of a side

and perpendicular to it.

The 3 perpendicular bisectors meet in a single point, it is called the triangles circumcentre;

this point is the centre of the circumcircle, the circle passing through the 3 vertices.


Classification of quadrilaterals

There are three types of quadrilaterals: parallelograms, trapeziums and trapezoids.

A parallelogram is a quadrilateral in which the opposite sides are parallel and equal in length. Also opposite angles are equal.

Rhomboids, rectangles, rhombuses and squares are parallelograms.

Rhomboid Rectangle Rhombus Square (right angles) (equal sides) (right angles

and equal sides)

A trapezium (called trapezoid in USA) has a pair of opposite sides parallel.

Trapezium Isosceles trapezium

A trapezoid (called trapezium in USA) is quadrilateral with no parallel sides:

An interesting trapezoid is the kite. It has two pair of sides. Each pair is made up of adjacent sides that are equal

in length. The angles are equal where the pairs meet.

Diagonals meet at a right angle, and one of the diagonal

bisects (cuts equally in half) the other.

Exercise 8

Draw two segments that intersect in their midpoints and that are perpendicular.

Join the endpoints and say what type of quadrilateral you get:

a) If both segments are equal in length.

b) If both segments are not equal in length.


Exercise 9

Draw a quadrilateral in each case.

a) Parallelogram with two lines of symmetry.

b) With four lines of symmetry.

c) Parallelogram with a line of symmetry.

d) Parallelogram without lines of symmetry.

e) Trapezoid with a line of symmetry.

Exercise 10

Name the following quadrilaterals.

Exercise 11

Construct a square in which the diagonal is 6 cm.

11.3.- REGULAR POLYGONS

A regular polygon is a polygon that has all sides equal and all interior angles

equal.

Properties of regular polygon

The apothem of a regular polygon is a line from the centre to the midpoint of a side. It is also the radius

of the incircle (see below).

The radius of a regular polygon is a line from the

centre to any vertex. It is also the radius of the

circumcircle (see below).


The incircle is the largest circle that fits inside a

regular polygon. Its radius is the apothem of the

polygon.

The circumcircle is the circle that passes through all the vertices of a regular polygon. Its radius is the

radius of the polygon.

In a regular polygon, the radius, the apothem and

half side make a right triangle.

11.4.- CIRCUMFERENCE

The circumference is the set of points that are a fixed

distance from another point. This point is called the

centre of the circumference.

You can also say that the circumference is the curved

line that goes around the circle.

The radius is the distance from the centre to the edge.

Tangents and circumferences

If a line is tangent to a circumference, it is perpendicular to the radius drawn to

the point of tangency.

Common tangents are lines or segments that are tangent to more than one circumference at the same time.

4 Common Tangents

(2 completely separate circumferences)

2 external tangents (blue)

2 internal tangents (black)


11.5.- PYTHAGORAS THEOREM

In a right triangle, the two smallest sides that make the right angle are called

legs and the longest side is called hypotenuse.

Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90) and you made a square on each of the

three sides, then the biggest square had the exact same area as the other two

squares put together.

It is called Pythagoras Theorem and can be

written in one short equation:

There are a lot of proofs of this theorem. Here you are one of them:

Since the two big squares are equal, their areas will be the same. If we leave out

the four equal triangles in each big square, we get:

2a in the first square 2 2b c+ in the second one

a = hypotenuse

b and c = legs

2 2 2a b c= +

In a right angled triangle: the square of the

hypotenuse is equal to the sum of the squares

of the other two sides.

2 2 2a b c= +


Exercise 13

Find the value of the area of the green square in each case.

a)

b)

Exercise 14

Find the missing side in each of these right-angled triangles. Round the answers

to 1 dp.

Exercise 15

Find the length of the diagonal of a square with side length 8 cm.

Exercise 16

In a computer catalogue, a computer monitor is listed as

being 19 inches. This distance is the diagonal distance across

the screen. If the screen measures 10 inches in height, what

is the actual width of the screen to the nearest inch?

Exercise 17

A ladder of length 5.5 m leans against a wall. The foot of the

ladder is 1 m from the wall. How far up the wall does the

ladder reach?

Exercise 18

Find the length of the side of a square with diagonal length 8 cm.

units 11+12 - theory and word problems (basic geometry 2d)

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