Download MESSINA OPERATIONS WITH SEGMENTS AND ANGLES CLIL 1

MESSINA
OPERATIONS WITH SEGMENTS AND ANGLES
CLIL
CLIL “OPERATIONS WITH SEGMENTS AND ANGLES”
PART 2: ANGLES
Today we want to introduce the concept of an angle.
AN ANGLE IS A PART OF A PLANE LIMITED BY TWO HALF-LINES, GOING OUT FROM THE SAME ORIGIN.
THEIR COMMON ORIGIN IS CALLED A VERTEX OF THE ANGLE AND THE TWO HALF-LINES ARE CALLED THE
SIDES OF THE ANGLE.
An angle is signed by the symbol ∠ and three capital letters: the first letter denotes a point on the first
side, the second one denotes the vertex and the third one denotes a point on the second side.
In the figure the angle is ∠ AOB
A.
(we use AÔB)
O
. B
You can see that there are two angles having the same sides and vertex: one of these is salient (or convex),
the other one is reentrant (or concave).
More generally, a figure is convex if it contains the whole segment AB, for any of its two points A and B,
otherwise the figure is concave.
It can be hard to check whether a figure is convex or concave, but not for an angle: if an angle doesn’t
contain the opposite half-lines of its sides, then it is convex, otherwise it is concave.
If the two sides of an angle are coincident one with the other, then we have a null angle and a round angle.
A round angle coincides with the whole plane, because all of the points of the plane are also points of it.
If the two sides of an angle are opposite half-lines, then we have a straight angle; it coincides with a halfplane.
A half of a straight angle is a right angle (or direct angle). Straight lines, forming a right angle, are called
mutually perpendicular lines. In symbols we write AB ⊥CD, for meaning that the straight line AB intersects
the straight line CD and they are mutually perpendicular, that is they form four right angles.
If an angle is less than a right angle it is called an acute angle; an angle greater than a right angle but less
than a straight angle is called an obtuse angle.
Two angles having in common the vertex and one side are called consecutive angles. Moreover, if their
other sides are opposite half-lines, then the two angles are called adjacent angles.
The unit of measurement of angles is a degree (its designation is ° or deg). A round angle measures 360°, so
1° is the 1/360 part of a round angle; a straight angle measures 180° because it is a half of a round angle; a
right angle measures 90° because it is a quarter of a round angle.
Similarly we have done with the segments we can compare angles by moving them and superposing one
on the other. For example, if I want to compare ∠ ABC and ∠ DEF then I move the angle ∠ ABC such as
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MESSINA
OPERATIONS WITH SEGMENTS AND ANGLES
CLIL
the vertex B results as superposed on the vertex E and the side BC is superposed on the side EF. Now one of
these following cases arises:
1. The side AB also results as superposed on DE: consequently the two given angles are congruent;
2. The side AB lies inside ∠ DEF: consequently ∠ ABC is lesser than ∠ DEF and ∠ AED is their
difference;
3. The side AB lies outside ∠ DEF: consequently ∠ ABC is greater than ∠ DEF and angle ∠ DEA is
their difference.
In order to find the sum of two given angles, we can move one of these, so that they result as consecutive.
In the same way, in order to find a multiple of a given angle by a number n, we can add that angle n times.
We can also divide a given angle into n equal parts, obtaining a sub-multiple of that angle by a number n.
The half-line dividing an angle into two equal parts is called the bisector of that angle.
A BISECTOR OF AN ANGLE IS A HALF-LINE DIVIDING THAT ANGLE IN TWO EQUAL PARTS.
Bisectors of adjacent angles are mutually perpendicular lines. Why? Demonstrate this theorem.
If the sum of two given angles is equal to a right angle, they are called complementary angles.
TWO ANGLES ARE COMPLEMENTARY IF THEIR SUM IS EQUAL TO A RIGHT ANGLE.
If the sum of two given angles is equal to a straight angle, they are called supplementary angles.
TWO ANGLES ARE SUPPLEMENTARY IF THEIR SUM IS EQUAL TO A STRAIGHT ANGLE.
I want to remark that complementary or supplementary angles can be non-consecutive angles, but two
adjacent angles are always supplementary.
At last, let’s consider two intersecting straight lines: they form four angles two by two vertically opposite
angles.
VERTICALLY OPPOSITE ANGLES (OR VERTICAL ANGLES) ARE CALLED SUCH TWO ANGLES WITH A COMMON
VERTEX, THAT SIDES OF ONE ANGLE ARE CONTINUATIONS OF THE OTHER.
Vertical angles are equal one to the other. Why? Demonstrate this theorem.
Bisectors of vertical angles are continuations one of the other. Why? Demonstrate this theorem.
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