POINTS AND LINES

ABRA VALLEY COLLEGES

BANGUED,ABRA The simplest objects in plane geometry are points and lines. Because they
are so simple, it is hard to give precise definitions of them, so instead we aim
to give students a rough description of their properties which are in line with
DEPARTMENT OF ENGINEERING AND ARCHITECTURE
our intuition. A  point marks a position but has no size. In practice, when we
draw a point it clearly has a definite width, but it represents a point in our
imagination. A  line has no width and extends infinitely in both directions.
MODULE IN When we draw a line it has width and it has ends, so it is not really a line, but

SOLID MENSURATION represents a line in our imagination. Given two distinct points  A  and B  then
there is one (and only one) line which passes through both points. We use
capital letters to refer to points and name lines either by stating two points
on the line, or by using small letters such as   and m. Thus, the given line

HALLEL JHON BUTAC below is referred to as the line AB or as the line  .

INSTRUCTOR

APPROVED BY:

DR. TERESITA J. GARCIA

VPEA
Given two distinct lines, there are two possibilities: They may either meet at
a single point or they may never meet, no matter how far they are extended
(or produced). Lines which never meet are called parallel. In the second
diagram, we write AB ||CD.
 The point  A  in the diagram divides the line
Three (or more) points that lie on a straight line are called collinear. into two pieces called rays. The ray AP is that ray which contains the
Three (or more) lines that meet at a single point are called concurrent. point P (and the point A).

Angles

Collinear Concurrent

INTERVALS, RAYS AND ANGLES

In the diagram, the shaded region between the rays OA and OB is called

the angle AOB or the angle BOA. The angle sign   is written so we write 
AOB.

Suppose  A  and B  are two points on a line.

The interval
AB is the part of the line between A and B, including the two endpoints.
 chose 360 since it was close to the number of days in a year.) Hence, the size
of a straight-angle is 180° and the size of a right-angle is 90°. Other angles
can be measured (approximately) using a protractor.

The shaded region outside is called

the reflex angle formed by OA and OB. Most of the time, unless we specify
the word reflex, all our angles refer to the area between the rays and not
outside them.
Straight angle Right angle Obtuse angle
THE SIZE OF AN ANGLE
Angles are classified according to their size. We say that an angle with
size α is acute
(a word meaning ‘sharp’) if 0° < α < 90°, α is obtuse (a word meaning
‘blunt’) if
90° < α < 180° and α is reflex if 180° < α < 360°.
Imagine that the ray OB is rotated about
the point O until it lies along OA. The amount of turning is called the size of Since the protractor has two scales, students need to be careful when

the angle AOB. We can similarly define the size of the reflex angle. drawing and
measuring angles. It is a worthwhile exercise to use a protractor to draw
We will often use small Greek letters, α, β, γ, ... to represent the size of an some angles such as 30°, 78°, 130°, 163°.
angle.

The size of the angle corresponding to one full revolution was divided (by the
Babylonians) into 360 equal parts, which we call degrees. (They probably
ANGLES AT A POINT  BOY is also the supplement of  BOX (straight angle),

we can conclude that these vertically opposite angles,

AOX and  BOY are equal. We thus have our first
important geometric statement:

Vertically opposite angles are equal.

A result in geometry (and in mathematics generally) is often called

Two angles at a point are said to be adjacent if they share a common ray. a theorem. A theorem is an important statement which can be proven by

Hence, in the diagram,  AOB and  BOC are adjacent. logical deduction. The argument above is a proof of the theorem; sometimes
proofs are presented formally after the statement of the theorem.
Adjacent angles can be added, so in the diagram
α = β + γ.

When two lines intersect, four angles are formed at the point of intersection.
In the diagram, the angles marked  AOX and  BOY are called vertically
If two lines intersect so that all four angles are
opposite.
right-angles, then the lines are said to be perpendicular.

Angles at a point − Geometric Arguments

The following reasons can be used in geometric arguments:

 Adjacent angles can be added or subtracted.

Since
 Angles in a revolution add to 360°.
 Angles in a straight line add to 180°.
 AOX is the supplement of  BOX (straight angle).
 Vertically opposite angles are equal. 

 Transversals and Parallel Lines

 A transversal is a line that meets two

other lines.
  
  
 Corresponding angles
 Various angles are formed by the transversal. In the diagrams below,
the two marked angles are called corresponding angles.

 We now look at what happens when
the two lines cut by the transversal are parallel.
 Inituitively, if the angle α were greater than β then CD would
cross AB  to the left of F and if it were less than β, it would cross to the
right of F. So since the lines do not cross at all, α can be neither less
nor more than β and so equals β.
 Alternatively, imagine translating the
angle QGD along GF until G coincides with F. Since the lines are
 parallel, we would expect that the angle α would coincide with the ‘Every multiple of 4 is an even number.’
angle β. This observation leads us to conjecture that:
 Co-interior Angles ‘Every even number is a multiple of 4.’

 Finally, in each diagram below, the two marked angles are called co-
and here, the first statement is true, but the second is false.
interior angles and lie on the same side of the transversal.

NGLE SUM OF A TRIANGLE

CONVERSE STATEMENTS

Many statements in mathematics have a converse, in which the implication

goes in the opposite direction. For example, the statement
The results from the previous section can
‘Every even number ends in 0, 2, 4, 6 or 8.’
be used to deduce one of the most important facts in geometry − the angle
sum of a triangle is 180° .
has converse

We begin with triangle ABC with angles α, β, γ as shown. Draw the

‘Every number that ends in 0, 2, 4, 6 or 8 is even.’
line DAE parallel to BC. Then,

This particular statement and its converse are both true, but this is by no
DAB = β (alternate angles, BC||DE)
means always
the case.For example, the following two statements are converses of each
EAC = γ (alternate angles, BC||DE)
other:
α+ β + γ = 180° (straight angle).
 3. Using only properties of parallel lines, find (with reasons) the missing
Thus, we have proven the theorem
angles in the following diagram.

ABRA VALLEY COLLEGES

Bangued Abra

SOLID MENSURATION 4. Which value of α will make AB parallel to CD?

PRELIM EXAM

Name: _________________________________ Date:_____________

Course:________________________________ Permit No: __________

ANSWER THE FOLLOWING QUESTIONS:

5. Write down:
1. By dividing the quadrilateral ABCD into two triangles, find the sum of the
I. a true geometrical statement whose converse is also true,
angles.
II.  false geometrical statement whose converse is true,
III.  a false geometrical statement whose converse is also false.

PREPARED BY: CHECKED BY:

HALLEL JHON B. BUTAC ENGR. JESSIE H. PALATTAO

2. Find the value of α in the following diagram.
INSTRUCTOR HEAD ENGINEERING

APPROVED BY:

DR. TERESITA J. GARCIA

VPEA