Hint: Starting with drawing a diagram using
all the information given in the question.
Assume two parallel lines as 'm' and 'n'.
And name all the angles formed by the
intersection of transverse with parallel
lines. Use the property that says that the
corresponding angles and opposite angles
are equal for a pair of parallel lines.
Combine the two hence formed equations
to get the required relationship.
Complete step-by-step answer:
Here in this problem, we are given two
parallel lines and a transversal intersects it.
Now within this situation, we need to prove
that each pair of alternate angles is equal.
Before starting with the solution to the
above problem, we must draw a figure
using all the information given to us.
Alternate angles are the two angles, other lines.
So according to the above figure, we
assumed that the two parallel lines be 'm
and 'n' with one transverse cutting both of
them at two points and forming eight
angles named as <1, <2, <3, <4, <5, <6,<7 and
<8
Here we need to prove that the pair
alternate angles, i.e. <3=<5 and <4=26
As we know that the corresponding angles
between two parallel lines are equal in
measure. A corresponding angle is the
angles that occupy the same relative
position at each intersection where a
straight line crosses two others.
i.e. <2=<6,<1= <5,43=<7 and <4=<8
•(i)
Also, the opposite angles formed in the
intersection of two lines are equal in
measure,
i.e. <1=43,42= <4,46= <8 and <5=<7
.. (11)
From relation (i) and (li), we can write two
pairs as <2=<6 and <2=<4 we can say <6=<4
Similarly, if we take two pairs from (i) and (li)
aS <1=<5
and <1=<3
From these above equations, we can
conclude that <3=<5
Therefore, we get the required relation
<3=45 and <4=<6
Hence, we proved that the alternate angles
of two parallel lines intersected by a
transverse are equal in measure.
Note: Remember that the original problem
was to prove that the alternate angles are
equal for two parallel lines and an
intersecting transverse. We here used our
assumption of two lines 'm' and 'n' and a
transverse using a figure. Naming the
angles formed on the intersection of these
lines helped us to form the relations and
equations. Questions like this method are
always useful. An alternative approach can
be to use the theorem that says that the From these above equations, we can
conclude that <3=<5
Therefore, we get the required relation
<3=25 and <4=26
Hence, we proved that the alternate angles
of two parallel lines intersected by a
transverse are equal in measure.
Note: Remember that the original problem
was to prove that the alternate angles are
equal for two parallel lines and an
intersecting transverse. We here used our
assumption of two lines 'm' and 'n' and a
transverse using a figure. Naming the
angles formed on the intersection of these
lines helped us to form the relations and
equations. Questions like this method are
always useful. An alternative approach can
be to use the theorem that says that the
sum of interior angles on the same side is
supplementary
