Parallel Lines And Transversals

• Given: Line q is transversal is to line m and line l.

• To find corresponding angles of

1) ∠ p

2) ∠ q

3) ∠ r

4) ∠ s

• Explanation:

If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.

So, now in the above given figure we have say, line q making transversal to line m and line l.

1) For ∠p, ∠w is the angle which is in the same side and same direction of transversal so ∠w is the corresponding angle to ∠p.

2) For ∠q, ∠x is the angle which is in the same side and same direction of transversal so ∠x is the corresponding angle to ∠q.

3) For ∠r, ∠y is the angle which is in the same side and same direction of transversal so ∠r is the corresponding angle to ∠y.

4) For ∠s, ∠z is the angle which is in the same side and same direction of transversal so ∠s is the corresponding angle to ∠z.

Now for Interior Alternate angles

Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.

When these angels are in the inner side they are called Interior alternate angels.

5) For ∠s the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠x. So ∠s and ∠x form pair of Interior Alternate angel.

6) For ∠w the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠r. So ∠w and ∠r form pair of Interior Alternate angel.

• Given: Line q is transversal is to line m and line l.

• To find: (1) Interior alternate angles

(2) Corresponding angles

(3) Interior angles

(1) Now for Interior Alternate angles

Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.

When these angels are in the inner side they are called Interior alternate angels.

1) For ∠b the angle which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠h. So ∠b and ∠h form pair of Interior Alternate angel.

2) For ∠c the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠e. So ∠c and ∠e form pair of Interior Alternate angel.

(2) Corresponding angles

If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.

So, now in the above given figure we have say, line q making transversal to line m and line l.

1)For ∠a, ∠e is the angle which is in the same side and same direction of transversal so ∠a is the corresponding angle to ∠e.

2)For ∠b, ∠f is the angle which is in the same side and same direction of transversal so ∠b is the corresponding angle to ∠f.

3)For ∠d, ∠h is the angle which is in the same side and same direction of transversal so ∠d is the corresponding angle to ∠h.

4)For ∠c, ∠g is the angle which is in the same side and same direction of transversal so ∠c is the corresponding angle to ∠g.

(3) Interior angles

A pair of angles which are on the same side of the transversal and inside the given lines is called a pair of interior angles.

So, we get only two such pairs of angels.

1) ∠b has ∠e on the same side of transversal and inside the given line. So ∠b and ∠e form pair of interior angels.

2) ∠c has ∠h on the same side of transversal and inside the given line. So ∠c and ∠h form pair of interior angels.

• Given: Line m || line n and line p is a transversal

• To find: The value of x.

Now in the given figure we have 3x and x. 3x and x form a pair of interior angle.

Now by the property of interior angels we know that, each pair of interior angles formed by two parallel lines and their transversal is of supplementary angles i.e. 180°.

∴ x+3x=180(∵ Property of interior angles.)

⇒ 4x=180

⇒

∴ The value of x is 45°.

• Given: Line a || line b and line l is a transversal

• To find: Value of x.

Now from the figure we can see

∠GHB=∠CHF (∵ opposite angles are same)

⇒ ∠GHB=2x

∠AGB+∠BHG=180° (∵ (∵ Property of interior angles.)

∴ 4x+2x=180°

⇒ 6x=180°

⇒

∴ The value of x is 30°

Option(D)

• Given: Line p || line q, line t and line s are transversals.

• To find: The measure of ∠x and ∠y.

Here we can see

∠KLD=∠HLD (∵ Opposite angles are equal)

⇒ ∠ KLD=70°

∠KLI+∠JIL=180° (∵ (∵ Property of interior angles.)

∴ 70+Y=180°

⇒ Y=110°

∴ The value of y is 110°

Also,

∠ BKL+∠ JKL =180° (Linear pair)

⇒ x+∠JKL = 180°

∴ ∠ JKL= 180-x…(1)

∠ KJI+∠ AJI =180° (Linear pair)

⇒ 40+∠KJI = 180°

∴ ∠ KJI= 140°…(2)

Now,

∠KJI+∠JKL=180° (∵ Property of interior angles.)

140+180-x=180 (From 1 and 2)

⇒-x=180-180-140

⇒ -x=-140°

∴ x=140°

∴ The value of x is 140°.

• Given: Line p || line q, line l || line m.

• To find: The measure of ∠a, ∠b and ∠c.

Now in this figure

∠CIJ+∠AJI=180° (∵ Exterior angles are supplementary)

⇒ 80° +a=180°

⇒ a = 100°

Also,

∠AJI+∠IJL=180° (Linear pair)

⇒100° +∠IJL=180°

∴ ∠IJL=80°

∠BLK=∠LJI=b(corresponding angles are equal)

∴ b=80°

∠EIK=∠CIJ=80° (Opposite angles are equal)

∠GKD=∠EIK=c(corresponding angles are equal)

∴ c=80°

∴ Values of a,b and c are 100° , 80° ,80° respectively.

• Given: Line a || line b, line l is transversal.

• To find: The measure of ∠x, ∠y and ∠z.

In, the figure above

∠AGE=z

Also,

∠AGE+∠EGB=180° (Linear pair)

z+105=180°

z=75°

∠GHD=x

∠EGB=∠GHD(corresponding angles are equal)

∴ x=105°

∠DHF=y

Also

∠DHF=∠GHD (Opposite angles are equal)

∠DHF=105°

∴ Values of x, y and z are 105° ,105° ,75° respectively.

• Given: line p || line l || line q.

• To find: Value x

Now, In the above figure ∠GHD=x

Also,

∠GHD=∠GHD+∠IHD

∠AGH=∠GHD (∵ Alternate angles are equal)…(1)

∠EIH=∠DHI (∵ Alternate angles are equal)…(2)

From (1) and (2) we get,

∠GHD=40°

∠DHI=30°

We know

∠GHD=∠GHD+∠IHD

∠GHD= 40°+30°

∠GHD= 70°

But,

∠ GHD = x

∴ x=70°

∴ The value of x is 70°.

Here we need draw a line l then take a point A outside the line and then through that point draw another line parallel to l.

Steps of construction:

1) Draw a line segment of any length. Mark it as CD.

2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point A.

3) Now take another perpendicular of same length as of AP, and in same direction.

4) Draw a line through those points.

5) This line is parallel to given line l.

Here we need draw a line l then take a point T outside the line and then through that point draw another line parallel to l.

Steps of construction:

1) Draw a line segment of any length. Mark it as CD.

2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point T.

3) Now take another perpendicular of same length as of TP, and in same direction.

4) Draw a line through those points.

5) This line is parallel to given line l.

Method : Draw a line parallel to line l at a distance 4 cm.

Steps of construction :

(1) Draw line l.

(2) Take two points A and B on the line l.

(3) Draw perpendiculars to the line l from points A and B.

(4) On the perpendicular lines take points P and Q at a distance of 4cm from

A and B respectively.

(5) Draw line PQ.

(6) Line PQ is a line parallel to the line l at a distance 4cm.