Question

<a and <b are the interior angles formed of two distinct lines. parallel lines. If <a = 72°, then find the measure of <b​

Answer 1

rectangle

AB and CD are two parallel lines intersected by a transversal L

X and Y are the points of intersection of L with AB and CD respectively. XP, XQ, YP and YQ are the angle bisectors of ∠AXY, ∠BXY, ∠CYX and ∠DYX

AB∣∣CD and L is transversal.

∴∠AXY= ∠DYX------- Pair of alternate angles

⟹

2

1

∠AXY=

2

1

∠DYX

⟹∠1=∠4 -----(∠1=

2

1

∠AXY and ∠4=

2

1

∠DYX)

⟹PX∣∣YQ (If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel)...(1)

Also ∠BXY=∠CYX (Pair of alternate angles)

⟹

2

1

∠BXY=

2

1

∠CYX

⟹∠2=∠3 -----(∠2=

2

1

∠BXY and ∠3=

2

1

∠CYX)

⟹PY∣∣XQ (If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are parallel)...(2)

From (1) and (2), we get

PXQY is a parallelogram ....(3)

∠CYD=180

∘

2

1

∠CYD=

2

180

∘

=90

∘

⟹

2

1

(∠CYX + ∠DYX)=90

∘

⟹

2

1

(∠CYX) +

2

1

(∠DYX)=90

∘

⟹∠3 + ∠4=90

∘

⟹∠PYQ=90

∘

....(4)

So, using (3) and (4), we conclude that PXQY is a rectangle.