Question

in each of the figures given below, ABCD is a rectangle.
find the values of x and y in each case​

Answer 1

Answer:

From the figure we know that the diagonals of a rectangle are equal and bisect each other.

Consider △ AOB

We get

OA = OB

We know that the base angles are equal,

∠ OAB = ∠ OBA

By using the sum property of triangle ∠ AOB + ∠ OAB + ∠ OBA = 180°

By substituting the values

110° + ∠ OAB + ∠ OBA = 180°

We know that ∠ OAB = ∠ OBA

So we get

2 ∠ OAB = 180o – 110o

By subtraction 2 ∠ OAB = 70°

By division ∠ OAB = 35°

We know that AB || CD and AC is a transversal

From the figure we know that ∠ DCA and ∠ CAB are alternate angles

∠ DCA = ∠ CAB = y° = 35°

Consider △ ABC

We know that

∠ ACB + ∠ CAB = 90°

So we get

∠ ACB = 90° – ∠ CAB

By substituting the values in above equation

∠ ACB = 90° – 35°

By subtraction

∠ ACB = x = 55°

Therefore, x = 55o and y = 35°.

Answer 2

Answer:

from the figure we know that the diagonals are equal and bisect at point O

Consider △AOB

we get

AO=OB

We know that the base angles are equal

∠OAB=∠OBA=35

By using the sum property of the triangle

∠AOB+∠OAB+∠OBA=180

By substituting the values we get

∠AOB+35 +35 =180

∠AOB=180 −70

∠AOB=110

from the figure, we know that the vertically opposite angles are equal

⇒∠DOC=∠AOB=y=110

In △ABC

We know that ∠ABC=90

Consider △OBC

We know that

∠OBC=x =∠ABC−∠OBA

By substituting the values

∠OBC=90 −35

∠OBC=55

therefore x=55 ,y=110