The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O such that $∠ D A C$ =30° and $∠ A O B$ =70°. Then, the measure of $∠ D B C$ is ____ °.

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Detailed Solution

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O such that

∠ D A C

=30° and

∠ A O B

=70°. Then, the measure of

∠ D B C

is 40°.
As, ABCD is a parallelogram, using alternate interior angle property,

∠ D A C = ∠ A C B = 30 o ⇒ ∠ D A C = 30 o

Now, as sum of angle on a straight line is

180 o

∠ A O B + ∠ A O D = 180 o 70 o + ∠ A O D = 180 o ∠ A O D = 180 o − 70 o ∠ A O D = 110 o

Now, as sum of interior angles of a triangle is

180 o

.
In triangle AOD using angle sum property,

∠ D A O + ∠ A O D + ∠ O D A = 180 o 30 o + 110 o + ∠ O D A = 180 o ∠ O D A = 180 o − 30 o − 110 o ∠ O D A = 40 o

Now using alternate interior angle property,

∠ O D A = 40 o ⇒ ∠ D B C = 40 o

The measure of

∠ D B C

40 o

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