Question

ABCD is a parallelogram AC = 15, m∠ BAC = 22° , m∠DAC = 27° Find: AB and BC

Answer 1

`AB=9.02\ units`

`BC=7.45\ units`

Explanation:

see the attached figure to better understand the problem

Remember that in a parallelogram opposites sides are parallel and congruent, opposites angles are congruent and consecutive angles are supplementary

step 1

Find the measure of angle ACB

we have

`m\angle BAC=22^o` ----> given problem

`m\angle ACB=m\angle DAC` ----> by alternate interior angles

`m\angle DAC=27^o` ----> given problem

so

`m\angle ACB=27^o`

step 2

Find the measure of angle ABC

The sum of the interior angles in any triangle must be equal to 180 degrees

In the triangle ABC of the figure

`m\angle BAC+m\angle ACB+m\angle ABC=180^o`

substitute the given values

`22^o+27^o+m\angle ABC=180^o`

`49^o+m\angle ABC=180^o`

`m\angle ABC=180^o-49^o`

`m\angle ABC=131^o`

step 3

Find the length side AB

In the triangle ABC

Applying the law of sines

`(AC)/(sin(ABC))=(AB)/(sin(ACB))`

substitute the given values

`(15)/(sin(131^o))=(AB)/(sin(27^o))`

`AB=(15)/(sin(131^o))(sin(27^o))`

`AB=9.02\ units`

step 4

Find the length side BC

In the triangle ABC

Applying the law of sines

`(AC)/(sin(ABC))=(BC)/(sin(BAC))`

substitute the given values

`(15)/(sin(131^o))=(BC)/(sin(22^o))`

`BC=(15)/(sin(131^o))(sin(22^o))`

`BC=7.45\ units`