Question

ABCD is a cyclic quadrilateral in which BC ll AD . angle ADC = 110 and angle BAC = 50 . find angle DAC

Answer 1

We have ABCD a cyclic quadrilateral and BC II AD
The sum of opposite angles of cyclic quadrilateral is 180 degree
This gives angle ABC=180-110=70 degree
The triangle ABC has angle ABC=70  and angle BAC=50
           Hence angle BCA=180 - 70 - 50 = 60 degree
 We know BC II AD and angles BCA and DAC are alternate interior angles
 Hence  angle BCA = DAC=60 degree
  Angle DAC=60 degree

Answer 2

$\red\bigstar$Answer:

• $\angle$DAC = 60°

$\pink\bigstar$Given:

• ABCD is a cyclic quadrilateral.
• BC || AD
• $\angle$ADC = 110°
• $\angle$ BAC = 50°

$\purple\bigstar$To find:

• $\angle$DAC

$\green\bigstar$ Solution:

In cyclic quadrilateral ABCD,

$\angle$ADC + $\angle$ ABC = 180°

( Sum of opposite angles in a cyclic quadrilateral is 180° )

$\implies$ 110° + $\angle$ABC = 180°

$\implies$ $\angle$ABC = 180° - 110°

$\implies$ $\angle$ ABC = 70°

Now,

$\because$ BC || AD and AB is the transversal,

$\implies$ $\angle$ABC + $\angle$DAB = 180°

( Sum of Co-intertior angles is 180° )

$\implies$ ABC + $\angle$BAC + $\angle$DAC = 180°

$\implies$70° + 50° + $\angle$DAC = 180°

$\implies$ 120° + $\angle$DAC = 180°

$\implies$ $\angle$DAC = 180° - 120°

$\implies$ $\angle$DAC= 60°

$\therefore$ $\angle$DAC = 60°

$\blue\bigstar$ Concepts Used:

• Sum of opposite angles in a cyclic quadrilateral is 180°

• Sum of Co-intertior angles is 180°

$\orange\bigstar$ More to Know:

• Corresponding angles are equal.

• If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary .

• When two lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles.

• The sum of angles in a linear pair is 180°

• If the sum of two angles is 90°, then both are complementary angles to each other.