Question

in give figure (15.90),PQRS is a cyclic quadrilateral .Find the measure of each of its angles.

Answer 1

Answer:

Since in cyclic quadrilateral the sum of opposite angles are 180°

Therefore, here Angle R + Angle P = 180°

y + 5y = 180°

6y = 180°

y = 30°

therefore Angle R = 30° and Angle P = 150°

Similarly, Angle S + angle Q = 180°

3x + x = 180°

4x = 180

x = 45°

therefore, angle S = 45×3 = 135 ° and angle Q = 45°

Answer 2

Given a cyclic quadrilateral PQRS, find its each angle

Explanation:

1. given quadrilateral ABCD having its four angles  'w', 'x', 'y' and 'z' respectively. Then, PQRS is said to be cyclic quadrilateral if the sum of its opposite angles is $180$°,    $w+y=x+z=180$°  
2. hence in given cyclic quadrilateral PQRS we have, angles '3x', '5y', 'x' and 'y' respectively, hence from above point we get                                   [tex]5y+y=180 \ \ \ \ \ \ \ \ \ 3x+x=180\\ y=30\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=45[/tex]            ------(i)  
3. now the measure of each  angle from (i) is given by ,                                            [tex]\angle P=3x=135\\ \angle Q=5x=150\\ \angle S=x=45\\ \angle R=y=30[/tex]      
4. hence, the angles of the quadrilateral are $135$°, $150$°, $45$° and $30$°.