Question

A puzzle in the form of a quadrilateral is inscribed in a circle. The vertices of the quadrilateral divide the circle into four arcs in a ratio of 1 : 2 : 5 : 4. Find the angle measures of the quadrilateral.

Answer 1

Answer:

Therefore,

∠A = 30°

∠B = 60°

∠C = 150°

∠D = 120°

Step-by-step explanation:

Given:

A puzzle in the form of a quadrilateral is inscribed in a circle.

The vertices A ,B ,C ,D of the quadrilateral divide the circle into four arcs in a ratio of 1 : 2 : 5 : 4.

Let the common multiple be "x" then the angles will be

∠A = 1x

∠B = 2x

∠C = 5x

∠D = 4x

To Find:

The angle measures of the quadrilateral = ?

Solution:

In a Quadrilateral inscribed in a Circle,

Sum of the measure of all the angles in a Quadrilateral is 360°

$m\angle A +m\angle B +m\angle C +m\angle D=360$

Substituting the values we get

$x+2x+5x+4x=360\\\\12x=360\\\\x=\dfrac{360}{12}=30$

Therefore the measures are

∠A = 30°

∠B = 2 × 30 = 60°

∠C = 5 × 30 = 150°

∠D = 4 × 30 = 120°

Therefore,

∠A = 30°

∠B = 60°

∠C = 150°

∠D = 120°

Answer 2

Answer:

m<1=135

m<2=75

m<3=45

m<4=105

Step-by-step explanation:

The arc measures:

x+2x+4x+5x=360

x=30

2(30)=60

4(30)=120

5(30)=150

The vertices of the quadrilateral are inscribed angles. You take the intercepted arc and divide it by 2. If you do this, you should get the angles of 135, 75, 45, and 105 as the angle measures of the quadrilateral.