Solving for Angles in a Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. A key property of a cyclic quadrilateral is that the sum of opposite angles is 180 degrees.

The angles of the cyclic quadrilateral are given in order as x°, (3x - 30)°, (y + 30)°, and (2x - y)°.

Let the angles be $A = x^\circ$, $B = (3x - 30)^\circ$, $C = (y + 30)^\circ$, and $D = (2x - y)^\circ$. Since the angles are taken in order, A and C are opposite angles, and B and D are opposite angles.

Setting up Equations for Opposite Angles

Using the property that opposite angles of a cyclic quadrilateral sum to 180 degrees, we can form two equations:

Equation 1: Sum of angles A and C

$\angle A + \angle C = 180^\circ$
$x + (y + 30) = 180$
$x + y + 30 = 180$
$x + y = 180 - 30$
$x + y = 150$ (Equation 1)

Equation 2: Sum of angles B and D

$\angle B + \angle D = 180^\circ$
$(3x - 30) + (2x - y) = 180$
$3x - 30 + 2x - y = 180$
$5x - y - 30 = 180$
$5x - y = 180 + 30$
$5x - y = 210$ (Equation 2)

Solving the System of Linear Equations

Now we have a system of two linear equations with two variables, x and y:

• $x + y = 150$ (Equation 1)
• $5x - y = 210$ (Equation 2)

We can solve this system using the elimination method. Notice that the coefficients of y are +1 and -1. Adding Equation 1 and Equation 2 will eliminate y:

$(x + y) + (5x - y) = 150 + 210$
$x + y + 5x - y = 360$
$6x = 360$

Now, solve for x:

$x = \frac{360}{6}$
$x = 60$

Now substitute the value of x = 60 into Equation 1 to find y:

$x + y = 150$
$60 + y = 150$
$y = 150 - 60$
$y = 90$

So, we have found $x = 60$ and $y = 90$.

Calculating the Measure of Each Angle

Now substitute the values of x and y back into the expressions for the angles:

• Angle 1: $x^\circ = 60^\circ$
• Angle 2: $(3x - 30)^\circ = (3 \times 60 - 30)^\circ = (180 - 30)^\circ = 150^\circ$
• Angle 3: $(y + 30)^\circ = (90 + 30)^\circ = 120^\circ$
• Angle 4: $(2x - y)^\circ = (2 \times 60 - 90)^\circ = (120 - 90)^\circ = 30^\circ$

The angles of the cyclic quadrilateral are 60°, 150°, 120°, and 30°.

Identifying the Smallest Angle

Comparing the four angle measures (60°, 150°, 120°, 30°), the smallest angle is 30°.

The smallest angle is 30°.

Revision Table: Cyclic Quadrilateral Angles

Additional Information: Properties of Cyclic Quadrilaterals

Beyond the sum of opposite angles being 180 degrees, here are some other properties of cyclic quadrilaterals:

• The perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent (they meet at a single point), which is the center of the circumscribed circle.
• Ptolemy's Theorem: For a cyclic quadrilateral ABCD, the sum of the products of the lengths of the opposite sides equals the product of the lengths of the diagonals. That is, $AB \cdot CD + BC \cdot AD = AC \cdot BD$.
• If one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the interior opposite angle. For example, if side AB of cyclic quadrilateral ABCD is extended to point E, then $\angle CBE = \angle ADC$.

Understanding these properties is crucial for solving problems involving cyclic quadrilaterals in geometry.