Understanding Cyclic Quadrilaterals and Angles

A cyclic quadrilateral is a four-sided figure (quadrilateral) inscribed in a circle. A key property of cyclic quadrilaterals is that their opposite angles are supplementary. This means that the sum of each pair of opposite angles is always $180^\circ$. For the cyclic quadrilateral PQRS, this property tells us:

• $\angle P + \angle R = 180^\circ$
• $\angle Q + \angle S = 180^\circ$

The question provides us with additional relationships between the angles:

• $\angle P$ is 4 times $\angle R$, which can be written as $\angle P = 4 \angle R$.
• $\angle S$ is 3 times $\angle Q$, which can be written as $\angle S = 3 \angle Q$.

Our goal is to find the average of $\angle Q$ and $\angle R$. The average of two values is their sum divided by 2, so we need to calculate $(\angle Q + \angle R) / 2$. To do this, we first need to find the values of $\angle Q$ and $\angle R$.

Calculating Angles R and P

We know that $\angle P + \angle R = 180^\circ$ and $\angle P = 4 \angle R$. We can substitute the second equation into the first equation to solve for $\angle R$:

$\angle P + \angle R = 180^\circ$

Substituting $4 \angle R$ for $\angle P$:

$4 \angle R + \angle R = 180^\circ$

Combine the terms involving $\angle R$:

$5 \angle R = 180^\circ$

Now, divide both sides by 5 to find the value of $\angle R$:

$\angle R = \frac{180^\circ}{5}$

$\angle R = 36^\circ$

With the value of $\angle R$, we can find $\angle P$ using $\angle P = 4 \angle R$:

$\angle P = 4 \times 36^\circ$

$\angle P = 144^\circ$

Let's quickly check if $\angle P + \angle R = 180^\circ$: $144^\circ + 36^\circ = 180^\circ$. This confirms our calculation for $\angle R$ and $\angle P$ is correct.

Calculating Angles Q and S

Similarly, we know that $\angle Q + \angle S = 180^\circ$ and $\angle S = 3 \angle Q$. We can substitute the second equation into the first equation to solve for $\angle Q$:

$\angle Q + \angle S = 180^\circ$

Substituting $3 \angle Q$ for $\angle S$:

$\angle Q + 3 \angle Q = 180^\circ$

Combine the terms involving $\angle Q$:

$4 \angle Q = 180^\circ$

Now, divide both sides by 4 to find the value of $\angle Q$:

$\angle Q = \frac{180^\circ}{4}$

$\angle Q = 45^\circ$

With the value of $\angle Q$, we can find $\angle S$ using $\angle S = 3 \angle Q$:

$\angle S = 3 \times 45^\circ$

$\angle S = 135^\circ$

Let's quickly check if $\angle Q + \angle S = 180^\circ$: $45^\circ + 135^\circ = 180^\circ$. This confirms our calculation for $\angle Q$ and $\angle S$ is correct.

Finding the Average of ∠Q and ∠R

We have found that $\angle Q = 45^\circ$ and $\angle R = 36^\circ$. Now we can calculate their average:

Average = $\frac{\angle Q + \angle R}{2}$

Average = $\frac{45^\circ + 36^\circ}{2}$

Average = $\frac{81^\circ}{2}$

Average = $40.5^\circ$

Summary of Angles

Check opposite angles sum: $\angle P + \angle R = 144^\circ + 36^\circ = 180^\circ$. $\angle Q + \angle S = 45^\circ + 135^\circ = 180^\circ$. The properties of a cyclic quadrilateral are satisfied.

The average of $\angle Q$ and $\angle R$ is $40.5^\circ$.

Revision Table: Cyclic Quadrilateral Properties

Additional Information: Related Concepts

Quadrilateral: A polygon with four edges (sides) and four vertices (corners). The sum of the interior angles of any quadrilateral is always $360^\circ$.

Supplementary Angles: Two angles are supplementary if their sum is $180^\circ$. This is a key property used in solving problems involving cyclic quadrilaterals.

Inscribed Polygon: A polygon is inscribed in a circle if all of its vertices lie on the circle. A cyclic quadrilateral is an inscribed quadrilateral.

Understanding the properties of cyclic quadrilaterals, especially the supplementary nature of opposite angles, is crucial for solving problems like this one involving angle relationships.