We are given a square inscribed in a circle. Its apothem and two radii drawn.

We want to find the measure of the angles $X P Y,$ $X P Q,$ and $P X Q .$ Let's find them one at a time.

Angle $X P Y$

The angle

X P Y

is a central angle of the square. The measure of each central angle of a regular

n - gon

is

\frac{3 6 0}{n} .

For a square, we have that

n = 4 .

Let's substitute this value in the formula and simplify.

\frac{3 6 0}{n}

Substitute

$n = 4$

\frac{3 6 0}{4}

CalcQuot

Calculate quotient

90

The measure of angle

X P Y

is

9 0^{\circ} .

Angle $X P Q$

The segment

P Q

is an apothem in the square. The apothem bisects the vertex angle of the isosceles triangle formed by the radii. Since we know that the measure of this angle is

9 0^{\circ},

we can divide

90

by

2

to obtain the measure of angle

X P Q .

\frac{9 0}{2} = 4 5^{\circ}

The measure of angle

X P Q

is

4 5^{\circ} .

Angle $P X Q$

As previously mentioned, the apothem bisects the vertex angle of the isosceles triangle formed by the radii. We can see in the diagram that the apothem makes a right angle with the side of the square. Therefore, we know the measures of two of the three interior angles of the right triangle $X P Q .$ The missing angle is $∠ P X Q .$

To find the measure of angle

P X Q

we will use the Triangle Sum Theorem. This theorem states that the sum of the three interior angles of a triangle adds to

18 0^{\circ} .

90 + 45 + m ∠ P X Q = 18 0^{\circ} ⇕ m ∠ P X Q = 4 5^{\circ}

The measure of angle

P X Q

is

4 5^{\circ} .

Angle XPY

Angle XPQ

Angle PXQ

Angle $X P Y$

Angle $X P Q$

Angle $P X Q$

Exercise