Inscribed Quadrilaterals: Examples and Real-Life Applications

	Pair $1$	Pair $2$
Opposite Angles	$∠ J$ and $∠ L$	$∠ K$ and $∠ M$
Sum	$9 9^{\circ} + 7 4^{\circ} = 17 3^{\circ} \times$	$10 5^{\circ} + 8 2^{\circ} = 18 7^{\circ} \times$

Pair

1

Pair

2

Opposite Angles

∠ J

and

∠ L

∠ K

and

∠ M

Sum

9 9^{\circ} + 7 4^{\circ} = 17 3^{\circ} \times

10 5^{\circ} + 8 2^{\circ} = 18 7^{\circ} \times

What are the values of $x$ and $y ?$

Inscribed quadrilateral ABCD in a circle

Notice that the quadrilateral ABCD is inscribed in a circle. By the Inscribed Quadrilateral Theorem, opposite angles, ∠ A and ∠ C as well as ∠ B and ∠ D, are supplementary.

This means that the sum of the angle measures equals 180^(∘). With this information, we can write two equations. 26y+ 2x=180^(∘) 3x+ 21y=180^(∘) By combining these equations, we get a system of equations which we can solve using the Method of Substitution.

Having solved for y, we can substitute this into the first equation and solve for x.

Consider the inscribed quadrilateral.

Quadrilateral KLMJ inscribed in a circle

Find

b .

Examining the diagram, we notice that ∠ K is the inscribed angle to the intercepted arc JML.

By the Inscribed Angle Theorem, we know that ∠ K is half the measure of JML. m∠ K = 1/2 mJML By substituting the measure of JML in this formula, we can determine the measure of ∠ K.

Let's add the measure of ∠ K to the diagram.

Notice that JKLM is an inscribed quadrilateral. According to the Inscribed Quadrilateral Theorem, opposite angles are supplementary. This means we can write the following equation. 4b+92^(∘)=180^(∘) Let's solve this equation for b.

What is the measure of $∠ A B C ?$

A quadrilateral DCBA inscribed in the circle O with an intercepted arc marked

To solve this exercise, we will add a third radius OD to the circle. All segments that marks a radius of the circle are congruent. OA≅ OD≅ OC Let's also add this information to the diagram.

A central angle always has the same measure as its intercepted arc. Notice that △ AOD and △ DOC by the Side-Side-Side Congruence Theorem are congruent triangles. Therefore, we know that m∠ AOD also equals 110^(∘).

Also since △ ADO and △ DOC both are isosceles triangles, by the Base Angles Theorem we know that the base angles are congruent.

Using the Interior Angles Theorem, we can determine the measure of the base angles for these triangles.

The measure of the base angles is 35^(∘). This means m∠ ADC, which is the sum of the measures of two base angles, must be 70^(∘).

Notice that the quadrilateral ABCD is inscribed in the circle. Therefore, we can use the Inscribed Quadrilateral Theorem to determine the measure of ∠ ABC, as it is the opposite angle to ∠ ADC.

Quadrilaterals Inscribed in a Circle

Catch-Up and Review

Investigating Quadrilaterals Inscribed in a Circle

Inscribed Quadrilateral Theorem

Inscribed Quadrilateral Theorem

Proof

Part $1$

Part $2$

Using the Inscribed Quadrilateral Theorem

Hint

Solution

Practice Using the Inscribed Quadrilateral Theorem

Cyclic Quadrilateral Exterior Angle Theorem

Cyclic Quadrilateral Exterior Angle Theorem

Proof

Using the Cyclic Quadrilateral Exterior Angle Theorem

Hint

Solution

Properties of Inscribed Quadrilaterals in a Circle

Inscribed Quadrilaterals and Polygons in Real Life

Quadrilaterals Inscribed in a Circle

Recommended exercises

	9 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Quadrilaterals Inscribed in a Circle

Catch-Up and Review

Proof

Part 1

Part 2

Hint

Solution

Proof

Hint

Solution

Quadrilaterals Inscribed in a Circle

Recommended exercises

Part $1$

Part $2$