4. Quadrilaterals Inscribed in a Circle
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Chapter 5
4.
Quadrilaterals Inscribed in a Circle
In the vast domain of geometry, inscribed quadrilaterals, often termed as cyclic quadrilaterals, have unique properties. One of the most notable characteristics is the relationship between the angles of these quadrilaterals when they are inscribed in a circle. This geometric concept is not just confined to theoretical discussions; cyclic quadrilaterals have real-life examples and applications. From architectural designs to art forms, the principles governing inscribed quadrilaterals are evident. Recognizing these properties and understanding their implications can provide valuable insights, especially when dealing with geometric designs and patterns in various fields.
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Lesson Settings & Tools
| | 9 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Quadrilaterals Inscribed in a Circle
Slide of 9
Polygons with a different number of sides can be inscribed in a circle. In this lesson, inscribed quadrilaterals, or polygons with four sides, will be explored. Furthermore, three main properties of inscribed quadrilaterals will be investigated.
Write the answer without the degree symbol.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
a Pair each geometric object with its definition.
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b In the circle, ∠ MON measures 70^(∘). Find the measure of the corresponding inscribed angle.
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c Calculate the sum of the arc measures.
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d On the circle, the measures of all arcs except one are given. Find the measure of that arc and the measure of the inscribed angle ∠ A.
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e Use the Polygon Interior Angles Theorem to calculate the missing angle measure of DEFG.
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Investigating Quadrilaterals Inscribed in a Circle
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Inscribed Quadrilateral Theorem
An inscribed quadrilateral is a quadrilateral whose vertices all lie on a circle. It can also be called a cyclic quadrilateral.
In the diagram above, all four vertices of ABCD lie on a circle. Therefore, ABCD is a cyclic quadrilateral.
By the Inscribed Angle Theorem, the measure of each of these inscribed angles is half the measure of its intercepted arc.
m∠ A=1/2mBCD [0.8em] m∠ C=1/2mBAD
The above equations can be simplified by multiplying both sides of each equation by 2.
2m∠ A=mBCD [0.2cm] 2m∠ C=mBAD
Next, 2m∠ A and 2m∠ C can be substituted for mBCD and mBAD, respectively, into the equation found earlier.
The sum of the measures of ∠ A and ∠ C is 180^(∘). This means that these angles are supplementary. By the same logic, it can be also proven that ∠ B and ∠ D are supplementary. This concludes the proof of Part 1. 
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Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Based on the diagram above, the following relations hold true.
m∠ A + m∠ C = 180^(∘)
m∠ B + m∠ D = 180^(∘)
Proof
This theorem will be proven in two parts.
- If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary.
- If the opposite angles of a quadrilateral are supplementary, then it can be inscribed in a circle.
Part 1
Consider a circle and an inscribed quadrilateral ABCD.
Notice that BCD and BAD together span the entire circle. Therefore, by the Arc Addition Postulate, the sum of their measures is 360^(∘). mBCD+mBAD=360^(∘) From the diagram, it can be seen that BCD and BAD are intercepted by ∠ A and ∠ C, respectively.
mBCD+mBAD=360^(∘)
SubstituteII
mBCD= 2m∠ A, mBAD= 2m∠ C
2m∠ A+ 2m∠ C=360^(∘)
FactorOut
Factor out 2
2(m∠ A+m∠ C)=360^(∘)
DivEqn
.LHS /2.=.RHS /2.
m∠ A+m∠ C=180^(∘)
Part 2
This part of the proof will be proven by contradiction. Suppose that ABCD is a quadrilateral that has supplementary opposite angles, but ABCD is not cyclic.
Since ABCD is not cyclic, the circle that passes through A, B, and C, does not pass through D. Let E be the point of intersection of AD and the circle. Consider the quadrilateral ABCE.
Because ABCE is inscribed in a circle, it can be concluded that the opposite angles ∠ AEC and ∠ B are supplementary. m∠ AEC+m∠ B=180^(∘) It was assumed that ABCD has supplementary opposite angles. Therefore, ∠ D and ∠ B are supplementary angles. m∠ D+m∠ B=180^(∘) By the Transitive Property of Equality, the above equations imply that ∠ AEC and ∠ D have equal measures. lm∠ AEC+ m∠ B=180^(∘) m∠ D+ m∠ B=180^(∘) ⇓ m∠ AEC=m∠ D However, this is not possible. The reason is that the measure of the exterior angle ∠ AEC of △ EDC can not be the same as the measure of the interior angle ∠ D.
This contradiction proves that the initial assumption was false, and ABCD is a cyclic quadrilateral. Note that a similar argument can be used if D lies inside the circle. The proof of Part 2 is now complete.
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Using the Inscribed Quadrilateral Theorem
The Inscribed Quadrilateral Theorem can be used to identify whether a quadrilateral is cyclic.
Tiffaniqua is given a quadrilateral JKLM. She wants to draw a circle that passes through all the vertices, but she does not know if it is possible. For that reason, she decided to measure the angles of JKLM.
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Hint
Compare the sums of the opposite angles' measures.
Solution
The Inscribed Quadrilateral Theorem can be used to determine whether JKLM is cyclic. According this theorem, the opposite angles of the quadrilateral need to be supplementary. Calculate the sum of opposite angles' measures and see if it is true.
| Pair 1 | Pair 2 | |
|---|---|---|
| Opposite Angles | ∠ J and ∠ L | ∠ K and ∠ M |
| Sum | 99^(∘)+74^(∘)=173^(∘) * | 105^(∘)+82^(∘)=187^(∘) * |
The sum of the angle measures in each pair is not equal to 180^(∘). Therefore, neither ∠ J and ∠ L nor ∠ K and ∠ M are supplementary. This finding implies that JKLM is not a cyclic quadrilateral.
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Practice Using the Inscribed Quadrilateral Theorem
Find the measure of ∠ D. Write your answer without the degree symbol.
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Cyclic Quadrilateral Exterior Angle Theorem
On the diagram below, one side of a cyclic quadrilateral ABCD is extended to E. As a result, ∠ ADE — the exterior angle of ABCD — is formed.
In this case, ∠ ABC is said to be the opposite interior angle. The relationship between these angles is described by the Cyclic Quadrilateral Exterior Angle Theorem.
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Cyclic Quadrilateral Exterior Angle Theorem
If a side of a cyclic quadrilateral is extended, then the exterior angle is congruent to the opposite interior angle.
Based on the diagram above, the following relation holds true.
∠ ABC ≅ ∠ ADE
Proof
Consider an inscribed quadrilateral with one side extended to point E.
From the diagram, it can be observed that ∠ ADE and ADC form a linear pair. Therefore, these angles are supplementary, which means that the sum of their measures is 180^(∘). m∠ ADE+ m∠ ADC=180^(∘) Also, by the Inscribed Quadrilateral Theorem, the opposite angles ∠ ABC and ∠ ADC are also supplementary.
Therefore, the following relation is true. m∠ ABC+ m∠ ADC=180^(∘) After analyzing the equations, ∠ ABC and ∠ ADE show to be supplementary to the same angle ∠ ADC. Therefore, by the Congruent Supplements Theorem, they are congruent.
∠ ABC ≅ ∠ ADE
This relation is illustrated on the diagram below.
By the same logic, this theorem can be proven for any other extended side of ABCD. The proof is now complete.
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Using the Cyclic Quadrilateral Exterior Angle Theorem
Davontay wants to go to a concert, but his parents say that he has to finish his homework first. In the last math exercise, he is asked to find the values of all variables.
Help Davontay solve the last exercise so that he can go to the concert.
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Hint
Identify the exterior angles to the inscribed quadrilateral WXYZ and the opposite interior angles. Then use the property that states these angles are congruent.
Solution
By observing the diagram, ∠ VXY and ∠ YXW can be recognized to form a linear pair, so they are supplementary angles.
m∠ VXY + m∠ YXW=180^(∘)
By substituting 84^(∘) for m∠ VXY and a^(∘) for m∠ YXW into the equation, the value of a can be calculated.
The value of a is 96. Next, notice that ∠ VXY is the exterior angle of WXYZ, while ∠ WZY is the opposite interior angle. 
Therefore, these angles are congruent. ∠ VXY≅ ∠ WZY ⇓ m∠ VXY=m∠ WZY
The measure of VXY is 84^(∘) and the measure of ∠ WZY is 3b^(∘). By substituting these values and solving the equation, the value of b can be found.
Therefore, the value of b is 28. Similarly, ∠ YZU and ∠ WXY are the exterior angle and the opposite interior angle, respectively. 
By the property mentioned earlier, these angles are congruent.
∠ YZU≅ ∠ WXY ⇓ m∠ YZU=m∠ WXY
The measure of ∠ YZU is 2c^(∘) and the measure of ∠ WXY is a^(∘), which is equal to 96^(∘). This information can be used to determine the value of c.
The value of c is 48.
m∠ VXY + m∠ YXW=180^(∘)
SubstituteII
m∠ VXY= 84^(∘), m∠ YXW= a^(∘)
84^(∘) + a^(∘)=180^(∘)
SubEqn
LHS-84^(∘)=RHS-84^(∘)
a^(∘)=96^(∘)
m∠ VXY=m∠ WZY
SubstituteII
m∠ VXY= 84^(∘), m∠ WZY= 3b^(∘)
84^(∘)= 3b^(∘)
DivEqn
.LHS /3.=.RHS /3.
28^(∘)=b^(∘)
RearrangeEqn
Rearrange equation
b^(∘)=28^(∘)
m∠ YZU=m∠ WXY
SubstituteII
m∠ YZU= 2c^(∘), m∠ WXY= 96^(∘)
2c^(∘)= 96^(∘)
DivEqn
.LHS /2.=.RHS /2.
c^(∘)=48^(∘)
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Properties of Inscribed Quadrilaterals in a Circle
Consider an inscribed quadrilateral EFGH. Draw a perpendicular bisector to each side of the polygon. 
As can be observed, all the perpendicular bisectors intersect at the center of the circle. This property is true for all cyclic quadrilaterals.
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Inscribed Quadrilaterals and Polygons in Real Life
It is worth mentioning that not only quadrilaterals can be inscribed in a circle. There can also be inscribed polygons with a different number of sides. Stonehenge is a real-world example of an inscribed polygon. Unfortunately, only some parts of it remain to this day.
However, when Stonehenge was built by ancient peoples about 5000 years ago, it had a cyclic polygon structure, as illustrated on the diagram below.
Quadrilaterals Inscribed in a Circle
Exercise 3.1
What is the measure of x?
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Notice that each circle contains the center of the other circles. Since every circle has a radius that can be drawn to the centers of each of the other circles', they must have the same dimension.
Let's add some more radii to the diagram which matches the sides of the trapezoid.
As we can see, x is an angle in an equilateral triangle. In such a triangle, the angles are all congruent.
We find x by using the Interior Angles Theorem. x+x+x =360^(∘) ⇕ x=60 ^(∘)
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Quadrilaterals Inscribed in a Circle
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