CorrectionKey=NL-C;CA-C Name Class Date 15.2 Angles in ...The converse of the Inscribed...

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Explore Investigating Inscribed QuadrilateralsThere is a relationship among the angles of a quadrilateral that is inscribed in a circle. You can use a protractor and compass to explore the angle measures of a quadrilateral inscribed in a circle.

A Measure the four angles of quadrilateral ABCD and record their values to the nearest degree on the diagram.

B Find the sums of the indicated angles.

m∠DAB + m∠ABC = ° m∠ABC + m∠BCD = °

m∠DAB + m∠BCD = ° m∠ABC + m∠CDA = °

m∠DAB + m∠CDA = ° m∠BCD + m∠CDA = °

C Use a compass to draw a circle with a diameter greater than the circle in Step A. Plot points E, F, G, and H consecutively around the circumference of the circle so that the center of the circle is not inside quadrilateral EFGH. Use a straightedge to connect each pair of consecutive points to draw quadrilateral EFGH.

D Measure the four angles of EFGH to the nearest degree and record their values on your diagram.

E Find the sums of the indicated angles.

m∠HEF + m∠EFG = ° m∠EFG + m∠FGH = °

m∠HEF + m∠FGH = ° m∠EFG + m∠GHE = °

m∠HEF + m∠GHE = ° m∠FGH + m∠GHE = °

Module 15 793 Lesson 2

15.2 Angles in Inscribed Quadrilaterals

Essential Question: What can you conclude about the angles of a quadrilateral inscribed in a circle?

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Reflect

1. Discussion Compare your work with that of other students. What conclusions can you make about the angles of a quadrilateral inscribed in a circle?

2. Based on your observations, does it matter if the center of the circle is inside or outside the inscribed quadrilateral for the relationship between the angles to hold? Explain.

Explain 1 Proving the Inscribed Quadrilateral TheoremThe result from the Explore can be formalized in the Inscribed Quadrilateral Theorem.

Inscribed Quadrilateral Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Example 1 Prove the Inscribed Quadrilateral Theorem.

Given: Quadrilateral ABCD is inscribed in circle O.

Prove: ∠A and ∠C are supplementary.∠B and ∠D are supplementary.

Step 1 The union of ⁀ BCD and ⁀ DAB is circle O.

Therefore, m ⁀ BCD + m ⁀ DAB = °

Step 2 ∠A is an inscribed angle and its intercepted arc is .

∠ is an inscribed angle and its intercepted arc is ⁀ DAB .

By the Inscribed Angle Theorem, m∠A = _ m ⁀ and

m∠C = _

m ⁀

.

Step 3 So, m∠A + m∠C = Substitution Property of Equality

= Distributive Property

= Substitution Property of Equality

= Simplify.

So, ∠A and ∠C are supplementary, by the definition of supplementary. Similar reasoning shows that ∠B and ∠D are also supplementary.


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(15y + 17)°

(y2 + 53)°

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The converse of the Inscribed Quadrilateral Theorem is also true. That is, if the opposite angles of a quadrilateral are supplementary, it can be inscribed in a circle. Taken together, these statements can be stated as the following biconditional statement. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Reflect

3. What must be true about a parallelogram that is inscribed in a circle? Explain.

4. Quadrilateral PQRS is inscribed in a circle and m∠P = 57°. Is it possible to find the measure of some or all of the other angles? Explain.

Explain 2 Applying the Inscribed Quadrilateral Theorem

Example 2 Find the angle measures of each inscribed quadrilateral.

A PQRS

Find the value of y.

m∠P + m∠R = 180° PQRS is inscribed in a circle.

(5y + 3) + (15y + 17) = 180 Substitute.

20y + 20 = 180 Simplify.

y = 8 Solve for y.

Find the measure of each angle.

m∠P = 5 (8) + 3 = 43° Substitute the value of y into each angle expression and evaluate.

m∠R = 15 (8) + 17 = 137°

m∠Q = 8 2 + 53 = 117°

m∠S + m∠Q = 180° Definition of supplementary

m∠S + 117° = 180° Substitute.

m∠S = 63° Subtract 117 from both sides.

So, m∠P = 43°, m∠R = 137°, m∠Q = 117°, and m∠S = 63°.



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(12z - 27)°

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B JKLM

Find the value of x.

m∠J + m∠ = ° JKLM is inscribed in a circle.

( + ) + ( - ) = Substitute.

x + = Simplify.

x = Subtract 24 from both sides.

x = Divide both sides by 13.

Find the measure of each angle.

m∠J = 39 + 7 ( ) = ° Substitute the value of x into each angle expression and evaluate.

m∠L = 6 ( ) - 15 = °

m∠K = 20 ( ) _ 3 = °

m∠M + m∠ = ° Definition of supplementary

m∠M + ° = ° Substitute.

m∠M = ° Subtract 80 from both sides.

So, m∠J = °, m∠L = °, m∠K = °, and m∠M = °.

Your Turn

5. Find the measure of each angle of inscribed quadrilateral TUVW.



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Explain 3 Constructing an Inscribed SquareMany designs are based on a square inscribed in a circle. Follow the steps to construct rectangle ACBD inscribed in a circle. Then show ACBD is a square.

Example 3 Construct an inscribed square.

Step 1 Use your compass to draw a circle. Mark the center, O.Draw diameter

_ AB using a straightedge.

Step 2 Use your compass to construct the perpendicular bisector of _ AB . Label the points

where the bisector intersects the circle as C and D.

Step 3 Use your straightedge to draw _ AC , _ CB , _ BD , and

_ DA .

Step 4 To show that ACBD is a square, you need to show that it has 4 sides and 4 angles.

Step 5 Complete the two-column proof to prove that ACBD has four congruent sides.

Statements Reasons

_ OA ≅

_

≅

_

≅

_

Radii of the circle O

m∠AOC = m∠COB = m∠ = m∠ = ° ̄ CD is the perpendicular bisector of _ AB .

△AOC ≅ △COB ≅ △BOD ≅ △DOA

_ AC ≅

_ CB ≅ _ BD ≅ _ DA

Use the diagram to complete the paragraph proof in Steps 6 and 7 that ACBD has four right angles.

Step 6 Since △AOC ≅ △COB, then ∠1 ≅ ∠ by CPCTC. By reasoning similar to

that in the previous proof, it can be shown that △BOC ≅ △COB. Therefore,

by the Transitive Property of Congruence, △AOC ≅ △ , and ∠1 ≅ ∠4 by

CPCTC. Also by the Transitive Property of Congruence, ∠ ≅ ∠4. Similar

arguments show that ∠1 ≅ ∠ , ∠5 ≅ ∠ , and ∠7 ≅ ∠ .



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Find the measures of the angles ofquadrialatral ABCD, which can beinscribed in a circle.

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Homework Problem

(2x - 28)°

(6z - 1)°(x - 2)°

(10z + 5)°

Marcus’s Work

x - 2 + 6z - 1 + 2x - 28 + 10z + 5 = 3603x + 16z - 26 = 360

3x + 16z = 386Cannot solve for two

di�erent variables!

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Step 7 The sum of all the angle measures in a triangle is °, so m∠1 + m∠2 + m∠ = 180°.

Since m∠AOC = °, m∠1 + m∠2 + 90° =180°. This means that m∠1 + m∠2 = °.

Since m∠1 = m∠2, it can be concluded that m∠1 = m∠2 = °. By similar reasoning, it is

shown that the measure of each of the congruent numbered angles is °. Therefore, the measure

of each of the four angles of quadrilateral ACBD is the of the measures of two of the adjacent

numbered angles, which is °.

Reflect

6. How could reflections be used to construct an inscribed square?

Your Turn

7. Finish the quilt block pattern by inscribing a square in the circle. Shade in your square.

Elaborate

8. Critique Reasoning Marcus said he thought some information was missing from one of his homework problems because it was impossible to answer the question based on the given information. The question and his work are shown. Critique Marcus’s work and reasoning.


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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

You use geometry software to inscribe quadrilaterals ABCD and GHIJ in a circle as shown in the figures. You then measure the angle at each vertex.Use the figure for Exercises 1−2. Use the figure for Exercises 3−4.

1. Suppose you drag the vertices of ∠A and ∠C to new positions on the circle and then measure ∠A and ∠C again. Does the relationship between ∠A and ∠C change? Explain.

3. Suppose m∠HIJ = 65° and that m∠H = m∠J. Can you find the measures of all the angles? Explain.

2. Suppose you know that m∠B is 74°. Is m∠D = 74°? Explain.

4. Justify Reasoning You have found that m∠H = m∠J, but then you drag the vertex of ∠G so that m∠H changes. Is the statement m∠H = m∠J still true? Justify your reasoning.

9. What must be true about a rhombus that is inscribed in a circle? Explain.

10. Essential Question Check-In Can all types of quadrilaterals be inscribed in a circle? Explain.


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Use the figure for Exercices 5−6. Find each measure using the appropriate theorems and postulates.

5. m∠B

6. m ⁀ DAB

7. GHIJ is a quadrilateral. If m∠HIJ + m∠HGJ = 180° and m∠H + m∠J = 180°, could the points G, H, I, and J points of a circle? Explain.

8. LMNP is a quadrilateral inscribed in a circle. If m∠L = m∠N, is

_ MP a diameter of the circle?

Explain.

9. Rafael was asked to construct a square inscribed in a circle. He drew a circle and a diameter of the circle. Describe how to complete his construction. Then, complete the construction.



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12. 13.

Multi-Step Find the angle measures of each inscribed quadrilateral.



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14. Critical Thinking Haruki is designing a fountain that consists of a square pool inscribed in a circular base represented by circle O. He wants to construct the square so that one of its vertices is point X. Construct the square and then explain your method.

For each quadrilateral, tell whether it can be inscribed in a circle. If so, describe a method for doing so using a compass and straightedge. If not, explain why not.

15. a parallelogram that is not a rectangle

16. a kite with two right angles

17. Represent Real-World Problems Lisa has not yet learned how to stop on ice skates, so she just skates straight across the circular rink until she reaches a wall. If she starts at P, turns 75° at Q, and turns 100° at R, find how many degrees she must turn at S to go back to her starting point.

18. In the diagram, C is the center of the circle and ∠YXZ is inscribed in the circle. Classify each statement.

a. _ CX ≅

_ CY True/False/Cannot be determined

b. _ CZ ≅ _ XY True/False/Cannot be determined

c. △CXZ is isosceles. True/False/Cannot be determined

d. △CYZ is equilateral. True/False/Cannot be determined

e. _ XY is a diameter of circle C. True/False/Cannot be determined


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H.O.T. Focus on Higher Order Thinking

19. Multi-Step In the diagram, m ⁀ JKL = 198° and m ⁀ KLM = 216°. Find the measures of the angles of quadrilateral JKLM.

20. Critical Thinking Explain how you can construct a regular octagon inscribed in a circle.

21. Represent Real-World Problems A patio tile design is constructed from a square inscribed in a circle. The circle has radius 5 √

_ 2 feet.

a. Find the area of the square.

b. Find the area of the shaded region outside the square.



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Here are some facts about the baseball field shown here:

• ABCD is the baseball “diamond,” a square measuring 90 feet on a side.

• Points A, B, E, H are collinear.• The distance from third base (Point B) to the left

field fence (Point E) equals the distance from first base (point D) to the right field fence (Point G) .

a. Is ⁀ EA congruent to ⁀ AG ? Explain why or why not.

b. Find m∠F. Explain your reasoning.

c. Identify an angle congruent to ∠HEF. Explain your reasoning.

Lesson Performance Task

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PARKING



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