14.2 Angles in Inscribed Quadrilaterals

Transcription

1 Name lass ate 14.2 ngles in Inscribed Quadrilaterals Essential Question: What can you conclude about the angles of a quadrilateral inscribed in a circle? Explore G.12. pply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems. lso G.5. Investigating Inscribed Quadrilaterals Resource Locker There is a relationship among the angles of a quadrilateral that is inscribed in a circle. hoose appropriate tools such as geometry software or a compass, straightedge, and protractor to explore the angle measures of a quadrilateral inscribed in a circle. onstruct a circle with an inscribed quadrilateral similar to the one shown. Measure the four angles of quadrilateral and record their values to the nearest degree on the diagram. Find the sums of the indicated angles. m + m = m + m = m + m = m + m = m + m = m + m = onstruct another circle with a diameter greater than the circle in Step. 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. Measure the four angles of EFGH to the nearest degree and record their values on your diagram. 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 Lesson 2

2 Reflect 1. iscussion ompare your work with that of other students. What conclusions can you make about the angles of a quadrilateral inscribed in a circle? 2. ased 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 Theorem The 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 is inscribed in circle O. Prove: and are supplementary. and are supplementary. O Step 1 The union of and is circle O. Therefore, m + m = Step 2 is an inscribed angle and its intercepted arc is. is an inscribed angle and its intercepted arc is. y the Inscribed ngle Theorem, m = _ m m = _ m Step 3 So, m + m =. and Substitution Property of Equality = istributive Property = Substitution Property of Equality = Simplify. So, and are supplementary, by the definition of supplementary. Similar reasoning shows that and are also supplementary. Module Lesson 2

3 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. 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 pplying the Inscribed Quadrilateral Theorem Example 2 Find the angle measures of each inscribed quadrilateral. PQRS Find the value of y. (y ) Q R (15y + 17) m P + m R = 180 PQRS is inscribed in a circle. P (5y + 3) S (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 = = 117 m S + m Q = 180 efinition of supplementary m S = 180 Substitute. m S = 63 Subtract 117 from both sides. So, m P = 43, m R = 137, m Q = 117, and m S = 63. Module Lesson 2

4 JKLM Find the value of x. (39 + 7x) J m J + m = JKLM is inscribed in a circle. 20x 3 ( + ) + ( - ) = Substitute. M L K x + = Simplify. (6x - 15) x = Subtract 24 from both sides. x = ivide both sides by 13. Find the measure of each angle. m J = ( ) = Substitute the value of x into each angle expression and evaluate. m L = 6 ( ) - 15 = 20 ( ) m K = _ = 3 m M + m = efinition 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. U (125 - z) T W (37 + 5z) (12z - 27) V Module Lesson 2

5 Explain 3 onstructing an Inscribed Square Many designs are based on a square inscribed in a circle. Follow the steps to construct rectangle inscribed in a circle. Then show is a square. Example 3 onstruct an inscribed square. Step 1 Use your compass to draw a circle. Mark the center, O. raw diameter _ using a straightedge. Step 2 Use your compass to construct the perpendicular bisector of _. Label the points where the bisector intersects the circle as and. Step 3 Use your straightedge to draw _,,, and _. Step 4 To show that is a square, you need to show that it has 4 sides and 4 angles O Step 5 omplete the two-column proof to prove that has four congruent sides. Image redits: Kevin Hsieh/Shutterstock Statements O Radii of the circle O Reasons m O = m O = m = m = is the perpendicular bisector of _. O O O O Use the diagram to complete the paragraph proof in Steps 6 and 7 that has four right angles. Step 6 Since O O, then 1 by PT. y reasoning similar to that in the previous proof, it can be shown that O O. Therefore, by the Transitive Property of ongruence, O, and 1 4 by PT. lso by the Transitive Property of ongruence, 4. Similar arguments show that 1, 5, and 7. Module Lesson 2

6 Step 7 The sum of all the angle measures in a triangle is, so m 1 + m 2 + m = 180. Since m O =, m 1 + m =180. This means that m 1 + m 2 =. Since m 1 = m 2, it can be concluded that m 1 = m 2 =. y 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 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. ritique 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. ritique Marcus s work and reasoning. Homework Problem Find the measures of the angles of quadrialatral, which can be inscribed in a circle. (x - 2) (10z + 5) (6z - 1) (2x - 28) Marcus s Work x z x z + 5 = 360 3x + 16z - 26 = 360 3x + 16z = 386 annot solve for two different variables! Module Lesson 2

7 9. What must be true about a rhombus that is inscribed in a circle? Explain. 10. Essential Question heck-in an all types of quadrilaterals be inscribed in a circle? Explain. Evaluate: Homework and Practice hoose appropriate tools such as geometry software or a compass, straightedge, and protractor to inscribe quadrilaterals and GHIJ in a circle as shown in the figures. Then measure the angle at each vertex. Use the figure for Exercises 1 2. Use the figure for Exercises 3 4. Online Homework Hints and Help Extra Practice I H G J 1. Suppose you drag the vertices of and to new positions on the circle and then measure and again. oes the relationship between and change? Explain. 2. Suppose you know that m is 74. Is m = 74? Explain. 3. Suppose m HIJ = 65 and that m H = m J. an you find the measures of all the angles? Explain. 4. Justify Reasoning You have found that m H = m J, but then you drag the vertex of H so that m H changes. Is the statement m H = m J still true? Justify your reasoning. Module Lesson 2

8 Use the figure for Exercices 5 6. Find each measure using the appropriate theorems and postulates. 5. m m 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. I 8. LMNP is a quadrilateral inscribed in a circle. If m L = m N, is _ MP a diameter of the circle? Explain. L P H N G J M 9. Rafael was asked to construct a square inscribed in a circle. He drew a circle and a diameter of the circle. escribe how to complete his construction. Then, complete the construction. Module Lesson 2

9 Multi-Step Find the angle measures of each inscribed quadrilateral. 10. P 11. Q (5x + 20) 10x R (7x - 8) S (4z - 10) (6z - 5) (10 + 5z) (z - 59) E z 2 z (14 + 4x) U V T (15y - 4) W (6x - 14) (12y - 5) Module Lesson 2

10 14. ritical 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. onstruct the square and then explain your method. O X 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. Q P S R 18. In the diagram, is the center of the circle and YXZ is inscribed in the circle. lassify each statement. X Z Y a. _ X _ Y True/False/annot be determined b. _ Z _ XY True/False/annot be determined c. XZ is isosceles. True/False/annot be determined d. YZ is equilateral. True/False/annot be determined e. _ XY is a diameter of circle. True/False/annot be determined Module Lesson 2

11 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. J K M L 20. ritical Thinking Explain how you can construct a regular octagon inscribed in a circle. 21. Represent Real-World Problems 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. Module Lesson 2

12 Lesson Performance Task Here are some facts about the baseball field shown here: is the baseball diamond, a square measuring 90 feet on a side. Points,, E, H are collinear. The distance from third base (Point ) to the left field fence (Point E) equals the distance from first base (point ) to the right field fence (Point G). H PRKING Left Field Fence F Right Field Fence E G 90 ft 90 ft 90 ft 90 ft a. Is E congruent to G? Explain why or why not. b. Find m F. Explain your reasoning. c. Identify an angle congruent to HEF. Explain your reasoning. Image redits: SuperStock/age fotostock Module Lesson 2