Angle Bisectors In A Triangle- Teacher

Angle Bisectors in a Triangle-TeacherA n g l e B i s e c t o r s i n a Tr i a n g l e Te a c h e rConcepts Relationship between an anglebisector and the arms of the angleApplying the Angle Bisector Theoremand its converseMaterials TI-Nspire Math and ScienceLearning HandheldPTE-Geom AngleBis EN.tnsOverviewIn this activity, students will explore the relationships between an anglebisector and segments in a triangle. They will determine the distances from anangle bisector to the sides of the bisected angle. In a triangle, proportionalrelationships occur when an angle bisector divides the opposite side into twoparts.Teacher PreparationThis activity is designed to be used in a highschool or middle school geometry classroom.The Angle Bisector Theorem states “If a point ison the bisector of an angle, then it is equidistantfrom the sides of the angle.”In a triangle, when an angle bisector divides theopposite side into two parts, the segmentscreated will be proportional to the adjacent sides.For the diagram used in Problem 2, the followingproportions will both be true:BD CDBD ABand AB ACCD ACProblem 3 is an optional extension involving theincenter of the triangle, which is the point ofconcurrency of all three angle bisectors.The screenshots within the activity demonstrateexpected student results. Refer to the screenshotsat the end of this activity for a preview of thestudent TI-Nspire .tns document.T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-TeacherClassroom ManagementThis activity is designed to be student-centeredwith the teacher acting as a facilitator whilestudents work cooperatively. Use the followingpages as a framework as to how the activity willprogress.The student worksheet helps guide studentsthrough the activity and provides a place forstudents to record their answers andobservations.Problem 1- The Angle Bisector Theorem1. Have students open the filePTE Geom AngleBis EN.tns and read thedirections on page 1.2.2. On page 1.3, BAC has been constructed.3. Students are to construct the angle bisector of BAC using the Angle Bisector tool(b 9 4 for Menu 9:Construction,4:Angle Bisector) (Figure 1).4. Direct students to place a new point on theFigure 1angle bisector.5. Label this point X (Figure 2). If the point was not labeled as it was created,select the Text tool by pressingb15 for Menu, 1:Actions, 5:Text.Figure 2T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-Teacher6. Have students measure angles BAX and CAS by pressingb 7 4 forMenu 7:Measurement, 4:Angle (Figure 3).7. Drag point B or C, and observe the results. Does your observation confirm the definitionof an angle bisector?8. If you want, hide the angle measures with theHide/Show tool by pressing b 1 2Figure 3for Menu 1:Actions, 2:Hide/Show. The distance from point X to the sides of theangle must be measured perpendicularly.9. Students will construct a line through Xperpendicular to AB using the Perpendiculartool (b 9 1 for Menu9:Construction, 1:Perpendicular) (Figure 4).10. Repeat to construct a line through XFigure 4perpendicular to AC (See Figure 4).11. Students should use the Intersection Point(s)tool (b 6 3 for Menu 6:Points &Lines, 3:Intersection Point(s)) to place pointsJJJGat the intersection of AB and itsperpendicular line and the intersection ofJJJGAC and its perpendicular line.Figure 512. Hide the perpendicular lines (Figure 5).13. Have students use the Segment tool (b6 5 for Menu 6: Points & Lines,5:Segment) to connect X to each intersectionpoint (Figure 6).14. Measure the lengths of each segment usingthe Length tool (b 7 1 for Menu7:Measurement, 1:Length).Figure 6T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-Teacher15. Students should drag point X and observe thechanges in the measurements (Figure 7).Figure 716. Then drag point B or C to change the size ofthe angle, and observe the results (Figure 8).17. Record observations on the worksheet.Problem 2 – One Angle Bisector in aTriangle18. Students should advance to page 2.1 and readFigure 8the directions.19. On page 2.2, students are to construct theangle bisector of BAC (Figure 9).20. Before continuing, have students drag avertex of UABC so that everyone getsdifferent data.21. Have students find the intersection point ofthe angle bisector and side BC .Figure 922. Label this point D (Figure 10).Figure 10T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-Teacher23. Direct students to measure the lengths ofAB , AC , BD , and CD using the Lengthtool (Figure 11).24. Record these values on the worksheet.25. Drag a vertex of the triangle, and recordmore data.26. Have students use the Text tool to put thepexpression on the screen (Figure 12).q Figure 11They will use this expression to calculateratios of the measurements.27. Divide pairs of the measurements using theCalculate tool (b 1 7 for Menu1:Actions, 7:Calculate).Figure 1228. Examine the ratios that result (Figure 13).29. Drag a vertex of UABC and examine theratios again. What do you notice?30. Ask students to identify a set of ratios thatare equal to each other.Figure 1331. Drag a vertex of the triangle to confirm this(Figure 14).32. Record observations on the worksheet.Figure 14T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-TeacherExtension: One Angle Bisector and theIncenter of a Triangle33. Students should advance to page 3.1 and readthe directions. On page 3.2, UDEF is constructed with allthree angle bisectors created (Figure 15). The point at which the three angle bisectorsFigure 15intersect is called the incenter and is labeledI.34. Students should hide angle bisectors EI andFI using the Hide/Show tool (Figure 16).35. Use the Intersection tool to find the point ofintersection of the remaining angle bisectorwith the opposite side and label it point G.36. Direct students to measure the lengths of DIFigure 16and DG with the Length tool (Figure 17).37. Also measure sides DE and DF and theperimeter of the triangle. The expressionsDIDE DFandare onDGPthe screen.38. Students should use the Calculate tool toevaluate these expressions (Figure 18).Figure 1739. Compare the quotients to each other.Figure 18T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-Teacher40. Students should drag a vertex of UDEF andexamine the quotients again (Figure 19). What do you notice?Figure 19T3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

Angle Bisectors in a Triangle-TeacherAngle Bisectors in a Triangle(Student)TI-Nspire Document: Geom AngleBis Student.tnsT3 PROFESSIONAL DEVELOPMENT SERVICES FROM TEXAS INSTRUMENTSPRE-SERVICE TEACHER EDUCATION WITH TI-NSPIRE TECHNOLOGY 2008 TEXAS INSTRUMENTS INCORPORATED

angle bisector to the sides of the bisected angle. In a triangle, proportional relationships occur when an angle bisector divides the opposite side into two parts. Teacher Preparation This activity is designed to be used in a high school or middle school geometry classroom. The Angle Bisector Theorem states “If a point is on the bisector of .