Name: Date: Period: Constructing Angle Bisectors and Parallel Lines TASK A: 1) Complete the following steps below. a. Draw a circle centered on point P. b. Mark any two points on the circle that are not the endpoints of a diameter. Label those points X and Y. c. Draw PX and PY. P 2) Measure the length of PX and of PY. What do you notice about the lengths? 3) Provide a convincing argument (proof) that what you noticed must be true. 4) Connect X and Y. What type of triangle did you construct? Provide a convincing proof. Radii of a circle are ; therefore, circles help construct segments. IMP Activity: Angle Bisector and Parallel Lines 1
TASK B: 5) Complete the following steps below. a. Draw a circle centered on point M. b. Mark a point outside of Circle M, but within a distance of one radius from Circle M. Label this point N. c. Construct a congruent circle centered at N by keeping your compass fixed at the same size opening as Circle M and centered on point N. d. If your point N was chosen correctly, your new circle will intersect the first circle in 2 places. Label those two intersections X and Y. e. Draw quadrilateral MXNY. M 6) What type of quadrilateral is MXNY? 7) What is true about the opposite sides of a rhombus? They are and 8) Connect points M and N. 9) What do you notice about the two triangles created (ΔMXN and ΔMYN)? 10) Provide a convincing argument (proof) that what you noticed must be true. 11) What does MN do to XMY? Give a convincing argument. Diagonals of a rhombus opposite angles. IMP Activity: Angle Bisector and Parallel Lines 2
TASK C: 11) Now construct the bisector of an angle by constructing a rhombus; follow the steps for each angle and then complete the explanation below. a) Draw a circle with its center on the vertex so that it intersects both rays of the angle. Label these points A and B. b) Using the compass at the same opening as the circle in part a), find the point equidistant from points A and B. c) Connect this point to the vertex. Explain why your constructions led to the bisection of each angle. IMP Activity: Angle Bisector and Parallel Lines 3
TASK D: 12) Use what you know to develop a compass and straightedge procedure for drawing a parallel line through a point off a given line. Hint: given the line L and the point P off the line, start by picking an arbitrary point Q on L; then join P and Q to form segment PQ. Use the rhombus construction to construct the line through P parallel to L (remember: opposite sides of a rhombus are parallel). Practice this procedure twice below, and then write step-by-step directions for your procedure. P 1. 2. 3. 4. IMP Activity: Angle Bisector and Parallel Lines 4
Teacher Directions: Angle Bisector and Parallel Lines Objective: Students will learn how to construct a rhombus and how to use its properties to construct an angle bisectors and parallel lines. Materials: Compass- 1 per student Straightedge 1 per student Activity Notes: Each page should flow through a cycle of students working in pairs or small teams followed by a whole class discussion with students presenting and defending their ideas and solutions. TASK A: 2-3: The lengths are congruent because they are both radii of the circle. 4: The triangle is an isosceles triangle by definition of isosceles. The key fact is that all radii of a circle are congruent; however, students will need to extend this idea to congruent circles also having congruent radii. This will be needed for TASK B. TASK B: 6: MXNY is a rhombus because all four sides are congruent (each side is a radius). 7. Sides of a rhombus are congruent and opposite sides are PARALLEL (students will use this property to construct parallel lines in TASK D). 8-10: The triangles are congruent isosceles triangles. They are congruent by S-S-S and they are isosceles because two legs are congruent since they are sides of a rhombus. 11: MN bisects XMY because the adjacent angles are congruent base angles of congruent triangles. (Students may need to be reminded about base angles of isosceles triangles.) The key conclusion is that diagonals of a rhombus bisect opposite angles. Students will use this to construct angle bisectors in TASK C: TASK C: Students should explain that since the distances were all equal, the shape formed was a rhombus. Recreating the rhombus and its diagonal (similar to what they did in TASK B) will create the angle bisector. TASK D: Again the rhombus can be recreated to construct a parallel line. Given: a line L and point P, not on the line 1. Choosing any point Q on line L, join Q and P to form PQ 2. Copy PQ along line L, label that vertex S. 3. Using the same length, construct the remaining vertex, R, of the rhombus by swinging arcs from both P and S and noting their intersection (R). 4. PQSR is a rhombus so PS is parallel to line L. IMP Activity: Angle Bisector and Parallel Lines Teacher Directions 5