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Segment and Angle Bisectors During today’s lesson, you will: bisect a segment bisect an angle solve problems involving segment and angle bisectors Mrs. McConaughy Geometry 1 DEFINITION: Midpoint Recall, the midpoint of a segment is ___________________________________ a point which divides (or bisects) the segment into two congruent segments. ___________________________________. A segment bisector is _________________ a segment, ray, line, or plane which intersects a segment at its ___________________________________ midpoint ___________________________________. Mrs. McConaughy Geometry 2 Draw a segment bisector given AY with midpoint M. How many segment bisectors can be drawn through the midpoint M? A M Mrs. McConaughy Geometry Y 3 DEFINITION: Perpendicular Lines Perpendicular lines ___________________ are two lines that intersect to form congruent right angles. ___________________________________. Mrs. McConaughy Geometry 4 Origami Exploration: Paper Folding Perpendicular Bisectors Step 1: On a piece of paper, draw HT. Step 2: Fold the paper so that B is on top of A. Step 3: Label the point where the fold intersects HT as point O. Step 4: Measure HO and HT. Step 5: Measure the four angles which are formed. Mrs. McConaughy Geometry 5 DEFINITION: Perpendicular You are now ready to write a good definition of a perpendicular bisector. Bisector A perpendicular bisector of a segment is __________________________________ a segment, ray or line which a. intersects a segment at its midpoint and b. forms ___________________________________ congruent right angles. ___________________________________. For additional information on paper-folding geometry, click on the pencil in the upper right-hand corner. Mrs. McConaughy Geometry 6 Origami Exploration: Paper Folding Angle Bisectors Step 1: On a piece of paper, draw ABC. Step 2: Fold the paper so that CB is directly on top of CA. Step 3: Draw any point on the fold and label the point D. Step 4: Use a protractor to measure angles ACD and BCD. You are now ready to write a good definition of an angle bisector. Mrs. McConaughy Geometry 7 DEFINITION: Angle Bisector An angle bisector is __________________ a ray (or segment) which divides an angle into two congruent ___________________________________ angles. The vertex of the segment or ray is ___________________________________ at the angle’s vertex. ___________________________________. Mrs. McConaughy Geometry 8 Example: Finding Angle Measures The ray FH bisects angle EFG. Given that m< EFG = 120, what are the measures of the two angles which are formed? Mrs. McConaughy Geometry 9 Example: Finding Angle Measures Given KN bisects angle JKL so that m< JKN = 5x - 25 and m< KNL = 3x + 5. Solve for x and find the measure of < JKN. Mrs. McConaughy Geometry 10 Work Space: Mrs. McConaughy Geometry 11 Final Checks for Understanding 1. What two geometric figures are formed when you paper-fold a perpendicular bisector? 2. How do you indicate right angles in a diagram? 3. Given QS is the bisector of < PQR, determine m< SQR and m< PQR when m < PQS = 40. Mrs. McConaughy Geometry 12 Homework Assignment Pages 37-38: 9-12 all OR Segment and Angle Bisectors WS 31 – 42 all; 44- 49 all Mrs. McConaughy Geometry 13