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Aim #8: How do we copy an angle and bisect an angle? CC Geometry H Do Now Define: Angle ____________________________________________________ _________________________________________________________ A half-plane is the part of a plane on one side of a straight line. Shade the half-plane B of AC that contains B. B A Shade the half-plane of AB that contains C. C What word describes the A intersection of the two shadings as it relates to the angle? ANGLE BISECTOR A C O OC bisects ≮AOB. B Definition (1): Angle bisector If C is in the interior of ≮AOB, and ≮AOC = ≮COB, then OC bisects ≮AOB, and OC is called the bisector of ≮AOB. Definition (2): Angle bisector: An angle bisector is a ray that divides an angle into two equal angles. • When we say m≮AOC = m≮COB, we mean that the angle measures are equal. • If ≮AOC ≅ ≮COB, then m≮AOC = m≮COB. C Angle Measurement Assumptions 1) To every ≮AOB, there corresponds a real number called the degree or measure of the angle. 2) ANGLE ADDITION POSTULATE: If C is a point on the interior of ≮AOB, then m≮AOC + m≮COB = m≮AOB C A "A whole is equal to the sum of its parts." O 3) LINEAR PAIR B C ≮BAC A B and ≮DAC are a linear pair. D Definition (1): Linear pair Two angles ≮BAC and ≮CAD form a linear pair if AB and AD are opposite rays on a line, and AC is any other ray. Definition (2): Linear pair Two angles form a linear pair if they are adjacent and their non-common sides are collinear. 4) If two angles ≮BAC and ≮CAD form a linear pair, then they are supplementary. 0 If two angles ≮BAC and ≮CAD form a linear pair, then they sum to 180 . C 0 m≮BAC + m≮CAD = 180 "Therefore" B A D ∴Angles that form a linear pair are supplementary. ∴Angles that form a linear pair have a sum of 1800. All CONSTRUCTIONS must be done using a compass and straightedge. Construction 1: Bisect an Angle http://www.mathsisfun.com/geometry/constructanglebisect.html A. Bisect the angle: What steps did you take to bisect an angle? List the steps below: Label vertex of angle as A. 1) ___________________________________________________________________ Draw circle A, radius less than length of given rays. 2)___________________________________________________________________ Label intersection of circle A with rays as B and C. 3)___________________________________________________________________ 4) Draw circle B, radius BC. ___________________________________________________________________ Draw circle C, radius BC. 5)___________________________________________________________________ 6) Label intersection pt. of circle B and C that lies in the interior of angle as D. ___________________________________________________________________ ___________________________________________________________________ Draw AD. 7)__________________________________________________________________ Practice: Bisect the following angles. Bisect the reflex angle with arc shown. Construction 2: Copy an Angle (Construct an equal angle.) http://www.mathsisfun.com/geometry/constructanglesame.html or "D" here Steps Needed to Copy an Angle: Label the vertex of the original angle as B. 1) __________________________________________________________________________________ Draw EG as one side of the angle to be drawn. 2) _______________________________________________________________________________ Draw circle (arc) B: center B, any radius. 3) _______________________________________________________________________________ Label the intersections of circle B with the sides of the angle as A and C. 4) _______________________________________________________________________________ Draw circle (arc) E: center E, radius BA. 5) _______________________________________________________________________________ Label the intersection of circle E with EG as F. 6) _______________________________________________________________________________ Draw circle (arc) C: center C, radius CA. 7) _______________________________________________________________________________ Draw circle (arc) F: center F, radius CA. 8) _______________________________________________________________________________ 9) Label either intersection of circle E and circle F as D. __________________________________________________________________________________ 10) Draw ED. _________________________________________________________________________________ Practice: Copy the angle. Let's Sum it Up!! Choose the correct fill-in from the box to the right. o (The is an abbreviation for “degrees”.) 0 90 360 equal arcs of a circle. 1. 1o is the measure of one of ______ 180 360 o ray 2. A 0 angle looks like a ______. adjacent o line o 0 90 3. An acute angle measures > ______ and < ______ . equal o 90 4. A right angle measures______ . ray o supplementary o 180 . 90 and < ____ 5. An obtuse angle measures >____ complementary line 180 o and looks like a _______. 6. A straight angle measures ____ supplementary 7. If two angles form a linear pair, the angles are _______________________. complementary 8. Two angles with a sum of 90o are called _________________. 9. Two angles that share a common ray, but no common interior points are called adjacent ______________ s. supplementary 10. Two angles with a sum of 180 are called __________________. o equal 11. An angle bisector divides an angle into two __________ angles. CC Geometry H Name ___________________ Date _____________________ HW #8 #1- 4 Bisect each angle below (the angle that is marked with an arc). The arc shown is not part of your construction. 2) 1) 3) 4) 5) Copy the angle below. 6) Construct and label BD, the bisector of Construct ABC. XYZ, such that m ABD = m XYZ. C A B Y Review 7) Find the measures of the three interior angles of the triangle. Show all steps. (120 x)0 (3x)0 (x2 5x)0 8. ΔABC is shown below. Use your compass to determine if it is an equilateral triangle. Justify your response. 9. Find the measures of x in each diagram. Write reasons for your answers. a) b) 1+14x 23x5 12x+17 21x+5 d) c) 3x 600 8x4 2800 P 7x