Download Mod 1 - Aim #8 - Manhasset Schools

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