Download Lesson 6.3 • Arcs and Angles

DG4GSP_897_06.qxd
12/20/06
1:32 PM
Page 88
Lesson 6.3 • Arcs and Angles
Investigation 1: Inscribed Angle Properties
In this investigation you’ll discover a relationship between an inscribed
angle and the arc it intercepts.
Sketch
Step 1
In a new sketch, construct circle AB.
Step 2
!, where point C is a point on the circle.
Construct BC
Step 3
!!, where point D is another point on the circle.
Construct CD
Step 4
Measure !DCB.
Step 5
Select, in order, point B, point D, and the circle, and choose
Construct ⏐ Arc On Circle. Change its line width to thick.
Step 6
m!DCB ! 50.38°
arc angle A ! 100.77°
D
A
B
C
Select the arc and choose Measure ⏐ Arc Angle.
Investigate
1. Drag point D (but not past points C or B) and look for a relationship
between the arc measure (called Arc Angle in Sketchpad) and the
measure of the inscribed angle. Make a conjecture (Inscribed
Angle Conjecture).
2. Drag point C. As long as you don’t drag it past points B and D, the
measurements don’t change. Is your computer broken? Well, dragging
point C doesn’t do anything to the arc. What does that mean for all the
inscribed angles that intercept that arc? If you’re not sure, construct
!. Write a
and measure another inscribed angle that intercepts BD
conjecture about inscribed angles that intercept the same arc (Inscribed
Angles Intercepting Arcs Conjecture).
3. Construct segment DB and change its line width to dashed. Drag
! passes through the circle’s center. What is the
point D until DB
measure of !DCB? Now drag point C to see if m!DCB changes.
Write a conjecture about angles inscribed in a semicircle (Angles
Inscribed in a Semicircle Conjecture).
Investigation 2: Cyclic Quadrilaterals
Now you’ll apply your previous discoveries to the angles of a quadrilateral
inscribed in a circle, which is called a cyclic quadrilateral.
C
D
Sketch
Step 1
In a new sketch, construct circle AB.
Step 2
Use the Segment tool to construct quadrilateral BCDE, where
points C, D, and E are also points on the circle.
Step 3
A
B
E
Measure the four angles of the quadrilateral.
(continued)
88
CHAPTER 6
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
DG4GSP_897_06.qxd
12/20/06
1:32 PM
Page 89
Lesson 6.3 • Arcs and Angles (continued)
Investigate
1. Look for relationships between pairs of angles in the quadrilateral.
Use the calculator to check any relationships you discover, then write
a conjecture (Cyclic Quadrilateral Conjecture).
2. Explain why the Cyclic Quadrilateral Conjecture is true. (What kinds
of angles did you measure? What is the sum of the arc measures of
the two arcs intercepted by opposite angles in the quadrilateral?)
Investigation 3: Arcs by Parallel Lines
Now you’ll discover a relationship between arcs formed when parallel
lines intersect a circle.
Sketch
Step 1
In a new sketch, construct circle AB.
Step 2
!"#, where point C is a point on the circle.
Construct BC
(Drag points B and C to make sure the line is
attached correctly.)
C
B
A
D
Step 3
Construct point D on the circle.
Step 4
!"#.
Construct a line through point D parallel to BC
Step 5
Construct point E where the new line intersects the circle.
Step 6
Select, in order, point E, point B, and the circle. Choose Construct ⏐
Arc On Circle. Change the line width of this arc to thick.
! the same way. Make sure you select your points in
Construct CD
Step 7
E
counterclockwise order.
Investigate
1. Select each arc and measure its arc angle. Drag point C and observe
the arcs and their measurements. Make a conjecture about the arcs
intercepted by parallel lines (Parallel Lines Intercepted Arcs Conjecture).
EXPLORE MORE
Given a circle and a point outside the circle, find a method for
constructing the two tangents from that point. Describe how you
made your construction. (Hint: Start by constructing a segment from
the point to the circle’s center. Then construct the midpoint of that
segment. You’re on your own from here.)
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
CHAPTER 6
89