Download Lesson 4.2 • Properties of Isosceles Triangles

Lesson 4.2 • Properties of Isosceles Triangles
In this activity you’ll learn how to construct an isosceles triangle
(a triangle with at least two sides the same length). Then you’ll
discover properties of isosceles triangles.
Investigation 1: Base Angles in an Isosceles Triangle
Sketch
Step 1
In a new sketch, construct a circle with center A and
control point B.
Step 2
Construct radius AB.
Step 3
Construct point C on the circle, and then construct radius AC.
Step 4
!.
Construct BC
Step 5
Hide circle AB.
A
C
B
Investigate
1. Drag each vertex of your triangle. Explain why the triangle is always
isosceles.
2. Measure !ACB and !ABC. Make sure the vertex is the second point
you select. Angles ACB and ABC are the base angles of the isosceles
triangle. Angle CAB is the vertex angle. Drag the vertices of your
triangle and observe the measures of !ABC and !ACB. What do you
observe about the measures?
3. Write a conjecture about the base angles of an isosceles triangle
(Isosceles Triangle Conjecture).
Investigation 2: Is the Converse True?
In this sketch you will construct a triangle that has base angles of equal
measure, then investigate the measures of the sides.
Sketch
Step 1
"!#.
In a new sketch, construct AB
Step 2
!# to form acute !CAB.
Choose the Ray tool and construct AC
Step 3
Mark !CAB as an angle of rotation by selecting, in order,
points C, A, and B, then choosing Transform  Mark Angle.
Step 4
Double-click on point B to mark it as a center for rotation.
C
A
B
Steps 1 and 2
(continued)
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
CHAPTER 4
55
Lesson 4.2 • Properties of Isosceles Triangles (continued)
Step 5
"!# and choose Transform  Rotate. Rotate AB
"!# by the
Select AB
marked angle.
Step 6
!#.
Construct the point of intersection D of this line and AC
D
C
A
B
Steps 5 and 6
Step 7
Hide the ray and the two lines, and leave all of the points showing.
Step 8
Choose the Segment tool and construct "ADB.
D
C
A
B
Steps 7 and 8
Investigate
1. Drag point C to change the measure of !CAB. What do you observe
about "ADB?
2. Measure the sides of "ADB by selecting them and choosing Measure 
!!
Length. Drag point C. What do you observe about the lengths of AD
!
and BD ?
3. What do you observe about a triangle that has two angles of equal
measure? Write your findings as a conjecture (Converse of the
Isosceles Triangle Conjecture).
EXPLORE MORE
Use Sketchpad to investigate the statement “If a triangle is equiangular,
then it is equilateral” and its converse, “If a triangle is equilateral, then
it is equiangular.” Can you construct an equiangular triangle that
is not equilateral? Can you construct an equilateral triangle that is
not equiangular? Write a conjecture based on your investigation
(Equilateral/Equiangular Triangle Conjecture).
56
CHAPTER 4
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press