Download Lesson 4.5 • Are There Other Congruence Shortcuts?

DG4GSP_897_04.qxd
12/20/06
1:08 PM
Page 61
Lesson 4.5 • Are There Other Congruence Shortcuts?
In this investigation you will investigate two other potential
congruence shortcuts.
Investigation 1: Is ASA a Congruence Shortcut?
If two angles and the side between them in one triangle are congruent to
two angles and the side between them in another triangle (ASA), must the
two triangles be congruent?
Sketch
Step 1
Open the sketch Congruence Shortcuts.gsp to the page ASA.
C
C
C
C
A
B
A
B
Givens:
A
A
B
B
Step 2
Drag the points labeled C in the broken triangle on the left so that
they coincide to form a triangle.
Step 3
In the broken triangle on the right, try to make the points labeled C
coincide so that the triangle formed is not congruent to the triangle
on the left.
Step 4
Change the measure of one or more of the given angles or the given
side (the angles and segment below the triangles) and try the
experiment again.
Investigate
1. Can you form two triangles with different sizes or shapes given the two
angles and the side between them?
2. If you are given two triangles such that two angles and the side
between them in one triangle are congruent to two angles and the side
between them in another triangle (ASA), is that enough information to
determine that the triangles are congruent?
3. Write a conjecture that summarizes your findings (ASA Congruence
Conjecture).
(continued)
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
CHAPTER 4
61
DG4GSP_897_04.qxd
12/20/06
1:08 PM
Page 62
Lesson 4.5 • Are There Other Congruence Shortcuts? (continued)
Investigation 2: Is AAA a Congruence Shortcut?
If the three angles of one triangle are congruent to the three angles in
another triangle (AAA), must the two triangles be congruent?
Sketch
Step 1
Go to the page AAA.
C
A (drag)
B (drag)
Givens:
A
B
C
(Angles A and B determine angle C.)
Step 2
Drag the points labeled A and B in the triangle.
Step 3
Change the measure of one or more of the given angles (the free
angles below the triangles) and try the experiment again.
Investigate
1. Can you form two or more triangles with different sizes or shapes
given the three angles?
2. If you are given two triangles such that the three angles in one triangle
are congruent to the three angles in the other triangle (AAA), is that
enough information to determine that the triangles are congruent?
3. Summarize your findings.
62
CHAPTER 4
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press