Download Lesson 4.1 • Triangle Sum Conjecture

Lesson 4.1 • Triangle Sum Conjecture
In this activity, you’ll investigate and make a conjecture about the sum of
the measures of the angles in a triangle.
Investigation: The Triangle Sum
Sketch
Step 1
In a new sketch, construct !ABC.
Step 2
Measure "CAB, "CBA, and "ACB. Make sure that
the vertex is the second point you select on each
angle.
Step 3
Use the calculator (choose Measure  Calculate) to
calculate m"CAB ! m"CBA ! m"ACB.
C
B
m!CAB ! 56.15°
m!CBA ! 26.97°
m!ACB ! 96.88°
A
Investigate
1. Drag points A, B, and C. What do you observe about the sum of the
three angles?
2. Write a conjecture about the sum of the three angles of a triangle
(Triangle Sum Conjecture).
EXPLORE MORE
1. Construct a triangle, and construct a line through one vertex parallel to
the opposite side.
a. How do the three angles formed at the vertex compare with the
angles in the triangle?
b. Explain how the Triangle Sum Conjecture follows logically from the
Parallel Lines Conjecture.
2. Select the three vertices and choose Construct  Triangle Interior.
Select vertex A and then vertex C, and choose Transform  Mark
Vector. Finally, select the interior of the triangle and choose
Transform  Translate. The dialog box should indicate that you
are translating by the marked vector from point A to point C.
Press Translate.
a. What would fill the space between your two triangles? Construct
!, mark it as center, and rotate the interior of
the midpoint of BC
!ABC 180° about this point.
b. What can you say about the three angles that now meet at point C?
c. How does this confirm the Triangle Sum Conjecture?
3. Investigate angle sums in other polygons.
4. Find another way to demonstrate the Triangle Sum Conjecture.
54
CHAPTER 4
Discovering Geometry with The Geometer’s Sketchpad
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