Download Advanced Geometry LT 3.1 – Triangle Sum and Exterior Angle

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Transcript
Advanced Geometry
LT 3.1 – Triangle Sum and Exterior Angle Theorems
Exploring the Triangle Sum Theorem
Activity:
1. Each member of your group should draw a different triangle on the half sheets of paper provided. Make
sure you have at least one acute, obtuse and right triangle in you group.
2. Label the angles in your triangle with 1 – 3.
3. Use the tracing paper to trace the 3 angles of the triangle as 3 adjacent angles. (Trace
, then trace
adjacent to it, then . They should all share the same vertex.)
4. Note what is formed once all three angles are traced side by side.
Reflect:
1. Compare your tracing paper with your group. What always seems to be true about the three angles of a
triangle when they are placed together? Does it matter what type of triangle it is?
2. Make a conjecture: What can you say about the sum of the angle measures in all triangles?
3. An equiangular triangle has three congruent angles. What do you think is true about the angles in an
equiangular triangle? Why?
4. In a right triangle, what is the relationship of the measures of the two acute angles? Why?
The Triangle Sum Theorem
The sum of the angle measures in a triangle is _______.
In order to prove the Triangle Sum Theorem we need the following postulate:
The Parallel Postulate
Through a point P not on a line l, there is exactly one line parallel to l.
Proving the Triangle Sum Theorem
Given:
Prove:
Statements
Reasons
1.
1. Given
2. Draw line l through point B
parallel to
(label the 2 new
angles 4 and 5)
2. Parallel Postulate
3. __________________________
3. _____________________________
4. __________________________
4. Angle Addition Postulate and
definition of a straight angle
5. __________________________
5. _____________________________
Reflect:
5. Use the Triangle Sum Theorem to explain why it is not possible for a triangle to have two right angles.
Exploring the Exterior Angle Theorem
When you extend the sides of a polygon, the original angles are called interior angles and the angles that form
linear pairs with the interior angles are called the exterior angles. Each exterior angle of a triangle has two
remote interior angles. A re mote interior angle is an interior angle that is not adjacent to the exterior angle. For
example:
Original Triangle
Triangle With One Side Extended
Activity:
1. Extend one side of the triangle you drew earlier.
2. Label the exterior angle as
3. Determine which angles are the remote interior angles to
4. Use your tracing paper to trace the remote interior angles adjacent to each other
Reflect:
6. What do you notice about the two remote interior angles when they are placed together ? Is it true for all of
your group members’ triangles?
Proving the Exterior Angle Theorem
Given:
Prove:
Statements
Reasons
1.
1. Given
2.
2. _______________________
3. __________________________
3. Triangle Sum Theorem
4.
4. _______________________
5.
5. _______________________
Reflect:
7. If two angles of one triangle are congruent to two angles of another triangle, must the third angles of the
triangles also be congruent? Why or why not?
8. According to the definition of an exterior angle, one of the sides of the triangle mus t be extended in order to
see it. How many ways can this be done for any vertex? How many exterior angles is it possible to draw for one
triangle?