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Transcript
Today you will
Apply the triangle angle-sum theorem and
the exterior angle theorem
4.2 Angles of Triangles
Triangle Angle-Sum Theorem:
The sum of the interior angles of a triangle is 180
o
A
o
m A + m B + m C = 180
C
B
Proof of the Triangle Angle-Sum Theorem:
This proof uses an Auxiliary Line (an extra line drawn to
help analyze geometric relationships).
Given:
ABC
Prove: m 1 + m
2+m
3 = 180
5. Def. of Supplementary
s
1.
2. Draw AD through A parallel to BC
2.
3.
4 and
BAD are a linear pair 3.
4.
4 and
BAD are suppl.
BAD = m 2 + m 5
6.
7. m 4 + m 2 + m 5 = 180
7.
1 and
5
9. m 4 = m 1, m 5 = m
10.
>
4.
5.
4
1
Reasons
ABC
8.
2 5
C
Statements
6. m
4
D
3
B
1.
A
>
3
8.
3
9.
10.
Find each missing angle measure.
60
70
55
4
interior angles:
1,
2,
3
exterior angles:
4,
5,
6
5
6
remote interior:
interior angle that is not adjacent to the exterior angle.
So angle five's remote interior angles are
A
1
C
B
If m 1 = 140, find m ACB
Now find m A + m B
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior angles.
Corollary: a theorem in which the proof follows directly from
another theorem.
Triangle Angle-Sum Corollaries:
The acute angles of a right triangle are complementary.
There can be at most one right or one obtuse angle in a
triangle.
Proof of Triangle Angle Sum Corollary:
Proof:
Given:
Prove:
ABC is right with rt
A comp. C
Statements
B
Reasons