Stephanie Lalos
Theorem 50
 The sum of measures of the three angles of a
triangle is 180o
B
60
A
80
40
C
mA  mB  mC  180
o
Proof
 According to the parallel postulate, there exists exactly one
line through point A parallel to BC
o
 Because of the straight angle, we know that m1  m2  3  180
A
1
2
3
B
C
 Since 1  B and 3  C (by
lines  alt int . s )
we may substitute to obtain B  2  C  180
Hence, mA  mB  mC  180
o
o
Other Proofs
Lemma
If ABCD is a quadrilateral and <)CAB = <)DCA then AB
and DC are parallel.
Proof
Assume to the contrary that AB and DC are not
parallel.
Draw a line trough A and B and draw a line trough D
and C.
These lines are not parallel so they cross at one point.
Call this point E.
Notice that <)AEC is greater than 0.
Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees.
Hence <)AEC + <)CAE + <)ACE is greater than 180
degrees.
Contradiction. This completes the proof.
Definition
Two Triangles ABC and A'B'C' are congruent if and only
if
|AB| = |A'B'|, |AC| = |A'C'|, |BC| = |B'C'| and,
<)ABC = <)A'B'C', <)BCA = <)B'C'A', <)CAB = <)C'A'B'.
Right triangles are used to prove the sum of the angles of a
triangle in a youtube video that can be seen here.
Definition
 Exterior angle – an angle of a polygon that is adjacent to
and supplementary to an interior angle of the polygon
 Examples -  1 is an exterior angle to the below triangles
1
For alternative exterior angle help visit…
1
Regents Prep
Theorem 51
 The measure of an exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles
B
1
A
C
m1  mB  mA
Theorem 52
 A segment joining the midpoints of two sides of a triangle
is parallel to the third side, and its length is one-half the
length of the third side. (Midline Theorem)
B
Given: D & E are midpoints
Therefore, AD  DB & BE  EC
Prove: a. DE
AC
D
E
1
b. DE = (AC)
2
A
C
B
Extend DE through to a point F
so that EF  DE. F is now
established, so F and C
determine FC.
F
E
D
BED  CEF
(vertical
A
angles are congruent)
C

DFCA is a parallelogram, one
BED
CEF (SAS)
pair of opposite sides is both
congruent and parallel,
therefore, DF AC
B  FCE (CPCTC)
Opposite sides of a
FC DA (alt. int.s  Lines)
parallelogram are congruent, so
DF=AC,
since EF=DE,
1

FC DA (transitive)
DE= 2 (EF) and by substitution
DE= 1 (AC).
2
Sample Problems
80
100
z
55
x + 100 + 60 = 180
x + 160 = 180
x = 20
x
y
55 + 80 + y = 180
135 + y = 180
y = 45
substitution
60
x + y + z = 180
20 + 45 + z = 180
z = 115
The measures of the three angles of a triangle are in the
ratio 2:4:6. Find the measure of the smallest angle.
4x
2x
6x
2x + 4x + 6x = 180
12x = 180
x = 15
2x = 30
A
80
Bisectors BD and CD meet at D
LetABC = 2x andABC = 2y
D
y
x
x
y
B
C
In
ABC,
2x + 2y + 80 = 180
2x + 2y = 100
x + y = 50
In
EBC,
x + y + mE = 180
50 + mE = 180 (substitution)
mE = 130
, and the measure of  B is twice that of
Find the measure of each angle of the triangle.
1  150
o
Let
A
= x and  B = (2x)
According to theorem 51,
150 = x + 2x
150 = 3x
50 = x
o
o
1 is equal to B +A
A
B
B
= 50
1
C
o
= 100
BCA
A
o
= 30
o
A
Practice Problems
Find the measures of the numbered angles.
1.
86o
o
69
2
3
0
5
o
47
4
65o
1
95 o
71o
2
2.
1
3
25o
40o
3.
E
B
G
4.
H
D
F
D
E
16
Find: GH
A
5.
Three triangles are in the ratio
3:4:5.
Find the measure of the largest
angle.
70 o
C
Find: mB , mBDE ,
and mBED
R
6.
7.
2x+4
B
50o
D
4x+6
Q
8.
x+24
Find: mR
S
A
C
Find:
mD
Always, Sometimes, Never
a. The acute angles of a right triangle are complementary.
b. A triangle contains two obtuse angles.
o
c. If one angle of an isosceles triangle is 60 , it is
equilateral.
d. The supplement of one of the angles in a triangle is
equal in measure to the sum of the other two angles.
Answer Key
1. 1 = 47
2 = 40
3 = 93
4 = 40
5 = 140
2. 1 = 75
2 = 85
3 = 70
3. GH = 8
4.
mB = 20
mBDE = 90
mBED = 70
5. 75
6. 48
7. 115
8. a.
b.
c.
d.
A
N
A
A
Works Cited
"Exterior Angles of a Triangle." Regents Prep. 2008. 29
May 2008
<http://regentsprep.org/REgents/math/triang/LExtAn
g.htm>.
Rhoad, Richard, George Milauskas, and Robert Whipple.
Geometry for Enjoyment and Challenge. Boston:
McDougal Littell, 1991.
"Triangle." Apronus. 29 May 2008
<http://www.apronus.com/geometry/triangle.htm>.