Download Angle Relationships in Triangles

Angle
Relationships in
Triangles
Geometry (Holt 4-3)
K. Santos
Triangle Sum Theorem
(4-3-1)
The sum of the angle measures of a triangle is 180°.
m< A + m<B + m<C =180°.
A
B
C
Corollary
A corollary is a theorem whose proof follows directly
from another theorem.
“Mini-theorem”
For example—there are two corollaries to the Triangle
Sum Theorem
Corollary 1 (4-3-2)
The acute angles of a right triangle are
complementary.
E
m< D and m< E are complementary
m< D + m< E = 90°
D
F
Example
One of the acute angles in a right triangle measures
27°. What is the measure of the other acute angle?
27 + x = 90
x = 63°
Example
Find x, y and z
yx
54°
62°
Z + 90 =180
Z = 90
Z + 62 + x = 180
90 + 62 + x = 180
152 + x = 180
x = 28°
y + 54 + 90 = 180
y + 144 = 180
y = 36°
Corollary 2 (4-3-3)
The measure of each angle of an equiangular triangle
is 60°.
A
B
m< A = m< B = m< C = 60°
180
3
= 60°
C
Interior vs Exterior
2
Exterior
Interior
1
3
4
Exterior angle
<4
Interior angles
<1, < 2, < 3
Remote interior angles to <4
<1 and < 2
Exterior Angle Theorem (4-3-4)
The measure of an exterior angle of a triangle is equal
to the sum of the measures of its remote interior
angles.
2
1
3
4
Exterior angle = sum of remote interior angles
m< 4 = m< 1 + m< 2
Example 1 (exterior angle)
Find x.
53°
x
90 + 53 = x
143° = x
Example 2 (remote interior angle)
Find x.
x
110
x + 110 = 145
x = 35°
145
Example 3 (algebraic)
H
Find m< J and m< H.
F
m< H + m< J= m< FGH
4x + 18 + 5x + 2 = 128
9x + 20 = 128
9x = 108
x= 12
m< H = 4(12) + 18
m< J = 5(12) + 2
m< H = 66°
m< J = 62°
(check 66 + 62 = 128)
128°
G
4x + 18
5x + 2
J
Third Angles Theorem (4-3-5)
If two angles of one triangle are congruent to two
angles of another triangle, then the third pair of
angles are congruent.
L
R
N
If: < L ≅ <R
<M≅<S
Then: < N ≅ <T
M
S
T
Example—Third angles
Find m< C and m< F.
m< C = 4y - 8
m< F = 2y +10
<C ≅ <F
4y – 8 = 2y +10
2y – 8 = 10
2y = 18
y=9
m< C = 4y – 8
m< C = 4(9) - 8
m< C = 28°
F
A
B
C
m< F = 2y +10
m< F = 2(9) +10
m< F = 28°
D
E