Download A B C

5.7: Sums of Angle Measures in Polygons
Brinkman
SKILLS TO AQUIRE
 Use the Triangle-, Quadrilateral-, and Polygon-Sum Theorems to determine angle measures.
 Know history and impact postulates had on parallel lines and the development of geometry.
In the picture, connect points A, B, and C on
the cheese head at the right. Fill in the blanks
below with the correct angle measurement.
A
B
𝑚∠𝐵𝐴𝐶 98
𝑚∠𝐶𝐵𝐴 35
𝑚∠𝐵𝐶𝐴 47
Find 𝑚∠𝐵𝐴𝐶 + 𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐵𝐶𝐴 = 𝟏𝟖𝟎! Sweet…
C
Definition: Triangle – Sum Theorem
The sum of the measures of the angles of a triangle is 180o.
Prove this is true! Okay…
Given: △ 𝑨𝑩𝑪
Prove: 𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180
a.) Draw line through A that is parallel to 𝐵𝐶 .
(Put protractor on 𝐴𝐵 and point A,
measure 90 degrees)
b.) On line just drawn, put points P and Q
on opposite sides of A, and where Q and C
are on the same side of line AB.
c.) What do lines AB and AC act as now? AB is the perp. Bisector of BC and AC is a
transversal
d.) Therefore 𝑚∠𝑃𝐴𝐵 = 𝒎∠𝟐
𝑎𝑛𝑑 𝑚∠𝑄𝐴𝐶 = 𝒎∠𝟑
e.) We know that the angle measure of a line is 180o
f.) Therefore 𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180!
Practice!
Laura Bora was working on her triangle sum homework and noticed that in triangle ABC, the angles
are in the extended ration 1:2:3. This means for some real number x, the angles have measures 1x,
2x, and 3x. Find their measures!
2x B
1x + 2x + 3x = 180
6x = 180
x = 30
𝒎∠𝑨= 30
1x
3x
A
𝒎∠𝑩 = 60
C
𝒎∠𝑪 = 90
When the Grinch wasn’t stealing Christmas, he was trying to figure out the sum of a quadrilateral. He
was stumped, because math wasn’t really his thing. But then he had an “Aha” moment! He noticed
that if he took a Quadrilateral, he could turn this into 2 TRIANGLES! Oh happy day!
He knew the sum of a triangle was 180o
Since there are 2 TRIANGLES in the quadrilateral,
he multiplied 2 x 180 and got 360o.
Definition: Quadrilateral – Sum Theorem
The sum of the measures of the angles of a quadrilateral is 360o.
Notice that there are two triangles when there are 4 sides and 4 angles.
Sums of Angles in Convex Polygons! Draw the number of triangles in each polygon…
There are 3 triangles
in a 5-gon.
There are 4 triangles
in a 6-gon.
There are 5 triangles
in a 7-gon.
Angle measure: 540o
Angle measure: 720 o
Angle measure: 900 o
Definition: Polygon – Sum Theorem
The sum of the measures of the angles of a convex n-gon is 180(n – 2)o.