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Algebra III
Lesson 12
Angles and Diagonals in Polygons Proofs of the Chord-Tangent Theorem
Angles & Diagonals in Polygons
At every corner of a polygon there is an inside and an outside angle.
O
I
-The inside angle can be greater
than 180° if the polygon is concave.
O
-At any given corner the inside and
outside angles add to 180°.
I
Interior angles are the same as inside angles in
convex polygons.
I
E
An exterior angle is the supplement to
a given interior angle.
E
A diagonal connects two corners not already
connected by a side.
Finding the Sum of the Interior Angles for any Polygon
1st – a triangle
Σ(Interior Angles) = 180°
2nd – a rectangle
- Put a diagonal in to create two triangles.
- Each triangle has 180°.
Σ(Interior Angles) = 2(180°) = 360°
3rd – a pentagon
- Draw all possible diagonals from one corner.
- This makes three triangles.
Σ(Interior Angles) = 3(180°) = 540°
Finally, a n-gon (a polygon with n sides)
- Draw all possible diagonals from one corner.
How many diagonals is this? Find the pattern:
3 sides → 0 diagonals
4 sides → 1 diagonal
5 sides → 2 diagonals
n sides → (n – 3) diagonals
- How many triangles are made? Find the pattern:
3 sides → 1 triangle
4 sides → 2 triangles
5 sides → 3 triangles
n sides → (n – 2) triangles
Σ(Interior Angles) = (n – 2)180°
For a convex polygon, the sum of the exterior angles always
equals 360°. Σ(Exterior Angles) = 360°
Finding the Total Number of Diagonals in a Polygon
1) In the polygon, pick a corner and draw all possible diagonals.
2) Go to the next corner, and again draw all possible
diagonals. Don’t draw a diagonal if it is already there from a
previous corner.
3) Repeat this procedure until all corners have been done.
4) Count the number of diagonals.
A Formula
# of diagonals =
n (n - 3)
2
n = the number of sides
Cross check with a rectangle
Example 12.1
Find the sum of the measures of (a) the interior angles and (b) the
exterior angles of a seven sided convex polygon.
a) Σ(Interior Angles) = (n – 2)180° = (7 – 2)180°
= (5)180° = 900°
b) Σ(Exterior Angles) = 360°
Proof of the Chord-Tangent Theorem
Refers to the angle made by the intersection of a
chord and a tangent.
Postulate – An inscribed angle equals one-half
the measure of its arc.
50°
25°
25°
Three Cases to the Chord-Tangent Theorem
B
B
B
X
O
X
X
O
A
Case 1
C
O
A
Case 2
C
A
Case 3
C
Example 12.2
B
Prove Case 1 of the Chord-Tangent Theorem
X
O
1) A tangent and a radii are perpendicular
by postulate.
2) So, m∠OAC = 90°, since OA is a radii.
3) Since BOA is a diameter, mBXA = 180°
4) ∴, m∠BAC =
1
mBXA
2
A
C
Example 12.3
Y
Prove Case 2 of the Chord-Tangent Theorem
B
X
O
1) Draw a diameter to A
1
mBY
2
1
3) m∠YAC = mYXA
2
2) m∠BAY =
A
By Postulate
By Case 1
1
1
Adding. 4) m∠BAY + m∠YAC = mBY + mYXA
2
2
1
5) m∠BAC = mBXA
2
C
Example 12.4
Y
Prove Case 3 of the Chord-Tangent Theorem
B
X
O
1) Put in diameter YOA
1
m∠YXA
2
1
3) m∠YAB = mYB
2
2) m∠YAC =
Subtract. 4) m∠YAC − m∠YAB =
A
By Case 1
By Postulate
1
(mYXA − mYB)
2
1
5) ∴ m∠BAC = mBXA
2
Practice
a) Find the sum of the measures of the interior angles and the sum of
the measures of the exterior angles of a six-sided convex polygon.
1) Σ(Interior Angles) = (n – 2)180° = (6 – 2)180°
= (4)180° = 720°
2) Σ(Exterior Angles) = 360°
b) How many diagonals can be drawn in a nine-sided convex polygon?
# of diagonals =
n (n - 3)
2
=
9(9 - 3)
2
=
9⋅6
2
= 27