Ninth Grade Mathematics: Exterior Angle Theorem & Applications
When any side of a triangle is extended beyond a vertex, it creates an exterior angle that lies outside the boundary of the polygon. The Exterior Angle Theorem reveals a fundamental relationship linking this outer angle directly to the interior angles on the opposite side of the triangle.
Definition of an Exterior Angle
An exterior angle of a triangle is formed by extending one side of the triangle past a vertex and measuring the angle between this extended line and the adjacent side.
A
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X B C Y
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v Exterior Angle ACD
At any vertex, extending a side creates an exterior angle that forms a linear pair with the adjacent interior angle. Because a triangle has three vertices and two sides meet at each vertex, there are six possible exterior angles (two congruent vertical pairs at each of the three vertices).
Statement and Proof of the Exterior Angle Theorem
The interior angles that do not touch the exterior angle are called the interior opposite angles (or remote interior angles).
The Exterior Angle Theorem:
If a side of a triangle is produced, then the exterior angle so formed
is equal to the sum of the two interior opposite angles.
In Triangle ABC (side BC produced to D):
Exterior Angle ACD = Interior Angle A + Interior Angle B
A
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*-------------*-------------> D
B C
Exterior Angle ACD
Formal Deductive Proof
We prove this theorem by combining the Angle Sum Property with the Linear Pair Axiom:
Given: Triangle ABC with side BC extended to point D, forming exterior angle ACD.
To Prove: Angle ACD = Angle A + Angle B.
Proof:
Step 1: In Triangle ABC, by the Angle Sum Property:
Angle A + Angle B + Angle ACB = 180 degrees (Equation 1)
Step 2: Line BCD is a straight line segment.
The adjacent angles Angle ACB and Angle ACD form a linear pair:
Angle ACB + Angle ACD = 180 degrees (Equation 2)
Step 3: Equate Equation 1 and Equation 2 since both equal 180 degrees:
Angle ACB + Angle ACD = Angle A + Angle B + Angle ACB
Step 4: Subtract Angle ACB from both sides of the equation:
Angle ACD = Angle A + Angle B.
The proof is complete: the measure of an exterior angle is always equal to the sum of its two remote interior angles.
Corollaries and the Polygon Exterior Angle Sum
Several powerful geometric principles flow directly from this theorem:
Corollary 1: An exterior angle of a triangle is strictly greater than
either of its interior opposite angles:
Angle ACD > Angle A and Angle ACD > Angle B.
Corollary 2: Sum of Exterior Angles of a Triangle:
If one exterior angle is taken at each vertex, their sum is always 360 degrees:
Ext Angle 1 + Ext Angle 2 + Ext Angle 3 = 360 degrees.
To see why the exterior sum is 360 degrees, notice that each exterior angle and its interior angle form a linear pair (180 degrees). Summing all three pairs yields 3 * 180 = 540 degrees. Subtracting the interior angle sum (180 degrees) leaves exactly 540 - 180 = 360 degrees.
Chapter Exercises
Exercise 1: An exterior angle of a triangle measures 110 degrees, and one of its interior opposite angles measures 45 degrees. Find the measure of the other interior opposite angle, as well as the adjacent interior angle.
Exercise 2: The exterior angle of a triangle is 130 degrees, and the two interior opposite angles are in the ratio 2 : 3. Find the measures of all three interior angles of the triangle.
Exercise 3: Side BC of triangle ABC is produced to D. If the angle bisector of Angle A and the angle bisector of exterior angle ACD intersect at point E, prove that Angle E = (1/2) * Angle B.
Solutions and Explanations
Solution 1: Let the exterior angle be 110 degrees, and let one interior opposite angle be 45 degrees. By the Exterior Angle Theorem, Exterior Angle = Interior Opposite Angle 1 + Interior Opposite Angle 2: 110 = 45 + Angle 2. Subtracting 45 from both sides gives Angle 2 = 110 - 45 = 65 degrees. The adjacent interior angle forms a linear pair with the exterior angle: Adjacent Interior Angle = 180 - 110 = 70 degrees. Verification: the three interior angles sum to 45 + 65 + 70 = 180 degrees.
Solution 2: Let the two interior opposite angles be represented by 2x and 3x degrees. By the Exterior Angle Theorem, their sum equals the exterior angle: 2x + 3x = 130. Combine like terms: 5x = 130. Divide by 5: x = 26. The two interior opposite angles measure 2(26) = 52 degrees and 3*(26) = 78 degrees. The third interior angle forms a linear pair with the 130-degree exterior angle: 180 - 130 = 50 degrees. The three interior angles of the triangle are 52 deg, 78 deg, and 50 deg. Verification: 52 + 78 + 50 = 180 degrees.
Solution 3: In triangle ABC, by the Exterior Angle Theorem, exterior Angle ACD = Angle A + Angle B. Since ray CE bisects Angle ACD, Angle ECD = (1/2) * Angle ACD = (1/2) * (Angle A + Angle B) = (1/2)Angle A + (1/2)Angle B (Equation 1). Now consider triangle AEC with side AC produced to D. Angle ECD is an exterior angle to triangle AEC, so by the Exterior Angle Theorem: Angle ECD = Angle E + Angle EAC. Because ray AE bisects Angle A, Angle EAC = (1/2) * Angle A. Therefore, Angle ECD = Angle E + (1/2)Angle A (Equation 2). Equating Equation 1 and Equation 2: Angle E + (1/2)Angle A = (1/2)Angle A + (1/2)Angle B. Subtracting (1/2)*Angle A from both sides yields Angle E = (1/2) * Angle B.