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Answer :
Final answer:
The Angle Addition Postulate in geometry combines with analytical methods of vector addition and subtraction, using the Pythagorean Theorem and trigonometric identities. Given two vectors, their sum can be computed along with the angle they form. This concept enables to find the resultant vector even if only angles of the two vectors are known.
Explanation:
The Angle Addition Postulate is a concept in geometry which states that if point B lies in the interior of angle AOC, then the measure of angle AOB + the measure of angle BOC is equal to the measure of angle AOC. To apply this in examples involving vectors, we can consider the analytical methods of vector addition and subtraction. This involves using the Pythagorean Theorem and trigonometric identities to determine the magnitude and direction of a resultant vector. For instance, given a vector A, we might want to find what two perpendicular vectors, Ax and Ay, add to produce it, making use of the Pythagorean theorem (x² + y² = h²) for finding the hypotenuse in this instance.
In a more practical scenario, considering two vectors A and B, their resulting vector R can be found using the formula R = √R² + R². The angle is determined as Θ = tan⁻¹ (Ry/Rx).
Finally, it's noteworthy to know that if only the angles of two vectors are known, one can find the angle of their resultant addition vector, which brings us back to the fundamental concept of Angle Addition Postulate.
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The measure of the angles are :
- 25°
- 129°
- x = 107° ; y = z = 73°
- 167°
- 52°
- Angle 1 = 109° ; Angle 2 = 71°
To find x :
A right angle = 90°
65 + x = 90
x = 90° - 65°
x = 25°
(2) :
Vertically opposite angles are equal :
Sum of angles at a point = 360°
[(51 × 2) + (2x)] = 360°
106° + 2x = 360°
2x = 360° - 102°
2x = 258°
x = 254° ÷ 2
x = 129°
(3.)
Verticaly opposite angles are equal :
107° = x°
y° = z°
(2y + (2 × 107)) = 360°
2y + 214° = 360°
2y = 360° - 214°
2y = 146°
y = 146° ÷ 2
y = 73°
y = z = 73°
4.)
Measure of supplementary angles = 180°
Let the measure of it's supplement = x
x + 13° = 180°
x = 180° - 13°
x = 167°
5.)
Complementary angles add up to 90°
Let the measure of it's complement = x
x + 38° = 90°
x = 90° - 38°
x = 52°
6.)
Sum of linear pair of angles = 180°
(5x + 9) + (3x + 11) = 180°
5x + 3x + 9 + 11 = 180
8x + 20 = 180
8x = 180 - 20
8x = 160
x = 160/8
x = 20°
Angle 1 = 5(20) + 9 = 109°
Angle 2 = 3(20) + 11 = 71°
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