If triangle ABC and triangle XYZ,if angle A and angle X are acute angles such thay cosa=cosx then show that angle A=angle X?
since cosine has a period of 2pi, either A=X or |A-X| = k*2pi
Since both A and X are less than pi/2, A=X
cosA=cosx
To prove that angle A is equal to angle X, given that cos(A) = cos(X) and both angles are acute, we can use the fact that the cosine function is a decreasing function on the interval (0, π).
1. Start by assuming that A ≠ X. This means that the two angles are not equal.
2. As both angles are acute, we can write 0 < A, X < π/2.
3. Since the cosine function is decreasing on (0, π), if two angles have the same cosine value, their measures must be equal. Therefore, if cos(A) = cos(X), then A = X.
4. This contradicts our assumption that A ≠ X, hence our assumption must be incorrect.
5. Therefore, we conclude that A = X.
Hence, if cos(A) = cos(X) and both A and X are acute angles, it follows that angle A is equal to angle X.
To prove that angle A = angle X, we need to use the fact that cos A = cos X, and that angles A and X are acute angles.
Here's the step-by-step proof:
1. Assume that triangle ABC and triangle XYZ are such that angle A and angle X are acute angles, and cos A = cos X. We'll prove that angle A = angle X.
2. Since cos A = cos X, we know that the measures of angle A and angle X are equal. However, this alone does not prove that the angles themselves are equal.
3. Let's assume that angle A ≠ angle X. Without loss of generality, let's assume that angle A is greater than angle X (A > X).
4. Since angles A and X are acute, it means that both angles are less than 90 degrees. Therefore, angle A > angle X implies that angle A is obtuse (greater than 90 degrees), which contradicts the fact that angle A is acute.
5. Since our assumption in step 3 leads to a contradiction, we conclude that our initial assumption (angle A ≠ angle X) is false.
6. Therefore, we can conclude that angle A must be equal to angle X.
By using the fact that cos A = cos X and considering the nature of acute angles, we were able to prove that angle A = angle X.