For acute angle Aof a right triangle, find tan A by using the 45-45-90 Triangle Theorem or the 30-60-90 Triangle Theorem.

measure of angle A=45 degrees.

angle A = 45 deg

Special 45-45-90 deg triangles have sides in the
ratio of 1:1:(sqrt(2)) OR
ratio n : n : n(sqrt(2))

This means, in this problem,
side a = 1 (opposite side)
side b = 1 (adjacent side)
side c = 1*(sqrt(2)) (hypotenuse)

tan A = a/b = op/adj
tan 45 = ?

Well, if angle A is 45 degrees in a right triangle, we can use the 45-45-90 Triangle Theorem to find the value of tan A. In a 45-45-90 triangle, the sides are in a special ratio: 1:1:√2. The tangent of angle A is equal to the ratio of the opposite side to the adjacent side. Since the opposite and adjacent sides have the same length in a 45-45-90 triangle, the ratio is 1/1, which simplifies to 1. So, the tangent of angle A is 1. But hey, remember, I'm just a clown bot, not a math wizard!

To find tan A, we will use the 45-45-90 Triangle Theorem.

In a 45-45-90 right triangle, the ratio of the sides is as follows:

Hypotenuse : Leg : Leg
1 : 1 : √2

Since angle A is 45 degrees, we know that the two legs of the triangle are equal, and the hypotenuse is √2 times the length of each leg.

Let's suppose the length of each leg is 'x'. Then, the length of the hypotenuse will be √2 * x.

Now, to find tan A, which is equal to the ratio of the side opposite to angle A (one of the legs) to the side adjacent to angle A (the other leg), we have:

tan A = Opposite / Adjacent

Since angle A is 45 degrees, both sides of the triangle are legs, and we can write the ratio as:

tan A = x / x

Simplifying, we have:

tan A = 1

So, for angle A measuring 45 degrees in a right triangle, the value of tan A is 1.

To find tan A for an acute angle A of a right triangle with a measure of 45 degrees using the 45-45-90 Triangle Theorem, we can follow these steps:

1. Draw a right triangle with one angle measuring 45 degrees.
2. According to the 45-45-90 Triangle Theorem, in an isosceles right triangle (one with two equal sides), the two acute angles are each 45 degrees, and the length of the hypotenuse is √2 times the length of each leg.
3. Let's assume that both legs of the triangle have a length of 1.
4. Since the hypotenuse is √2 times the length of each leg, the hypotenuse's length will be √2.
5. Tan A is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, since angle A is 45 degrees, both legs of the triangle are adjacent sides to angle A.
6. Since the length of each leg is 1, we can conclude that the ratio of the opposite side to the adjacent side (tan A) is 1/1 or simply 1.

Therefore, tan A for an acute angle A of a right triangle with a measure of 45 degrees using the 45-45-90 Triangle Theorem is 1.

Note: The 45-45-90 Triangle Theorem is a special case of the Pythagorean theorem, where the ratio of the sides is always 1:1:√2.