The SAS (Side-Angle-Side) similarity theorem is a geometric theorem that states that if two triangles have two pairs of corresponding sides that are in proportion and the included angles are congruent, then the triangles are similar. Using this theorem, we can determine what value of x will make triangle ONM similar to triangle SRQ.

Let’s begin by looking at the two triangles in question. Triangle ONM has sides of length 5x, 8x, and 10x, while triangle SRQ has sides of length 8, 10, and x. To determine what value of x will make these two triangles similar by the SAS similarity theorem, we need to find a pair of corresponding sides that are in proportion and the included angle that is congruent.

First, we can see that the side of length 8 in triangle SRQ corresponds to the side of length 10x in triangle ONM. We can set up a proportion between these two sides:

8/10x = a/b

where a and b are the corresponding sides in triangle SRQ and triangle ONM that are in proportion. Solving for a, we get:

a = 4x/5

Next, we can see that the side of length 10 in triangle SRQ corresponds to the side of length 8x in triangle ONM. We can set up another proportion between these two sides:

10/8x = c/d

where c and d are the corresponding sides in triangle SRQ and triangle ONM that are in proportion. Solving for c, we get:

c = 5x/4

Finally, we need to find the included angle that is congruent in both triangles. This angle is the angle between the sides of length 10x and 8x in triangle ONM, which is opposite to the side of length 5x. To find this angle, we can use the Law of Cosines:

(5x)^2 = (8x)^2 + (10x)^2 – 2(8x)(10x)cosθ

where θ is the angle we are trying to find. Simplifying this equation, we get:

25x^2 = 164x^2 – 160x^2cosθ

or

cosθ = (139x^2)/(160x^2) = 139/160

Now that we have found the biographypark corresponding sides in proportion and the included angle that is congruent, we can use the SAS similarity theorem to find what value of x will make triangle ONM similar to triangle SRQ. According to the SAS similarity theorem, the two triangles are similar if and only if the corresponding sides are in proportion and the included angle is congruent. Therefore, we need to check if the ratio of the corresponding sides is equal to the ratio of the included angles:

a/c = (4x/5)/(5x/4) = 16/25

and

sinθ = √(1 – cos^2θ) = √(1 – (139/160)^2) ≈ 0.467

Since the ratios are equal, we have found that triangle ONM is similar to triangle SRQ if x satisfies the equation:

16/25 = 0.467

Solving for x, we get:

x ≈ 9.38

Therefore, if x is approximately equal to 9.38, then triangle ONM is similar to triangle SRQ by the SAS similarity theorem.