Question

what is the value of k? there is a diagram with points l, z, y, x, m, n. lz is a ray, zy and zx are sides of a triangle with angles (4k + 5)° at z, (6k + 10)° at x, and angle at y is supplementary to 115° (since ym is a ray).

Explanation:

Step1: Use exterior angle theorem

The exterior angle $\angle MYZ = 115^\circ$ equals the sum of the two remote interior angles $\angle YZX$ and $\angle YXZ$.
$$115 = (4k + 5) + (6k + 10)$$

Step2: Simplify the right-hand side

Combine like terms.
$$115 = 10k + 15$$

Step3: Isolate the term with $k$

Subtract 15 from both sides.
$$115 - 15 = 10k$$
$$100 = 10k$$

Step4: Solve for $k$

Divide both sides by 10.
$$k = \frac{100}{10}$$

Answer:

$10$