Given two triangles, $\mathrm{ABC}$ and $\mathrm{XYZ}$, for which pair of triangles is $cos\left(b\right)$ equal to $cos\left(z\right)$?

• A. When triangle $\mathrm{ABC}$ and triangle $\mathrm{XYZ}$ are congruent with $b$ corresponding to $z$
• B. When both triangles are right triangles with $b$ and $z$ as right angles
• C. When triangle $\mathrm{ABC}$ is isosceles and triangle $\mathrm{XYZ}$ is scalene
• D. When triangle $\mathrm{ABC}$ and triangle $\mathrm{XYZ}$ are similar but $b$ does not correspond to $z$

Answer: (A)

Verified step by step guidance

1. Recall that the cosine of an angle in a triangle depends on the measure of that angle. Therefore, for \(\cos(b)\) to equal \(\cos(z)\), the angles \(b\) and \(z\) must be equal or have the same cosine value.
2. Understand that in congruent triangles, corresponding angles are equal. So if triangles \(ABC\) and \(XYZ\) are congruent and angle \(b\) corresponds to angle \(z\), then \(\cos(b) = \cos(z)\) naturally holds.
3. Consider the case of right triangles where both \(b\) and \(z\) are right angles (90 degrees). Since \(\cos(90^\circ) = 0\), \(\cos(b)\) will equal \(\cos(z)\) regardless of other triangle properties.
4. Note that if triangles are similar but \(b\) does not correspond to \(z\), then even though the triangles have the same shape, the angles \(b\) and \(z\) may differ, so \(\cos(b)\) may not equal \(\cos(z)\).
5. Recognize that if one triangle is isosceles and the other is scalene, their angles differ in measure, so \(\cos(b)\) will generally not equal \(\cos(z)\) unless by coincidence, which is not guaranteed.