Question

1. what is the measure of angle ( d_1^circ )?
2. what is the tangent ratio of angle ( c_2 )?

Explanation:

Step1: Recall rhombus angle properties

In a rhombus, consecutive angles are supplementary, so $\angle ADC + \angle BCD = 180^\circ$. Also, the diagonal bisects the angles. The given angle at $D$ (between diagonal and side) is $50^\circ$, so $\angle ADC = 2 \times 50^\circ = 100^\circ$.

Step2: Calculate $\angle d_1$

$\angle d_1$ is supplementary to $\angle ADC$:
$\angle d_1 = 180^\circ - 100^\circ = 80^\circ$

---

Step1: Find half-diagonals length

In a rhombus, diagonals bisect each other at right angles. Side length $s=5$ cm, half of diagonal $AC$ is $\frac{3.8}{2}=1.9$ cm. Let half of diagonal $BD$ be $x$. Use Pythagoras: $x^2 + 1.9^2 = 5^2$
$x^2 = 25 - 3.61 = 21.39$
$x = \sqrt{21.39} \approx 4.625$ cm

Step2: Identify sides for $\tan(c_2)$

$\angle c_2$ is in a right triangle with adjacent side $x \approx 4.625$ cm, opposite side $1.9$ cm.
$\tan(c_2) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.9}{4.625} = \frac{3.8}{9.25} = \frac{76}{185} \approx 0.4108$

Answer:

1. $\boldsymbol{80^\circ}$
2. $\boldsymbol{\frac{76}{185}}$ (or approximately $\boldsymbol{0.41}$)