Question

find the value of c in rhombus qrst.
angle at s: 4c - 18°
angle at q: 5c - 53°
c = \square°

Explanation:

Step1: Recall rhombus adjacent angle property

Adjacent angles in a rhombus are supplementary (sum to $180^\circ$).

Step2: Set up the equation

$$(5c - 53) + (4c - 18) = 180$$

Step3: Simplify left side of equation

Combine like terms:
$$9c - 71 = 180$$

Step4: Solve for $c$

Add 71 to both sides, then divide by 9:
$$9c = 180 + 71 = 251$$
$$c = \frac{251}{9} \approx 27.89$$
Correction: Recheck simplification

Step2 (Revised): Set up correct equation

$$(5c - 53) + (4c - 18) = 180$$

Step3 (Revised): Combine like terms

$$9c - 71 = 180$$

Step4 (Revised): Isolate $9c$

$$9c = 180 + 71 = 251$$

Step5 (Revised): Solve for $c$

$$c = \frac{251}{9} \approx 27.89$$
Final check: Verify angle sum
$5(\frac{251}{9}) -53 = \frac{1255}{9} - \frac{477}{9} = \frac{778}{9} \approx 86.44^\circ$
$4(\frac{251}{9}) -18 = \frac{1004}{9} - \frac{162}{9} = \frac{842}{9} \approx 93.56^\circ$
$\frac{778}{9} + \frac{842}{9} = \frac{1620}{9} = 180^\circ$, which is correct.

Answer:

$c = \frac{251}{9}$ or approximately $27.9^\circ$