一个十边形中，其中四个外角的大小比例为 1 : 2 : 2 : 3。其余的外角每个大小均为 $36^\circ$。求该十边形的最大内角的大小。 三角形 ABC 的面积为 35 cm^2，D 是一个使得 CD 平行于 AB 的点。CD : AB 的比值为 1 : 4。计算三角形 ACD 的面积。

1. Find the total of the four exterior angles in the given ratio.

A decagon has $10$ exterior angles. Since $6$ of them are each $36^\circ$,
$\text{sum of remaining 4 exterior angles}=360^\circ-6(36^\circ)=144^\circ.$

2. Use the ratio to find the smallest exterior angle, then the largest interior angle.

The ratio is $1:2:2:3$, so the total number of parts is
$1+2+2+3=8.$
Each part is
$144^\circ \div 8=18^\circ.$
The exterior angles are $18^\circ,36^\circ,36^\circ,54^\circ$.

The largest interior angle corresponds to the smallest exterior angle:
$180^\circ-18^\circ=162^\circ.$

1. Relate the areas of the two triangles using their parallel bases.

Since $CD \parallel AB$, the perpendicular height from $A$ to $CD$ is the same as the perpendicular height from $C$ to $AB$.

Therefore,
$\frac{\text{Area of }\triangle ACD}{\text{Area of }\triangle ABC}=\frac{CD}{AB}=\frac{1}{4}.$

2. Calculate the area of triangle ACD.

$\text{Area of }\triangle ACD=\frac{1}{4}\times 35=8.75$