The sizes of four of the exterior angles of a decagon are in the ratio 1 : 2 : 2 : 3. The remaining exterior angles are each of size $36^\circ$. Find the size of the largest interior angle of the decagon. The area of a triangle ABC is 35 cm^2 and D is a point such that CD is parallel to AB. The ratio of CD : AB is 1 : 4. Calculate the area of the triangle ACD.

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.

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