Question

part 1
find the sum of the measures of the interior angles of each convex polygon.

1. decagon
2. 15 - gon

use \\(\square abcd\\) to find each measure.

1. \\(m\angle adc\\)
2. \\(ad\\)
3. \\(ab\\)
4. \\(m\angle bcd\\)

algebra quadrilateral efgh is a rectangle.

1. if \\(m\angle feg = 57\\), find \\(m\angle geh\\).
2. if \\(m\angle hge = 13\\), find \\(m\angle fge\\).
3. if \\(fk = 32\\) feet, find \\(eg\\).
4. find \\(m\angle hef + m\angle efg\\).
5. if \\(ef = 4x - 6\\) and \\(hg = x + 3\\), find \\(ef\\).

Explanation:

Step1: Recall polygon angle sum formula

Sum of interior angles = $(n-2)\times180^\circ$, where $n$ = number of sides

Step2: Calculate for decagon ($n=10$)

$$\begin{align} (10-2)\times180^\circ&=8\times180^\circ\\ &=1440^\circ \end{align}$$

Step3: Calculate for 15-gon ($n=15$)

$$\begin{align} (15-2)\times180^\circ&=13\times180^\circ\\ &=2340^\circ \end{align}$$

Step4: Find $m\angle ADC$ (parallelogram consecutive angles supplementary)

$$\begin{align} m\angle ADC&=180^\circ - 115^\circ\\ &=65^\circ \end{align}$$

Step5: Find $AD$ (parallelogram opposite sides equal)

$AD = BC = 18$

Step6: Find $AB$ (parallelogram opposite sides equal)

$AB = DC = 12$

Step7: Find $m\angle BCD$ (parallelogram opposite angles equal)

$m\angle BCD = m\angle BAD = 115^\circ$

Step8: Find $m\angle GEH$ (rectangle corner angle = $90^\circ$)

$$\begin{align} m\angle GEH&=90^\circ - 57^\circ\\ &=33^\circ \end{align}$$

Step9: Find $m\angle FGE$ (rectangle alternate interior angles equal)

$m\angle FGE = m\angle HGE = 13^\circ$

Step10: Find $EG$ (rectangle diagonals bisect each other, so $EG=2\times FK$)

$$\begin{align} EG&=2\times32\\ &=64 \text{ feet} \end{align}$$

Step11: Find $m\angle HEF + m\angle EFG$ (rectangle interior angles are $90^\circ$ each)

$$\begin{align} m\angle HEF + m\angle EFG&=90^\circ + 90^\circ\\ &=180^\circ \end{align}$$

Step12: Solve for $x$ then find $EF$ (rectangle opposite sides equal: $EF=HG$)

$$\begin{align} 4x-6&=x+3\\ 4x-x&=3+6\\ 3x&=9\\ x&=3 \end{align}$$

Substitute $x=3$:

$$\begin{align} EF&=4(3)-6\\ &=12-6\\ &=6 \end{align}$$

Answer:

1. $1440^\circ$
2. $2340^\circ$
3. $65^\circ$
4. $18$
5. $12$
6. $115^\circ$
7. $33^\circ$
8. $13^\circ$
9. $64$ feet
10. $180^\circ$
11. $6$