Question

1. if (mangle dec=(12x - 3)^{circ}), (mangle bce=(7x - 26)^{circ}), and (mangle dae = 72^{circ}), find each angle measure.

(mangle dec=)
(mangle bce=)
(mangle ade=)
(mangle edb=)
(mangle dbc=)

Explanation:

Step1: Identify similar - triangles

Since $\overline{AD}=\overline{DB}$ and $\overline{AE}=\overline{EC}$, by the mid - point theorem, $DE\parallel BC$. So, $\triangle ADE\sim\triangle ABC$ and $\angle ADE=\angle ABC$, $\angle AED=\angle ACB$, $\angle DAE=\angle BAC$. Also, $\angle DEC+\angle BCE = 180^{\circ}$ (co - interior angles as $DE\parallel BC$).

Step2: Use the angle - relationship to find $x$

We know that $\angle DEC=(12x - 3)^{\circ}$ and $\angle BCE=(7x - 26)^{\circ}$. Since $\angle DEC+\angle BCE = 180^{\circ}$, we have the equation $(12x - 3)+(7x - 26)=180$.
Combining like terms: $12x+7x-3 - 26 = 180$, which simplifies to $19x-29 = 180$.
Adding 29 to both sides: $19x=180 + 29=209$.
Dividing both sides by 19: $x = 11$.

Step3: Find the angle measures

• $\angle DEC=(12x - 3)^{\circ}$. Substitute $x = 11$: $\angle DEC=12\times11-3=132 - 3=129^{\circ}$.
• $\angle BCE=(7x - 26)^{\circ}$. Substitute $x = 11$: $\angle BCE=7\times11-26=77 - 26 = 51^{\circ}$.
• Since $\triangle ADE\sim\triangle ABC$ and $\angle DAE = 72^{\circ}$, and $\angle ADE$ and $\angle ABC$ are corresponding angles, and $\angle AED$ and $\angle ACB$ are corresponding angles. In $\triangle ADE$, $\angle ADE+\angle AED+\angle DAE=180^{\circ}$. Since $\angle AED=\angle BCE = 51^{\circ}$ (corresponding angles for $DE\parallel BC$), then $\angle ADE=180^{\circ}-\angle DAE-\angle AED=180 - 72-51 = 57^{\circ}$.
• $\angle EDB$ and $\angle DBC$ are alternate - interior angles as $DE\parallel BC$. Let's first find $\angle ABC$. In $\triangle ABC$, $\angle BAC = 72^{\circ}$ and $\angle ACB = 51^{\circ}$, so $\angle ABC=180-(72 + 51)=57^{\circ}$. Then $\angle EDB=\angle DBC$. And $\angle DBC=\angle ABC-\angle ABD$. Since $\triangle ADE\sim\triangle ABC$, and $\angle ADE = 57^{\circ}$, $\angle EDB = 57^{\circ}$, $\angle DBC = 57^{\circ}$.

Answer:

$m\angle DEC = 129^{\circ}$
$m\angle BCE = 51^{\circ}$
$m\angle ADE = 57^{\circ}$
$m\angle EDB = 57^{\circ}$
$m\angle DBC = 57^{\circ}$