Question

1. if m∠dec=(12x - 3)°, m∠bce=(7x - 26)°, and m∠dae = 72°, find each angle measure. m∠dec = m∠bce = m∠ade = m∠edb = m∠dbc =

Explanation:

Step1: Note parallel - side property

Since $\overline{AD}=\overline{DB}$ and $\overline{AE}=\overline{EC}$, by the mid - point theorem, $DE\parallel BC$. Then $\angle DEC$ and $\angle BCE$ are same - side interior angles, so $\angle DEC+\angle BCE = 180^{\circ}$.

Step2: Set up the equation

We know that $m\angle DEC=(12x - 3)^{\circ}$ and $m\angle BCE=(7x - 26)^{\circ}$. So, $(12x - 3)+(7x - 26)=180$.

Step3: Simplify the left - hand side of the equation

Combine like terms: $12x+7x-3 - 26 = 180$, which gives $19x-29 = 180$.

Step4: Solve for $x$

Add 29 to both sides: $19x=180 + 29=209$. Then divide both sides by 19: $x=\frac{209}{19}=11$.

Step5: Find the measure of $\angle DEC$

Substitute $x = 11$ into the formula for $\angle DEC$: $m\angle DEC=12x-3=12\times11 - 3=132 - 3=129^{\circ}$.

Step6: Find the measure of $\angle BCE$

Substitute $x = 11$ into the formula for $\angle BCE$: $m\angle BCE=7x-26=7\times11 - 26=77 - 26 = 51^{\circ}$.

Step7: Find the measure of $\angle ADE$

Since $DE\parallel BC$, $\angle ADE=\angle ABC$ (corresponding angles). And in $\triangle ADE$, we don't need to use the parallel - line relationship for this angle calculation. But since $\angle ADE$ and $\angle DEC$ are corresponding angles for the transversal $DE$ with respect to parallel lines $DE$ and $BC$, $\angle ADE=\angle ABC$. Also, $\angle ADE$ and $\angle BCE$ are corresponding angles, so $m\angle ADE = 51^{\circ}$.

Step8: Find the measure of $\angle EDB$

$\angle ADB$ is a straight - angle, so $\angle ADE+\angle EDB = 180^{\circ}$. Then $m\angle EDB=180 - m\angle ADE=180 - 51=129^{\circ}$.

Step9: Find the measure of $\angle DBC$

Since $DE\parallel BC$, $\angle DBC=\angle EDB$ (alternate interior angles), so $m\angle DBC = 129^{\circ}$.

Answer:

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