Question

two angles are supplementary if the sum of their angle measures is equal to 180°. two angles are complementary if the sum of their angle measures is equal to 90°. if p is in the interior of ∠mnq, then m∠mnq = m∠mnp + m∠pnq. task 4: in the image, m∠abc = 156°. find m∠abd and m∠cbd. turn and talk: suppose ∠abc and ∠cbd are adjacent complementary angles and m∠abc=(3x + 11)° and m∠cbd=(6x - 2)°. what is the value of x? what are the measures of the angles? does the shared ray bisect the angle?

Explanation:

Step1: Set up equation for adjacent - complementary angles

Since $\angle ABC$ and $\angle CBD$ are adjacent complementary angles, $m\angle ABC + m\angle CBD=90^{\circ}$. Given $m\angle ABC=(3x + 11)^{\circ}$ and $m\angle CBD=(6x - 2)^{\circ}$, we have the equation $(3x + 11)+(6x - 2)=90$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $3x+6x+11 - 2=90$, which simplifies to $9x + 9=90$.

Step3: Solve for $x$

Subtract 9 from both sides: $9x=90 - 9=81$. Then divide both sides by 9, so $x = 9$.

Step4: Find the measures of the angles

For $m\angle ABC$, substitute $x = 9$ into the expression: $m\angle ABC=3x + 11=3\times9+11=27 + 11=38^{\circ}$.
For $m\angle CBD$, substitute $x = 9$ into the expression: $m\angle CBD=6x - 2=6\times9-2=54 - 2=52^{\circ}$.

Step5: Determine if the shared ray bisects the angle

Since $m\angle ABC
eq m\angle CBD$, the shared ray does not bisect the angle.

Answer:

$x = 9$
$m\angle ABC = 38^{\circ}$
$m\angle CBD = 52^{\circ}$
No, the shared ray does not bisect the angle.