Question

rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if m∠abd=(3r + 5)° and m∠dbc=(5r - 27)°, find m∠abd and m∠dbc.
m∠abd=

m∠dbc=

Explanation:

Step1: Use perpendicular - angle property

Since $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are perpendicular, $\angle ABC = 90^{\circ}$. And $\angle ABC=\angle ABD+\angle DBC$. So, $(3r + 5)+(5r-27)=90$.

Step2: Solve the equation for $r$

Combine like - terms: $3r+5r + 5-27=90$, which simplifies to $8r-22 = 90$. Add 22 to both sides: $8r=90 + 22=112$. Divide both sides by 8: $r=\frac{112}{8}=14$.

Step3: Find $m\angle ABD$

Substitute $r = 14$ into the expression for $\angle ABD$: $m\angle ABD=(3r + 5)^{\circ}=(3\times14 + 5)^{\circ}=(42 + 5)^{\circ}=47^{\circ}$.

Step4: Find $m\angle DBC$

Substitute $r = 14$ into the expression for $\angle DBC$: $m\angle DBC=(5r-27)^{\circ}=(5\times14-27)^{\circ}=(70 - 27)^{\circ}=43^{\circ}$.

Answer:

$m\angle ABD = 47^{\circ}$
$m\angle DBC = 43^{\circ}$