Question

angles cad and bad form a linear pair, shown below. if the $m\angle cad=(3x + 46)^{circ}$ and $m\angle bad=(4x + 15)^{circ}$, then the value of $x$ is, the $m\angle cad =$ and the $m\angle bad =$.

Explanation:

Step1: Recall linear - pair property

Since $\angle CAD$ and $\angle BAD$ form a linear pair, $m\angle CAD + m\angle BAD=180^{\circ}$. So, $(3x + 46)+(4x + 15)=180$.

Step2: Combine like - terms

Combine the $x$ terms and the constant terms: $3x+4x+46 + 15=180$, which simplifies to $7x+61 = 180$.

Step3: Isolate the variable term

Subtract 61 from both sides of the equation: $7x=180 - 61$, so $7x=119$.

Step4: Solve for $x$

Divide both sides by 7: $x=\frac{119}{7}=17$.

Step5: Find $m\angle CAD$

Substitute $x = 17$ into the expression for $m\angle CAD$: $m\angle CAD=3x + 46=3\times17+46=51 + 46=97^{\circ}$.

Step6: Find $m\angle BAD$

Substitute $x = 17$ into the expression for $m\angle BAD$: $m\angle BAD=4x + 15=4\times17+15=68 + 15=83^{\circ}$.

Answer:

$x = 17$, $m\angle CAD=97^{\circ}$, $m\angle BAD=83^{\circ}$