Question

1. sat/act point c is in the interior of ∠abd, and ∠abc≅∠cbd. if m∠abc = (\frac{5}{2}x + 18)° and m∠cbd=(4x)°, what is m∠abd? (a) 12° (b) 36° (c) 48° (d) 72 (e) 96

Explanation:

Step1: Set up the equation

Since $\angle ABC\cong\angle CBD$, we set $\frac{5}{2}x + 18=4x$.

Step2: Solve for $x$

Subtract $\frac{5}{2}x$ from both sides: $18 = 4x-\frac{5}{2}x$. Combine like - terms: $18=\frac{8x - 5x}{2}=\frac{3x}{2}$. Multiply both sides by $\frac{2}{3}$ to get $x = 12$.

Step3: Find the measure of $\angle ABC$ or $\angle CBD$

Substitute $x = 12$ into the expression for $\angle CBD$ (we could also use the expression for $\angle ABC$). $m\angle CBD=4x$, so $m\angle CBD = 4\times12 = 48^{\circ}$.

Step4: Find the measure of $\angle ABD$

Since $\angle ABD=\angle ABC+\angle CBD$ and $\angle ABC=\angle CBD = 48^{\circ}$, then $m\angle ABD=48^{\circ}+48^{\circ}=96^{\circ}$.

Answer:

E. $96^{\circ}$