Question

1. if $overrightarrow{bd}$ bisects $angle cbe$, $bcperp ba$, $mangle cbd=(3x + 25)^{circ}$, and $mangle dbe=(7x - 19)^{circ}$, find $mangle abd$.

Explanation:

Step1: BD bisects ∠CBE, so ∠CBD=∠DBE

$3x + 25 = 7x - 19$

Step2: Solve for x

$4x = 44 \implies x = 11$

Step3: Find ∠CBD

$∠CBD = 3(11) + 25 = 58^\circ$

Step4: BC⊥BA, so ∠ABC=90°

∠ABD = ∠ABC - ∠CBD = $90^\circ - 58^\circ = 32^\circ$

Answer:

$32^\circ$