Question

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

Explanation:

Step1: Determine the angle relationship

Since \( BA \) and \( BC \) are perpendicular, \( \angle ABC = 90^\circ \). And \( D \) is in the interior of \( \angle ABC \), so \( \angle ABD + \angle DBC=\angle ABC \). Substituting the given expressions, we get \((3r + 5)^\circ+(5r - 27)^\circ = 90^\circ\).

Step2: Solve for \( r \)

Simplify the left - hand side of the equation:

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Step3: Calculate \( m\angle ABD \)

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

Step4: Calculate \( m\angle DBC \)

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

Answer:

\( m\angle ABD = 47^\circ \), \( m\angle DBC = 43^\circ \)