Question

if $overline{ad} cong overline{cd}$ and $mangle abd = 66^circ$, what is $mangle abc$?
diagram: points a, b, c, d with right angles at a (between horizontal line through a and $overline{ad}$) and c (between horizontal line through c and $overline{cd}$); d connected to a, b, c; $overline{bd}$ is a segment. $mangle abc = square^circ$

Explanation:

Step1: Identify congruent triangles

Since \( \overline{AD} \cong \overline{CD} \), \( \angle BAD = \angle BCD = 90^\circ \) (right angles), and \( \overline{BD} \) is common, by HL (Hypotenuse - Leg) congruence, \( \triangle ABD \cong \triangle CBD \).

Step2: Find \( m\angle CBD \)

By congruence, \( \angle ABD \cong \angle CBD \), so \( m\angle CBD = m\angle ABD = 66^\circ \).

Step3: Calculate \( m\angle ABC \)

\( m\angle ABC = m\angle ABD + m\angle CBD = 66^\circ + 66^\circ = 132^\circ \).

Answer:

\( 132 \)