Question

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Explanation:

Step1: Identify congruent triangles

Triangles \( \triangle BCD \) and \( \triangle BAD \) are congruent (right triangles with \( CD = AD \) and \( BD \) common), so \( \angle CBD=\angle ABD \)? Wait, no, the angles at \( B \): \( \angle ABD=(9x - 22)^\circ \) and \( \angle CBD=(6x - 7)^\circ \)? Wait, no, actually, since \( \triangle BCD \cong \triangle BAD \) (HL, as \( \angle C=\angle A = 90^\circ \), \( CD = AD \), \( BD \) hypotenuse), so \( \angle CBD=\angle ABD \)? Wait, no, the diagram shows \( \angle ABD=(9x - 22) \) and \( \angle CBD=(6x - 7) \)? Wait, no, maybe \( \angle ABD \) and \( \angle CBD \) are equal? Wait, no, maybe the angles are equal because the triangles are congruent, so \( 9x - 22=6x - 7 \)? Wait, solve for \( x \):

\( 9x - 22=6x - 7 \)

Step2: Solve for \( x \)

Subtract \( 6x \) from both sides: \( 3x - 22=-7 \)

Add 22 to both sides: \( 3x = 15 \)

Divide by 3: \( x = 5 \)

Step3: Find \( m\angle ABD \)

Substitute \( x = 5 \) into \( 9x - 22 \):

\( 9(5)-22=45 - 22 = 23 \)? Wait, no, wait, maybe I mixed up the angles. Wait, no, if \( \triangle BCD \cong \triangle BAD \), then \( \angle CBD=\angle ABD \)? Wait, no, the labels: \( \angle ABD=(9x - 22) \), \( \angle CBD=(6x - 7) \). Wait, maybe the angles are equal, so \( 9x - 22=6x - 7 \), solving gives \( x = 5 \), then \( \angle ABD=9(5)-22=23 \)? Wait, but let's check again.

Wait, maybe the triangles are congruent, so \( \angle ABD=\angle CBD \)? Wait, no, the diagram: \( \angle at B \) is split into \( \angle ABD \) and \( \angle CBD \), and since \( \triangle BCD \cong \triangle BAD \), then \( \angle ABD=\angle CBD \). So set \( 9x - 22=6x - 7 \)

Solving:

\( 9x - 6x=-7 + 22 \)

\( 3x = 15 \)

\( x = 5 \)

Then \( m\angle ABD=9(5)-22=45 - 22 = 23 \)? Wait, no, that seems small. Wait, maybe I had the angles reversed. Wait, maybe \( \angle ABD \) is \( 9x - 22 \) and \( \angle CBD \) is \( 6x - 7 \), but actually, since the triangles are congruent, those angles are equal, so solving \( 9x - 22=6x - 7 \) gives \( x = 5 \), then \( \angle ABD=9*5 - 22=23 \). Wait, but let's check: \( 6x - 7=6*5 - 7=30 - 7=23 \), so both angles are 23, which makes sense.

Wait, but the problem is to find \( m\angle ABD \), so substituting \( x = 5 \) into \( 9x - 22 \) gives 23? Wait, no, wait, maybe I made a mistake. Wait, no, if \( x = 5 \), then \( 9x - 22=45 - 22=23 \), and \( 6x - 7=30 - 7=23 \), so \( \angle ABD=23^\circ \)? Wait, but that seems correct.

Answer:

\( 23 \)