Question

find the measure of \\( \angle a b d \\).

\\( m \angle a b d = ( 10 x - 48 ) ^ { \circ } \\)

\\( m \angle c b d = ( 5 x - 3 ) ^ { \circ } \\)

enter deg after any value that is in degrees.

Explanation:

Step1: Identify angle relationship

From the diagram, \( \angle ABC = 90^\circ \) (since \( \angle A \) and \( \angle C \) are right angles, and \( AB \) and \( CD \) are equal, \( AD \) and \( BC \) are equal, so \( ABCD \) is a rectangle? Wait, actually, \( \angle ABD + \angle CBD=\angle ABC = 90^\circ \). So we set up the equation: \( (10x - 48)+(5x - 3)=90 \).

Step2: Solve for \( x \)

Combine like terms: \( 10x+5x - 48 - 3 = 90 \) → \( 15x - 51 = 90 \).
Add 51 to both sides: \( 15x=90 + 51 \) → \( 15x = 141 \).
Divide by 15: \( x=\frac{141}{15}=9.4 \)? Wait, no, wait, maybe I made a mistake. Wait, the figure: \( \angle A \) and \( \angle C \) are right angles, \( AD = BC \) (marked with one tick), \( AB = CD \) (marked with one tick)? Wait, no, the angle at \( B \): \( \angle ABD \) and \( \angle CBD \) add up to \( \angle ABC \), which is a right angle (since \( \angle A \) and \( \angle C \) are right angles, so \( ABCD \) is a rectangle, so \( \angle ABC = 90^\circ \)). So equation: \( (10x - 48)+(5x - 3)=90 \).
Wait, let's recalculate: \( 10x + 5x - 48 - 3 = 90 \) → \( 15x - 51 = 90 \) → \( 15x = 141 \) → \( x = 9.4 \)? That seems odd. Wait, maybe the figure is a square? No, maybe I misread the angle. Wait, maybe \( \angle ABD \) and \( \angle CBD \) are equal? Wait, no, the diagram shows \( AD \) and \( BC \) with one tick, \( AB \) and \( CD \) with one tick? Wait, no, the ticks: \( AD \) has one tick, \( BC \) has one tick? Wait, no, the diagram: \( AD \) is horizontal with one tick, \( BC \) is horizontal with one tick? Wait, maybe \( ABCD \) is a rectangle, so \( AD = BC \), \( AB = CD \), and \( \angle A = \angle C = 90^\circ \), so \( \triangle ABD \) and \( \triangle CBD \) are congruent? Wait, no, the angle bisector? Wait, maybe \( \angle ABD = \angle CBD \)? No, the expressions are different. Wait, maybe the sum is 90 degrees. Wait, let's check again.

Wait, maybe I made a mistake in the equation. Let's do it again:

\( (10x - 48) + (5x - 3) = 90 \)

\( 10x + 5x - 48 - 3 = 90 \)

\( 15x - 51 = 90 \)

\( 15x = 90 + 51 = 141 \)

\( x = 141 / 15 = 9.4 \). Then \( m\angle ABD = 10(9.4) - 48 = 94 - 48 = 46 \) degrees. \( m\angle CBD = 5(9.4) - 3 = 47 - 3 = 44 \) degrees. 46 + 44 = 90, which works. Wait, but maybe the figure is a square? No, 46 and 44 add to 90. So that's correct.

Wait, but let's check again. Maybe the angle is 90 degrees. So step 1: set sum of angles to 90. Step 2: solve for x. Step 3: find \( m\angle ABD \).

Step3: Calculate \( m\angle ABD \)

Substitute \( x = 9.4 \) into \( 10x - 48 \): \( 10(9.4) - 48 = 94 - 48 = 46 \) degrees.

Wait, but maybe I made a mistake in the equation. Wait, maybe \( \angle ABC \) is 90 degrees, so the sum of \( \angle ABD \) and \( \angle CBD \) is 90. So the equation is correct. So \( x = 9.4 \), \( m\angle ABD = 46 \) degrees.

Wait, but let's check the arithmetic again. 10x - 48 + 5x - 3 = 90 → 15x - 51 = 90 → 15x = 141 → x = 141/15 = 9.4. Yes. Then 109.4=94, 94-48=46. 59.4=47, 47-3=44. 46+44=90. Correct.

Answer:

46 deg