Question

find ( mangle cad ) and ( mangle bad ).
there is a diagram with point a, and rays ab (vertical downward), ac (horizontal rightward), ad (at an angle), and another ray between ac and ad. angle between ab and the ray to ad is ( (5x + 57)^circ ), angle between ac and ad is ( (x + 15)^circ ), and angle between ab and ac is a right angle.

Explanation:

Step1: Identify the right angle

From the diagram, $\angle BAC$ is a right angle, so $m\angle BAC = 90^\circ$. Also, $m\angle BAD + m\angle CAD = m\angle BAC$. Wait, actually, looking at the angles: $m\angle BAD = (5x + 57)^\circ$ and $m\angle CAD = (x + 15)^\circ$, and since $\angle BAC$ is $90^\circ$, we have $m\angle BAD + m\angle CAD = 90^\circ$? Wait, no, wait. Wait, the angle between $AB$ and $AD$ is $(5x + 57)^\circ$, and between $AD$ and $AC$ is $(x + 15)^\circ$, and between $AB$ and $AC$ is $90^\circ$. So, $m\angle BAD + m\angle CAD = 90^\circ$? Wait, no: $m\angle BAD + m\angle CAD = m\angle BAC = 90^\circ$? Wait, no, actually, $\angle BAD$ is the angle between $AB$ and $AD$, and $\angle CAD$ is between $AD$ and $AC$, so $\angle BAD + \angle CAD = \angle BAC = 90^\circ$. So:

$(5x + 57) + (x + 15) = 90$

Step2: Solve for x

Combine like terms:

$5x + x + 57 + 15 = 90$

$6x + 72 = 90$

Subtract 72 from both sides:

$6x = 90 - 72$

$6x = 18$

Divide both sides by 6:

$x = \frac{18}{6} = 3$

Step3: Find m∠CAD

Substitute x = 3 into $m\angle CAD = (x + 15)^\circ$:

$m\angle CAD = 3 + 15 = 18^\circ$

Step4: Find m∠BAD

Substitute x = 3 into $m\angle BAD = (5x + 57)^\circ$:

$m\angle BAD = 5(3) + 57 = 15 + 57 = 72^\circ$

Answer:

$m\angle CAD = 18^\circ$, $m\angle BAD = 72^\circ$