Question

find m∠adc if m∠adc = (-6n + 243)°. diagram: point d with rays da, db, dc. angle adb is (-5n + 137)°, angle cdb is (-10n + 178)°.

Explanation:

Step1: Recognize the angle sum

Angles around point \( D \) on a straight line? Wait, no, \( \angle ADB \) and \( \angle CDB \) and \( \angle ADC \)? Wait, actually, \( \angle ADB + \angle CDB = \angle ADC \)? Wait, no, looking at the diagram, \( \angle ADB = (-5n + 137)^\circ \), \( \angle CDB = (-10n + 178)^\circ \), and \( \angle ADC = (-6n + 243)^\circ \). Wait, actually, the sum of \( \angle ADB \) and \( \angle CDB \) should equal \( \angle ADC \)? Wait, no, maybe it's a full angle? Wait, no, the diagram shows \( A \), \( D \), \( B \) and \( C \), \( D \), \( B \), so \( \angle ADB + \angle CDB = \angle ADC \)? Wait, let's check:

So, \( (-5n + 137) + (-10n + 178) = -6n + 243 \)

Step2: Solve for \( n \)

Combine like terms: \( -15n + 315 = -6n + 243 \)

Subtract \( -6n \) from both sides: \( -9n + 315 = 243 \)

Subtract 315: \( -9n = 243 - 315 = -72 \)

Divide by -9: \( n = \frac{-72}{-9} = 8 \)

Step3: Find \( m\angle ADC \)

Substitute \( n = 8 \) into \( -6n + 243 \):

\( -6(8) + 243 = -48 + 243 = 195 \)? Wait, that can't be right. Wait, maybe the sum of \( \angle ADB \), \( \angle CDB \), and \( \angle ADC \) is 360? Wait, no, the diagram: \( A \) and \( C \) are on opposite sides of \( DB \), so \( \angle ADB + \angle ADC + \angle CDB = 360 \)? Wait, no, maybe \( \angle ADB + \angle CDB = 360 - \angle ADC \)? Wait, I think I made a mistake. Let's re-examine.

Wait, the problem says "Find \( m\angle ADC \) if \( m\angle ADC = (-6n + 243)^\circ \)". Wait, maybe the other two angles \( \angle ADB = (-5n + 137)^\circ \) and \( \angle CDB = (-10n + 178)^\circ \) are supplementary to \( \angle ADC \)? No, maybe the sum of \( \angle ADB \), \( \angle CDB \), and \( \angle ADC \) is 360? Wait, no, the diagram: \( D \) is the vertex, with \( DA \), \( DB \), \( DC \). So \( \angle ADB + \angle BDC + \angle CDA = 360 \)? Wait, no, if \( DA \) and \( DC \) are on opposite sides of \( DB \), then \( \angle ADB + \angle CDB = \angle ADC \)? No, that would be if they are adjacent. Wait, maybe the correct equation is \( (-5n + 137) + (-10n + 178) + (-6n + 243) = 360 \)? Let's try that.

Combine like terms: \( -5n -10n -6n + 137 + 178 + 243 = 360 \)

\( -21n + 558 = 360 \)

\( -21n = 360 - 558 = -198 \)

\( n = \frac{-198}{-21} = \frac{66}{7} \approx 9.428 \), which doesn't seem right. Wait, maybe the diagram is a straight angle? Wait, no, the original problem might have \( \angle ADB \) and \( \angle CDB \) forming a linear pair with \( \angle ADC \)? Wait, I think I misread the problem. Wait, the problem says "Find \( m\angle ADC \) if \( m\angle ADC = (-6n + 243)^\circ \)". Wait, maybe the other two angles \( \angle ADB = (-5n + 137)^\circ \) and \( \angle CDB = (-10n + 178)^\circ \) are equal to \( \angle ADC \)? No, that doesn't make sense. Wait, maybe the sum of \( \angle ADB \) and \( \angle CDB \) is equal to \( \angle ADC \)? Let's check with the first calculation:

\( (-5n + 137) + (-10n + 178) = -15n + 315 \)

Set equal to \( -6n + 243 \):

\( -15n + 315 = -6n + 243 \)

\( -9n = -72 \)

\( n = 8 \)

Then \( \angle ADC = -6(8) + 243 = 195^\circ \), but that's more than 180, which is possible, but maybe the diagram is a full circle? Wait, no, maybe the correct approach is that \( \angle ADB + \angle CDB + \angle ADC = 360 \), but that would be a full angle. Wait, let's check with \( n = 8 \):

\( \angle ADB = -5(8) + 137 = -40 + 137 = 97^\circ \)

\( \angle CDB = -10(8) + 178 = -80 + 178 = 98^\circ \)

\( \angle ADC = 195^\circ \)

Sum: \( 97 + 98 + 195 = 390 \), which is more than 360. So that's wrong.

Wait, maybe the diagram is such that \( \angle ADB \) and \( \angle ADC \) are supplementary? No, the labels are \( A \), \( D \), \( B \) and \( C \), \( D \), \( B \), so \( DB \) is a common side, \( DA \) and \( DC \) are on opposite sides. So \( \angle ADB + \angle CDB = \angle ADC \)? No, that would be if \( DA \) and \( DC \) are on the same side. Wait, maybe the correct equation is \( \angle ADB + \angle ADC + \angle CDB = 360 \), but that's a full angle. Wait, maybe the problem has a typo, or I misread the angle expressions. Wait, the problem says "Find \( m\angle ADC \) if \( m\angle ADC = (-6n + 243)^\circ \)". Wait, maybe the other two angles are \( \angle ADB = (-5n + 137)^\circ \) and \( \angle CDB = (-10n + 178)^\circ \), and \( \angle ADB + \angle CDB = \angle ADC \)? Let's check the sum:

\( 97 + 98 = 195 \), which matches \( \angle ADC = 195 \). But 195 is more than 180, which is possible for a reflex angle. So maybe that's correct.

So, proceeding with \( n = 8 \), \( \angle ADC = -6(8) + 243 = 195^\circ \). Wait, but let's verify again.

Wait, maybe the diagram is a straight line, but no, three angles. Wait, perhaps the problem is that \( \angle ADB \) and \( \angle CDB \) are adjacent and form \( \angle ADC \), so their sum is \( \angle ADC \). So the equation is correct. So the answer is 195 degrees.

Answer:

\( 195^\circ \)