In the diagram below, A is on line segment CE, and AB bisects angle DAC (meaning that AB splits angle DAC into two equal angles). If DA is parallel to EF and angle AEF = 10 angle BAC - 12 degrees, then what is DAC in degrees?
The answer is 32 degrees.
Thank you helpful--at last
10*BAC -12+2BAC=180 right
AOPS is supposed to be difficult, it is best to work it out on your own so you actually understand the material
To find the value of DAC in degrees, we need to use the given information in the diagram and the given equation.
Let's break down the steps to solve the problem:
Step 1: Identify the given information:
- A is on line segment CE.
- AB bisects angle DAC.
- DA is parallel to EF.
- Angle AEF = 10 angle BAC - 12 degrees.
Step 2: Use angle bisector property:
Since AB bisects angle DAC, we can use the angle bisector property, which states that AB divides angle DAC into two equal angles. Therefore, angle DAB is equal to angle CAB.
Step 3: Use the property of parallel lines:
Since DA is parallel to EF, we can use the property of alternate interior angles. So, angle AEF is equal to angle DAE.
Step 4: Apply the given equation:
Angle AEF = 10 angle BAC - 12 degrees.
Step 5: Substitute the values:
Substituting angle AEF with angle DAE and angle DAB with angle CAB, we get:
Angle DAE = 10 angle CAB - 12 degrees.
Step 6: Use angle addition property:
Since angle DAE = angle DAB + angle BAE, we can rewrite the equation as:
Angle DAB + angle BAE = 10 angle CAB - 12 degrees.
Step 7: Use the angle bisector property again:
Angle DAB is equal to angle CAB. So, we can rewrite the equation as:
Angle CAB + angle BAE = 10 angle CAB - 12 degrees.
Step 8: Simplify the equation:
Rearranging the equation, we get:
angle BAE = 9 angle CAB - 12 degrees.
Step 9: Solve for angle CAB:
Since angle BAE = 9 angle CAB - 12 degrees, we can solve for angle CAB by isolating it.
Starting with angle BAE = 9 angle CAB - 12 degrees, we add 12 degrees to both sides:
angle BAE + 12 degrees = 9 angle CAB.
Next, we divide both sides by 9:
(angle BAE + 12 degrees) / 9 = angle CAB.
Step 10: Calculate angle CAB:
Now we substitute the value of angle BAE. Given that angle AEF = 10 angle BAC - 12 degrees and that angle AEF = angle DAE, we can write:
(angle DAE + 12 degrees) / 9 = angle CAB.
Step 11: Simplify and calculate angle CAB:
Substituting angle DAE = angle CAB, the equation becomes:
(angle CAB + 12 degrees) / 9 = angle CAB.
Simplifying further:
angle CAB + 12 degrees = 9 * angle CAB.
Now, solve for angle CAB:
8 * angle CAB = 12 degrees.
Dividing both sides by 8:
angle CAB = 1.5 degrees.
Step 12: Find angle DAC:
Since DAC is equal to twice angle CAB (due to the angle bisector property), we can find angle DAC:
angle DAC = 2 * angle CAB = 2 * 1.5 degrees = 3 degrees.
Therefore, DAC is equal to 3 degrees.