m∠1 is 4 times m∠2. ∠1 and ∠2 are complementary. ∠1 and ∠3 are vertical angles. ∠3 and ∠4 are supplementary. What are the measures of the four angles?

Answer 1

Let's first set up a system of equations from these clues:

m∠1 = 4 * m∠2

m∠1 + m∠2 = 90

m∠1 = m∠3

m∠3 + m∠4 = 180

And then solve the first pair of equations:

m∠1 = 4 * m∠2

m∠1 + m∠2 = 90

4 * m∠2 + m∠2 = 90

5 * m∠2 = 90

m∠2 = 18

m∠1 = 72

From the third equation, we have:

m∠1 = m∠3

m∠3 = 72

And finally, from the fourth equation:

m∠3 + m∠4 = 180

72 + m∠4 = 180

m∠4 = 108

So the angles are 72, 18, 72, and 108 degrees, respectively.