Exercise 37 Page 132

To find m∠ AEB, we first need to find the value of y. To do so, we recognize that the labeled angles are vertical angles. Therefore, they are congruent by the Vertical Angles Theorem.

This means we can equate the expressions for the two angle measures and solve for y.

3y+20=5y-16

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Solve for y

AddEqn

LHS+16=RHS+16

3y+36=5y

SubEqn

LHS-3y=RHS-3y

36=2y

DivEqn

.LHS /1.=.RHS /1.{2

18=y

RearrangeEqn

Rearrange equation

y=18

Now that we know the y-variable, we can substitute the value into the expression for either one of the marked vertical angles to find its value. Let's substitute the value of y into the expression for m∠ AEC.

3y+20

Substitute

y= 18

3( 18)+20

Multiply

54+20

AddTerms

Add terms

We have found that m∠ AEC=74. The other vertical angle has the same measure. Let's consider the diagram once more, this time including the simplified expressions as well as marking the unknown angle ∠ AEB.

Notice that the unknown angle forms a linear pair with both 74^(∘) angles. We can use this to create another equation. m∠ AEB+ 74=180 ⇕ m∠ AEB = 106 We have found that m∠ AEB=106.

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