Exercise 45 Page 428

We are given some measures of angles in a rectangle ABCD in terms of x- and y-terms, and we are asked to find the values of x and y. Let's start with graphing this rectangle and highlighting the given angles.

Let's notice that, since diagonals in a rectangle are congruent and bisect each other, AE=EB. This means that triangle AEB is an isosceles triangle and m∠EAB=m∠ABE.

We know that all four angles in a rectangle are right angles. This means that the sum of m∠ABE and m∠EBC is equal to 90. Let's use this information to find the value of x.

m∠ABE+m∠EBC=90

SubstituteII

∠ABE= 4x+6, ∠EBC= 60

4x+6+60=90

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Solve for x

AddTerms

Add terms

4x+66=90

SubEqn

LHS-66=RHS-66

4x=24

DivEqn

.LHS /4.=.RHS /4.

x=6

Now, as we know x=6, we can find the measure of both ∠EAB and ∠ABE.

4x+6

Substitute

x= 6

4( 6)+6

Multiply

Multiply

24+6

AddTerms

Add terms

30

These angles have measures of 30. Next let's notice that ∠DEC and ∠AEB are vertical angles, so they have the same measure. Therefore, we can write that m∠AEB is also 10-11y.

To find the value of y, we can use the Triangle Sum Theorem. Recall that, according to this theorem, the sum of the measures of angles in a triangle is always 180. Let's substitute 30 for m∠EAB and m∠ABE, and 10-11y for m∠AEB.

m∠AEB+m∠EAB+m∠ABE=180

SubstituteValues

Substitute values

10-11y+ 30+ 30=180

AddTerms

Add terms

70-11y=180

SubEqn

LHS-70=RHS-70

-11y=110

DivEqn

.LHS /(-11).=.RHS /(-11).

y=-10

Therefore, the values of x and y are 6 and -10 respectively.