Exercise 21 Page 559

We are asked to find the values of x and y and the measures of the interior angles of △ ABC. Let's deal with one thing at a time.

Values of x and y

We are given the following diagram.

As we can see, angle ∠ A intercepts the arc BC, which measures 6y^(∘). We can use the Measure of an Inscribed Angle Theorem, which states the following.

m∠ A=1/2mBC It is given that the measure of ∠ A is 2x^(∘) and the measure of BC is 6y^(∘). Let's substitute these values into the equation and solve it for x.

m∠ A=1/2mBC

SubstituteII

m∠ A= 2x, mBC= 6y

2x=1/2( 6y)

Multiply

Multiply

2x=3y

DivEqn

.LHS /2.=.RHS /2.

x=3/2y

Next, let's recall that the sum of the measures of all the arcs in a circle is 360^(∘). In our case, the circle consists of AB, BC, and AC. mAB+mBC+mAC=360^(∘) ⇓ 6y^(∘)+6y^(∘)+4x^(∘)=360^(∘) Let's substitute x with 32y and then solve the equation for y.

6y^(∘)+6y^(∘)+4x^(∘)=360^(∘)

Substitute

x= 32y

6y^(∘)+6y^(∘)+4( 32y)^(∘)=360^(∘)

Multiply

Multiply

6y^(∘)+6y^(∘)+6y^(∘)=360^(∘)

AddTerms

Add terms

18y^(∘)=360^(∘)

DivEqn

.LHS /18^(∘).=.RHS /18^(∘).

y= 20

Now that we know the value of y, we can calculate the value of x. x=3/2( 20)=30

Measures of the Interior Angles

From the diagram we know that the measure of ∠ A is 2x^(∘).

Let's substitute x with 30 and find the measure of this angle. m∠ A=2( 30^(∘))=60^(∘) Angle ∠ C intercepts arc AB whose measure is the same as the measure of BC intercepted by ∠ A. Thus, the measures of ∠ A and ∠ C are equal. m∠ A=∠ C ⇓ m∠ C =60^(∘) Finally, using the fact that the sum of interior angles of a triangle is 180^(∘), we can calculate the measure of the last angle, ∠ B. m∠ A+m∠ B+m∠ C=180^(∘) Let's substitute 60^(∘) for m∠ A and m∠ C and calculate the value of m∠ B.

m∠ A+m∠ B+m∠ C=180^(∘)

SubstituteII

m∠ A= 60^(∘), m∠ C= 60^(∘)

60^(∘)+m∠ B+ 60^(∘)=180^(∘)

AddTerms

Add terms

120^(∘)+m∠ B=180^(∘)

SubEqn

LHS-120^(∘)=RHS-120^(∘)

m∠ B=60^(∘)

Therefore, all the interior angles of △ ABC measure 60^(∘).