Exercise 43 Page 374

We are given that in a parallelogram $S T U V,$ the measure of $∠ T S U$ is $3 2^{\circ},$ the measure of $∠ U S V$ is $(x^{2})^{\circ},$ and $∠ T U V$ is an acute angle with a measure of $12 x^{\circ} .$ Let's take a look at the given diagram.

Let's recall that opposite angles in a parallelogram are congruent by the Parallelogram Opposite Angles Theorem. Using this fact, we can write an equation for the given angle measures.

m ∠ T S U + m ∠ U S V = m ∠ T U V 3 2^{\circ} + x^{2}^{\circ} = 12 x^{\circ}

Now, we can solve the above quadratic equation for

x .

To do this, we will use the Quadratic Formula.

a x^{2} + b x + c = 0 ⇕ x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Our first step will be to move all terms to one side of the equation. Then, we will identify the values of

a,

b,

and

c .

32 + x^{2} = 12 x ⇕ 1 x^{2} + (- 12) x + 32 = 0

We see that

a = 1,

b = - 12,

and

c = 32 .

Let's substitute these values into the Quadratic Formula.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

SubstituteValues

Substitute values

x = \frac{- ( - 1 2 ) \pm ( - 1 2 ) ^{2} - 4 ( 1 ) ( 3 2 )}{2 ( 1 )}

▼

Solve for

x

and Simplify

NegNeg

$- (- a) = a$

x = \frac{1 2 \pm ( - 1 2 ) ^{2} - 4 ( 1 ) ( 3 2 )}{2 ( 1 )}

CalcPow

Calculate power

x = \frac{1 2 \pm 1 4 4 - 4 ( 1 ) ( 3 2 )}{2 ( 1 )}

MultByOne

$a \cdot 1 = a$

x = \frac{1 2 \pm 1 4 4 - 4 ( 3 2 )}{2}

MultNegPos

$(- a) b = - a b$

x = \frac{1 2 \pm 1 4 4 - 1 2 8}{2}

SubTerm

Subtract term

x = \frac{1 2 \pm 1 6}{2}

CalcRoot

Calculate root

x = \frac{1 2 \pm 4}{2}

The solutions for this equation are

x = \frac{1 2 \pm 4}{2} .

Let's separate them into the positive and negative cases.

$x = \frac{1 2 \pm 4}{2}$
$x_{1} = \frac{1 2 + 4}{2}$	$x_{2} = \frac{1 2 - 4}{2}$
$x_{1} = \frac{1 6}{2}$	$x_{2} = \frac{8}{2}$
$x_{1} = 8$	$x_{2} = 4$

Using the Quadratic Formula, we found that the solutions of the given equation are

x_{1} = 8

and

x_{2} = 4 .

Since we want

∠ T U V

to be an acute angle, the value of

12 x

must be less than

90 .

Let's check for which solution this condition is satisfied.

12 x_{1} = 12 (8) = 96 \times 12 x_{2} = 12 (4) = 48 ✓

Only when

x = 4

is

∠ T U V

acute. Using this value, we can evaluate the exact measure of

∠ U S V .

m ∠ U S V = (x^{2})^{\circ} = (4^{2})^{\circ} = 1 6^{\circ}

The measure of

∠ U S V

is

1 6^{\circ} .