Let's begin by drawing $△ D E F$ and labeling the lengths of the sides. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.

First, we can tell that it is not a right triangle, as the sides do not satisfy the Pythagorean Theorem.

20^{2} + 16^{2} \neq = 21.6^{2}

Let's find the measures of

∠ D,

∠ E,

and

∠ F

one at a time.

Finding $m ∠ D$

The measures of all three sides of the triangle are given. Therefore, we can use the Law of Cosines to find

m ∠ D .

d^{2} = e^{2} + f^{2} - 2 e f cos D

Let's substitute

d = 21.6,

e = 20,

and

f = 16

to isolate

cos D .

d^{2} = e^{2} + f^{2} - 2 e f cos D

SubstituteValues

Substitute values

21.6^{2} = 20^{2} + 16^{2} - 2 (20) (16) cos D

▼

Solve for

cos D

CalcPow

Calculate power

466.56 = 400 + 256 - 2 (20) (16) cos D

Multiply

466.56 = 400 + 256 - 640 cos D

AddTerms

Add terms

466.56 = 656 - 640 cos D

SubEqn

$LHS - 656 = RHS - 656$

- 189.44 = - 640 cos D

DivEqn

$LHS / (- 640) = RHS / (- 640)$

\frac{- 1 8 9 . 4 4}{- 6 4 0} = cos D

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

\frac{1 8 9 . 4 4}{6 4 0} = cos D

CalcQuot

Calculate quotient

0.296 = cos D

RearrangeEqn

Rearrange equation

cos D = 0.296

Now, we can use the inverse cosine ratio and a calculator to find

m ∠ D .

m ∠ D = cos^{- 1} 0.296

UseCalc

Use a calculator

m ∠ D \approx 72.782489 \dots

RoundInt

Round to nearest integer

m ∠ D \approx 73

Finding $m ∠ E$

Now that we know the measure of

∠ D,

we can find

m ∠ E

using the Law of Sines.

\frac{sin D}{d} = \frac{sin E}{e}

Let's substitute

d = 21.6,

e = 20,

and

m ∠ D \approx 73

to isolate

sin E .

\frac{sin D}{d} = \frac{sin E}{e}

SubstituteValues

Substitute values

\frac{sin 7 3}{2 1 . 6} = \frac{sin E}{2 0}

▼

Solve for

sin E

MultEqn

$LHS \cdot 20 = RHS \cdot 20$

\frac{2 0 sin 7 3}{2 1 . 6} = sin E

RearrangeEqn

Rearrange equation

sin E = \frac{2 0 sin 7 3}{2 1 . 6}

Now we can use the inverse sine ratio to find

m ∠ E .

m ∠ E = sin^{- 1} \frac{2 0 sin 7 3}{2 1 . 6}

UseCalc

Use a calculator

m ∠ E \approx 62.309091 \dots

RoundInt

Round to nearest integer

m ∠ E \approx 62

Finding $m ∠ F$

Finally, to find

m ∠ F

we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to

180 .

m ∠ F + 73 + 62 = 180 \Leftrightarrow m ∠ F \approx 45

Completing the Triangle

With all of the angle measures, we can complete our diagram.

Extra

The Law of Cosines can be visualized by drawing a triangle and labeling angles and sides.

Similarly, the Law of Sines can also be visualized by drawing a triangle and labeling angles and sides.

Finding m∠D

Finding m∠E

Finding m∠F

Completing the Triangle

Extra

Finding $m ∠ D$

Finding $m ∠ E$

Finding $m ∠ F$

Exercise

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