Solved on Feb 27, 2024

Find the measure of angle D in a parallelogram with $\mathrm{m} \angle B=5x$ and $\mathrm{m} \angle C=2x+12$ . Answer: $\mathrm{m} \angle D = 180 - 5x - (2x+12) = 180 - 7x - 12$ . To check, sum the angles of a parallelogram is 360 degrees.

STEP 1

Assumptions
1. $ABCD$ is a parallelogram.
2. The measure of angle $B$ is $5x$ degrees.
3. The measure of angle $C$ is $2x + 12$ degrees.
4. The sum of the measures of angles in any parallelogram is 360 degrees.
5. Opposite angles in a parallelogram are equal.

STEP 2

Since opposite angles in a parallelogram are equal, we have:
$\angle A = \angle C$ $\angle B = \angle D$

STEP 3

Using the property that the sum of the measures of angles in a parallelogram is 360 degrees, we can write the equation:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$

STEP 4

Substitute the given expressions for $\angle B$ and $\angle C$ into the equation, and use the fact that $\angle A = \angle C$ and $\angle B = \angle D$ :
$\angle C + 5x + \angle C + 5x = 360^\circ$

STEP 5

Since $\angle C = 2x + 12$ , substitute this into the equation:
$(2x + 12) + 5x + (2x + 12) + 5x = 360^\circ$

STEP 6

Combine like terms:
$14x + 24 = 360^\circ$

STEP 7

Subtract 24 from both sides of the equation:
$14x = 360^\circ - 24$

STEP 8

Calculate the right side of the equation:
$14x = 336^\circ$

STEP 9

Divide both sides by 14 to solve for $x$ :
$x = \frac{336^\circ}{14}$

STEP 10

Calculate the value of $x$ :
$x = 24^\circ$

STEP 11

Now that we have the value of $x$ , we can find the measure of angle $D$ by substituting $x$ into the expression for $\angle B$ , since $\angle B = \angle D$ :
$\angle D = 5x$

STEP 12

Substitute the value of $x$ into the expression for $\angle D$ :
$\angle D = 5 \times 24^\circ$

STEP 13

Calculate the measure of angle $D$ :
$\angle D = 120^\circ$
The measure of angle $D$ is $120^\circ$ .
To check the answer, we can verify that the sum of all angles in the parallelogram adds up to $360^\circ$ .

STEP 14

Check the sum of the angles using the values of $x$ and the expressions for $\angle B$ and $\angle C$ :
$\angle A + \angle B + \angle C + \angle D = (2x + 12) + 5x + (2x + 12) + 5x$

STEP 15

Substitute the value of $x$ into the check equation:
$\angle A + \angle B + \angle C + \angle D = (2 \times 24^\circ + 12) + 5 \times 24^\circ + (2 \times 24^\circ + 12) + 5 \times 24^\circ$

STEP 16

Calculate the sum:
$\angle A + \angle B + \angle C + \angle D = (48^\circ + 12) + 120^\circ + (48^\circ + 12) + 120^\circ$

STEP 17

Combine like terms:
$\angle A + \angle B + \angle C + \angle D = 60^\circ + 120^\circ + 60^\circ + 120^\circ$

STEP 18

Calculate the total:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$
Since the sum of the angles equals $360^\circ$ , our answer for $\angle D$ is confirmed to be correct.
$\mathrm{m} \angle D= 120^\circ$

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Find the measure of angle D in a parallelogram with $\mathrm{m} \angle B=5x$ and $\mathrm{m} \angle C=2x+12$ . Answer: $\mathrm{m} \angle D = 180 - 5x - (2x+12) = 180 - 7x - 12$ . To check, sum the angles of a parallelogram is 360 degrees.

STEP 1

Assumptions
1. $ABCD$ is a parallelogram.
2. The measure of angle $B$ is $5x$ degrees.
3. The measure of angle $C$ is $2x + 12$ degrees.
4. The sum of the measures of angles in any parallelogram is 360 degrees.
5. Opposite angles in a parallelogram are equal.

STEP 2

Since opposite angles in a parallelogram are equal, we have:
$\angle A = \angle C$ $\angle B = \angle D$

STEP 3

Using the property that the sum of the measures of angles in a parallelogram is 360 degrees, we can write the equation:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$

STEP 4

Substitute the given expressions for $\angle B$ and $\angle C$ into the equation, and use the fact that $\angle A = \angle C$ and $\angle B = \angle D$ :
$\angle C + 5x + \angle C + 5x = 360^\circ$

STEP 5

Since $\angle C = 2x + 12$ , substitute this into the equation:
$(2x + 12) + 5x + (2x + 12) + 5x = 360^\circ$

STEP 6

Combine like terms:
$14x + 24 = 360^\circ$

STEP 7

Subtract 24 from both sides of the equation:
$14x = 360^\circ - 24$

STEP 8

Calculate the right side of the equation:
$14x = 336^\circ$

STEP 9

Divide both sides by 14 to solve for $x$ :
$x = \frac{336^\circ}{14}$

STEP 10

Calculate the value of $x$ :
$x = 24^\circ$

STEP 11

Now that we have the value of $x$ , we can find the measure of angle $D$ by substituting $x$ into the expression for $\angle B$ , since $\angle B = \angle D$ :
$\angle D = 5x$

STEP 12

Substitute the value of $x$ into the expression for $\angle D$ :
$\angle D = 5 \times 24^\circ$

STEP 13

Calculate the measure of angle $D$ :
$\angle D = 120^\circ$
The measure of angle $D$ is $120^\circ$ .
To check the answer, we can verify that the sum of all angles in the parallelogram adds up to $360^\circ$ .

STEP 14

Check the sum of the angles using the values of $x$ and the expressions for $\angle B$ and $\angle C$ :
$\angle A + \angle B + \angle C + \angle D = (2x + 12) + 5x + (2x + 12) + 5x$

STEP 15

Substitute the value of $x$ into the check equation:
$\angle A + \angle B + \angle C + \angle D = (2 \times 24^\circ + 12) + 5 \times 24^\circ + (2 \times 24^\circ + 12) + 5 \times 24^\circ$

STEP 16

Calculate the sum:
$\angle A + \angle B + \angle C + \angle D = (48^\circ + 12) + 120^\circ + (48^\circ + 12) + 120^\circ$

STEP 17

Combine like terms:
$\angle A + \angle B + \angle C + \angle D = 60^\circ + 120^\circ + 60^\circ + 120^\circ$

STEP 18

Calculate the total:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$
Since the sum of the angles equals $360^\circ$ , our answer for $\angle D$ is confirmed to be correct.
$\mathrm{m} \angle D= 120^\circ$

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STEP 1

Assumptions1. ABCDABCDABCD is a parallelogram.2. The measure of angle BBB is 5x5x5x degrees.3. The measure of angle CCC is 2x+122x + 122x+12 degrees.4. The sum of the measures of angles in any parallelogram is 360 degrees.5. Opposite angles in a parallelogram are equal.

STEP 2

Since opposite angles in a parallelogram are equal, we have:∠A=∠C\angle A = \angle C∠A=∠C ∠B=∠D\angle B = \angle D∠B=∠D

STEP 3

Using the property that the sum of the measures of angles in a parallelogram is 360 degrees, we can write the equation:∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ∠A+∠B+∠C+∠D=360∘

STEP 4

Substitute the given expressions for ∠B\angle B∠B and ∠C\angle C∠C into the equation, and use the fact that ∠A=∠C\angle A = \angle C∠A=∠C and ∠B=∠D\angle B = \angle D∠B=∠D:∠C+5x+∠C+5x=360∘\angle C + 5x + \angle C + 5x = 360^\circ∠C+5x+∠C+5x=360∘

STEP 5

Since ∠C=2x+12\angle C = 2x + 12∠C=2x+12, substitute this into the equation:(2x+12)+5x+(2x+12)+5x=360∘(2x + 12) + 5x + (2x + 12) + 5x = 360^\circ(2x+12)+5x+(2x+12)+5x=360∘

STEP 6

Combine like terms:14x+24=360∘14x + 24 = 360^\circ14x+24=360∘

STEP 7

Subtract 24 from both sides of the equation:14x=360∘−2414x = 360^\circ - 2414x=360∘−24

STEP 8

Calculate the right side of the equation:14x=336∘14x = 336^\circ14x=336∘

STEP 9

Divide both sides by 14 to solve for xxx:x=336∘14x = \frac{336^\circ}{14}x=14336∘​

STEP 10

Calculate the value of xxx:x=24∘x = 24^\circx=24∘

STEP 11

Now that we have the value of xxx, we can find the measure of angle DDD by substituting xxx into the expression for ∠B\angle B∠B, since ∠B=∠D\angle B = \angle D∠B=∠D:∠D=5x\angle D = 5x∠D=5x

STEP 12

Substitute the value of xxx into the expression for ∠D\angle D∠D:∠D=5×24∘\angle D = 5 \times 24^\circ∠D=5×24∘

STEP 13

Calculate the measure of angle DDD:∠D=120∘\angle D = 120^\circ∠D=120∘The measure of angle DDD is 120∘120^\circ120∘.To check the answer, we can verify that the sum of all angles in the parallelogram adds up to 360∘360^\circ360∘.

STEP 14

Check the sum of the angles using the values of xxx and the expressions for ∠B\angle B∠B and ∠C\angle C∠C:∠A+∠B+∠C+∠D=(2x+12)+5x+(2x+12)+5x\angle A + \angle B + \angle C + \angle D = (2x + 12) + 5x + (2x + 12) + 5x∠A+∠B+∠C+∠D=(2x+12)+5x+(2x+12)+5x

STEP 15

STEP 16

Calculate the sum:∠A+∠B+∠C+∠D=(48∘+12)+120∘+(48∘+12)+120∘\angle A + \angle B + \angle C + \angle D = (48^\circ + 12) + 120^\circ + (48^\circ + 12) + 120^\circ∠A+∠B+∠C+∠D=(48∘+12)+120∘+(48∘+12)+120∘

STEP 17

Combine like terms:∠A+∠B+∠C+∠D=60∘+120∘+60∘+120∘\angle A + \angle B + \angle C + \angle D = 60^\circ + 120^\circ + 60^\circ + 120^\circ∠A+∠B+∠C+∠D=60∘+120∘+60∘+120∘

STEP 18

Calculate the total:∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ∠A+∠B+∠C+∠D=360∘Since the sum of the angles equals 360∘360^\circ360∘, our answer for ∠D\angle D∠D is confirmed to be correct.m∠D=120∘\mathrm{m} \angle D= 120^\circm∠D=120∘

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Assumptions
1. $ABCD$ is a parallelogram.
2. The measure of angle $B$ is $5x$ degrees.
3. The measure of angle $C$ is $2x + 12$ degrees.
4. The sum of the measures of angles in any parallelogram is 360 degrees.
5. Opposite angles in a parallelogram are equal.

Since opposite angles in a parallelogram are equal, we have:
$\angle A = \angle C$ $\angle B = \angle D$

Using the property that the sum of the measures of angles in a parallelogram is 360 degrees, we can write the equation:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$

Substitute the given expressions for $\angle B$ and $\angle C$ into the equation, and use the fact that $\angle A = \angle C$ and $\angle B = \angle D$ :
$\angle C + 5x + \angle C + 5x = 360^\circ$

Since $\angle C = 2x + 12$ , substitute this into the equation:
$(2x + 12) + 5x + (2x + 12) + 5x = 360^\circ$

Combine like terms:
$14x + 24 = 360^\circ$

Subtract 24 from both sides of the equation:
$14x = 360^\circ - 24$

Calculate the right side of the equation:
$14x = 336^\circ$

Divide both sides by 14 to solve for $x$ :
$x = \frac{336^\circ}{14}$

Calculate the value of $x$ :
$x = 24^\circ$

Now that we have the value of $x$ , we can find the measure of angle $D$ by substituting $x$ into the expression for $\angle B$ , since $\angle B = \angle D$ :
$\angle D = 5x$

Substitute the value of $x$ into the expression for $\angle D$ :
$\angle D = 5 \times 24^\circ$

Calculate the measure of angle $D$ :
$\angle D = 120^\circ$
The measure of angle $D$ is $120^\circ$ .
To check the answer, we can verify that the sum of all angles in the parallelogram adds up to $360^\circ$ .

Check the sum of the angles using the values of $x$ and the expressions for $\angle B$ and $\angle C$ :
$\angle A + \angle B + \angle C + \angle D = (2x + 12) + 5x + (2x + 12) + 5x$

Calculate the sum:
$\angle A + \angle B + \angle C + \angle D = (48^\circ + 12) + 120^\circ + (48^\circ + 12) + 120^\circ$

Combine like terms:
$\angle A + \angle B + \angle C + \angle D = 60^\circ + 120^\circ + 60^\circ + 120^\circ$

Calculate the total:
$\angle A + \angle B + \angle C + \angle D = 360^\circ$
Since the sum of the angles equals $360^\circ$ , our answer for $\angle D$ is confirmed to be correct.
$\mathrm{m} \angle D= 120^\circ$