Math

Question Find the measure of angle D in a parallelogram with mB=5x\mathrm{m} \angle B=5x and mC=2x+12\mathrm{m} \angle C=2x+12. Answer: mD=1805x(2x+12)=1807x12\mathrm{m} \angle D = 180 - 5x - (2x+12) = 180 - 7x - 12. To check, sum the angles of a parallelogram is 360 degrees.

Studdy Solution

STEP 1

Assumptions
1. ABCDABCD is a parallelogram.
2. The measure of angle BB is 5x5x degrees.
3. The measure of angle CC is 2x+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 B=D\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:
A+B+C+D=360\angle A + \angle B + \angle C + \angle D = 360^\circ

STEP 4

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

STEP 5

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

STEP 6

Combine like terms:
14x+24=36014x + 24 = 360^\circ

STEP 7

Subtract 24 from both sides of the equation:
14x=3602414x = 360^\circ - 24

STEP 8

Calculate the right side of the equation:
14x=33614x = 336^\circ

STEP 9

Divide both sides by 14 to solve for xx:
x=33614x = \frac{336^\circ}{14}

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

Calculate the measure of angle DD:
D=120\angle D = 120^\circ
The measure of angle DD is 120120^\circ.
To check the answer, we can verify that the sum of all angles in the parallelogram adds up to 360360^\circ.

STEP 14

Check the sum of the angles using the values of xx and the expressions for B\angle B and C\angle 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

STEP 15

Substitute the value of xx into the check equation:
A+B+C+D=(2×24+12)+5×24+(2×24+12)+5×24\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:
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

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

STEP 18

Calculate the total:
A+B+C+D=360\angle A + \angle B + \angle C + \angle D = 360^\circ
Since the sum of the angles equals 360360^\circ, our answer for D\angle D is confirmed to be correct.
mD=120\mathrm{m} \angle D= 120^\circ

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