QuestionAlgebra: $\overrightarrow{B D}$ bisects $\angle A B C$ . Find $x$ and $m \angle A B C$ if $m \angle A B D=5x$ , $m \angle D B C=3x+10$ .

Studdy Solution

STEP 1

Assumptions1. Vector $\overrightarrow{B D}$ bisects $\angle A B C$ . . $m \angle A B D =5x$
3. $m \angle D B C =3x +10$

STEP 2

Since $\overrightarrow{B D}$ bisects $\angle A B C$ , it means that $m \angle A B D = m \angle D B C$ . We can set up the equation as follows $m \angle A B D = m \angle D B C$

STEP 3

Now, plug in the given values for $m \angle A B D$ and $m \angle D B C$ to set up the equation.
$5x =3x +10$

STEP 4

To solve for $x$ , we need to get all terms involving $x$ on one side of the equation and the constant term on the other side. We can do this by subtracting $3x$ from both sides of the equation.
$x -3x =3x +10 -3x$

STEP 5

implify the equation to find the value of $x$ .
$2x =10$

STEP 6

To solve for $x$ , divide both sides of the equation by2.
$x = \frac{10}{2}$

STEP 7

Calculate the value of $x$ .
$x =5$

STEP 8

Now that we have the value of $x$ , we can find the measure of $\angle A B C$ by adding $m \angle A B D$ and $m \angle D B C$ . $m \angle A B C = m \angle A B D + m \angle D B C$

STEP 9

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$m \angle A B C =5x +3x +$

STEP 10

Substitute the value of $x$ into the equation.
$m \angle A B C =5(5) +3(5) +10$

STEP 11

Calculate the measure of $\angle A B C$ .
$m \angle A B C =25 +15 +10 =50$ So, $x =5$ and $m \angle A B C =50$ degrees.

Was this helpful?

QuestionAlgebra: $\overrightarrow{B D}$ bisects $\angle A B C$ . Find $x$ and $m \angle A B C$ if $m \angle A B D=5x$ , $m \angle D B C=3x+10$ .

Studdy Solution

STEP 1

Assumptions1. Vector $\overrightarrow{B D}$ bisects $\angle A B C$ . . $m \angle A B D =5x$
3. $m \angle D B C =3x +10$

STEP 2

Since $\overrightarrow{B D}$ bisects $\angle A B C$ , it means that $m \angle A B D = m \angle D B C$ . We can set up the equation as follows $m \angle A B D = m \angle D B C$

STEP 3

Now, plug in the given values for $m \angle A B D$ and $m \angle D B C$ to set up the equation.
$5x =3x +10$

STEP 4

To solve for $x$ , we need to get all terms involving $x$ on one side of the equation and the constant term on the other side. We can do this by subtracting $3x$ from both sides of the equation.
$x -3x =3x +10 -3x$

STEP 5

implify the equation to find the value of $x$ .
$2x =10$

STEP 6

To solve for $x$ , divide both sides of the equation by2.
$x = \frac{10}{2}$

STEP 7

Calculate the value of $x$ .
$x =5$

STEP 8

Now that we have the value of $x$ , we can find the measure of $\angle A B C$ by adding $m \angle A B D$ and $m \angle D B C$ . $m \angle A B C = m \angle A B D + m \angle D B C$

STEP 9

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$m \angle A B C =5x +3x +$

STEP 10

Substitute the value of $x$ into the equation.
$m \angle A B C =5(5) +3(5) +10$

STEP 11

Calculate the measure of $\angle A B C$ .
$m \angle A B C =25 +15 +10 =50$ So, $x =5$ and $m \angle A B C =50$ degrees.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionAlgebra: BDundefined\overrightarrow{B D}BD bisects ∠ABC\angle A B C∠ABC. Find xxx and m∠ABCm \angle A B Cm∠ABC if m∠ABD=5xm \angle A B D=5xm∠ABD=5x, m∠DBC=3x+10m \angle D B C=3x+10m∠DBC=3x+10.

Studdy Solution

STEP 1

Assumptions1. Vector BDundefined\overrightarrow{B D}BD bisects ∠ABC\angle A B C∠ABC. . m∠ABD=5xm \angle A B D =5xm∠ABD=5x3. m∠DBC=3x+10m \angle D B C =3x +10m∠DBC=3x+10

STEP 2

Since BDundefined\overrightarrow{B D}BD bisects ∠ABC\angle A B C∠ABC, it means that m∠ABD=m∠DBCm \angle A B D = m \angle D B Cm∠ABD=m∠DBC. We can set up the equation as followsm∠ABD=m∠DBCm \angle A B D = m \angle D B Cm∠ABD=m∠DBC

STEP 3

Now, plug in the given values for m∠ABDm \angle A B Dm∠ABD and m∠DBCm \angle D B Cm∠DBC to set up the equation.5x=3x+105x =3x +105x=3x+10

STEP 4

To solve for xxx, we need to get all terms involving xxx on one side of the equation and the constant term on the other side. We can do this by subtracting 3x3x3x from both sides of the equation.x−3x=3x+10−3xx -3x =3x +10 -3xx−3x=3x+10−3x

STEP 5

implify the equation to find the value of xxx.2x=102x =102x=10

STEP 6

To solve for xxx, divide both sides of the equation by2.x=102x = \frac{10}{2}x=210​

STEP 7

Calculate the value of xxx.x=5x =5x=5

STEP 8

Now that we have the value of xxx, we can find the measure of ∠ABC\angle A B C∠ABC by adding m∠ABDm \angle A B Dm∠ABD and m∠DBCm \angle D B Cm∠DBC.m∠ABC=m∠ABD+m∠DBCm \angle A B C = m \angle A B D + m \angle D B Cm∠ABC=m∠ABD+m∠DBC

STEP 9

Substitute the given expressions for m∠ABDm \angle A B Dm∠ABD and m∠DBCm \angle D B Cm∠DBC into the equation.m∠ABC=5x+3x+m \angle A B C =5x +3x +m∠ABC=5x+3x+

STEP 10

Substitute the value of xxx into the equation.m∠ABC=5(5)+3(5)+10m \angle A B C =5(5) +3(5) +10m∠ABC=5(5)+3(5)+10

STEP 11

Calculate the measure of ∠ABC\angle A B C∠ABC.m∠ABC=25+15+10=50m \angle A B C =25 +15 +10 =50m∠ABC=25+15+10=50So, x=5x =5x=5 and m∠ABC=50m \angle A B C =50m∠ABC=50 degrees.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionAlgebra: $\overrightarrow{B D}$ bisects $\angle A B C$ . Find $x$ and $m \angle A B C$ if $m \angle A B D=5x$ , $m \angle D B C=3x+10$ .

Assumptions1. Vector $\overrightarrow{B D}$ bisects $\angle A B C$ . . $m \angle A B D =5x$
3. $m \angle D B C =3x +10$

Since $\overrightarrow{B D}$ bisects $\angle A B C$ , it means that $m \angle A B D = m \angle D B C$ . We can set up the equation as follows $m \angle A B D = m \angle D B C$

Now, plug in the given values for $m \angle A B D$ and $m \angle D B C$ to set up the equation.
$5x =3x +10$

To solve for $x$ , we need to get all terms involving $x$ on one side of the equation and the constant term on the other side. We can do this by subtracting $3x$ from both sides of the equation.
$x -3x =3x +10 -3x$

implify the equation to find the value of $x$ .
$2x =10$

To solve for $x$ , divide both sides of the equation by2.
$x = \frac{10}{2}$

Calculate the value of $x$ .
$x =5$

Now that we have the value of $x$ , we can find the measure of $\angle A B C$ by adding $m \angle A B D$ and $m \angle D B C$ . $m \angle A B C = m \angle A B D + m \angle D B C$

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$m \angle A B C =5x +3x +$

Substitute the value of $x$ into the equation.
$m \angle A B C =5(5) +3(5) +10$

Calculate the measure of $\angle A B C$ .
$m \angle A B C =25 +15 +10 =50$ So, $x =5$ and $m \angle A B C =50$ degrees.