Question $\overrightarrow{\mathrm{BD}}$ bisects $\angle ABC$ . If $m \angle ABD=(3x+15)^{\circ}$ and $m \angle DBC=(5x-12)^{\circ}$ , find $m \angle ABC$ .

Studdy Solution

STEP 1

Assumptions1. $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ . $m \angle A B D=(3 x+15)^{\circ}$
3. $m \angle D B C=(5 x-12)^{\circ}$

STEP 2

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B D$ is equal to the measure of $\angle D B C$ .
$m \angle A B D = m \angle D B C$

STEP 3

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$(3 x+15)^{\circ} = (5 x-12)^{\circ}$

STEP 4

olve the equation for $x$ .
$3x -x = -15 - (-12)$

STEP 5

implify the equation.
$-2x = -3$

STEP 6

Divide both sides of the equation by -2 to solve for $x$ .
$x = \frac{-3}{-2} =1.5$

STEP 7

Now that we have the value of $x$ , we can find the measure of $\angle A B D$ or $\angle D B C$ by substituting $x =1.5$ into either of the given expressions. Let's use $m \angle A B D = (3 x+15)^{\circ}$ .
$m \angle A B D =3(1.5) +15$

STEP 8

Calculate the measure of $\angle A B D$ .
$m \angle A B D =4.5 +15 =19.5^{\circ}$

STEP 9

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B C$ is twice the measure of $\angle A B D$ .
$m \angle A B C =2 \times m \angle A B D$

STEP 10

Substitute the measure of $\angle A B D$ into the equation.
$m \angle A B C =2 \times19.5^{\circ}$

STEP 11

Calculate the measure of $\angle A B C$ .
$m \angle A B C = \times19.5^{\circ} =39^{\circ}$ The measure of $\angle A B C$ is $39^{\circ}$ .

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Question $\overrightarrow{\mathrm{BD}}$ bisects $\angle ABC$ . If $m \angle ABD=(3x+15)^{\circ}$ and $m \angle DBC=(5x-12)^{\circ}$ , find $m \angle ABC$ .

Studdy Solution

STEP 1

Assumptions1. $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ . $m \angle A B D=(3 x+15)^{\circ}$
3. $m \angle D B C=(5 x-12)^{\circ}$

STEP 2

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B D$ is equal to the measure of $\angle D B C$ .
$m \angle A B D = m \angle D B C$

STEP 3

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$(3 x+15)^{\circ} = (5 x-12)^{\circ}$

STEP 4

olve the equation for $x$ .
$3x -x = -15 - (-12)$

STEP 5

implify the equation.
$-2x = -3$

STEP 6

Divide both sides of the equation by -2 to solve for $x$ .
$x = \frac{-3}{-2} =1.5$

STEP 7

Now that we have the value of $x$ , we can find the measure of $\angle A B D$ or $\angle D B C$ by substituting $x =1.5$ into either of the given expressions. Let's use $m \angle A B D = (3 x+15)^{\circ}$ .
$m \angle A B D =3(1.5) +15$

STEP 8

Calculate the measure of $\angle A B D$ .
$m \angle A B D =4.5 +15 =19.5^{\circ}$

STEP 9

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B C$ is twice the measure of $\angle A B D$ .
$m \angle A B C =2 \times m \angle A B D$

STEP 10

Substitute the measure of $\angle A B D$ into the equation.
$m \angle A B C =2 \times19.5^{\circ}$

STEP 11

Calculate the measure of $\angle A B C$ .
$m \angle A B C = \times19.5^{\circ} =39^{\circ}$ The measure of $\angle A B C$ is $39^{\circ}$ .

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QuestionBDundefined\overrightarrow{\mathrm{BD}}BD bisects ∠ABC\angle ABC∠ABC. If m∠ABD=(3x+15)∘m \angle ABD=(3x+15)^{\circ}m∠ABD=(3x+15)∘ and m∠DBC=(5x−12)∘m \angle DBC=(5x-12)^{\circ}m∠DBC=(5x−12)∘, find m∠ABCm \angle ABCm∠ABC.

Studdy Solution

STEP 1

Assumptions1. BDundefined\overrightarrow{\mathrm{BD}}BD bisects ∠ABC\angle A B C∠ABC . m∠ABD=(3x+15)∘m \angle A B D=(3 x+15)^{\circ}m∠ABD=(3x+15)∘3. m∠DBC=(5x−12)∘m \angle D B C=(5 x-12)^{\circ}m∠DBC=(5x−12)∘

STEP 2

Since BDundefined\overrightarrow{\mathrm{BD}}BD bisects ∠ABC\angle A B C∠ABC, the measure of ∠ABD\angle A B D∠ABD is equal to the measure of ∠DBC\angle D B C∠DBC.m∠ABD=m∠DBCm \angle A B D = m \angle D B Cm∠ABD=m∠DBC

STEP 3

Substitute the given expressions for m∠ABDm \angle A B Dm∠ABD and m∠DBCm \angle D B Cm∠DBC into the equation.(3x+15)∘=(5x−12)∘(3 x+15)^{\circ} = (5 x-12)^{\circ}(3x+15)∘=(5x−12)∘

STEP 4

olve the equation for xxx.3x−x=−15−(−12)3x -x = -15 - (-12)3x−x=−15−(−12)

STEP 5

implify the equation.−2x=−3-2x = -3−2x=−3

STEP 6

Divide both sides of the equation by -2 to solve for xxx.x=−3−2=1.5x = \frac{-3}{-2} =1.5x=−2−3​=1.5

STEP 7

STEP 8

Calculate the measure of ∠ABD\angle A B D∠ABD.m∠ABD=4.5+15=19.5∘m \angle A B D =4.5 +15 =19.5^{\circ}m∠ABD=4.5+15=19.5∘

STEP 9

Since BDundefined\overrightarrow{\mathrm{BD}}BD bisects ∠ABC\angle A B C∠ABC, the measure of ∠ABC\angle A B C∠ABC is twice the measure of ∠ABD\angle A B D∠ABD.m∠ABC=2×m∠ABDm \angle A B C =2 \times m \angle A B Dm∠ABC=2×m∠ABD

STEP 10

Substitute the measure of ∠ABD\angle A B D∠ABD into the equation.m∠ABC=2×19.5∘m \angle A B C =2 \times19.5^{\circ}m∠ABC=2×19.5∘

STEP 11

Calculate the measure of ∠ABC\angle A B C∠ABC.m∠ABC=×19.5∘=39∘m \angle A B C = \times19.5^{\circ} =39^{\circ}m∠ABC=×19.5∘=39∘The measure of ∠ABC\angle A B C∠ABC is 39∘39^{\circ}39∘.

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Question $\overrightarrow{\mathrm{BD}}$ bisects $\angle ABC$ . If $m \angle ABD=(3x+15)^{\circ}$ and $m \angle DBC=(5x-12)^{\circ}$ , find $m \angle ABC$ .

Assumptions1. $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ . $m \angle A B D=(3 x+15)^{\circ}$
3. $m \angle D B C=(5 x-12)^{\circ}$

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B D$ is equal to the measure of $\angle D B C$ .
$m \angle A B D = m \angle D B C$

Substitute the given expressions for $m \angle A B D$ and $m \angle D B C$ into the equation.
$(3 x+15)^{\circ} = (5 x-12)^{\circ}$

olve the equation for $x$ .
$3x -x = -15 - (-12)$

implify the equation.
$-2x = -3$

Divide both sides of the equation by -2 to solve for $x$ .
$x = \frac{-3}{-2} =1.5$

Calculate the measure of $\angle A B D$ .
$m \angle A B D =4.5 +15 =19.5^{\circ}$

Since $\overrightarrow{\mathrm{BD}}$ bisects $\angle A B C$ , the measure of $\angle A B C$ is twice the measure of $\angle A B D$ .
$m \angle A B C =2 \times m \angle A B D$

Substitute the measure of $\angle A B D$ into the equation.
$m \angle A B C =2 \times19.5^{\circ}$

Calculate the measure of $\angle A B C$ .
$m \angle A B C = \times19.5^{\circ} =39^{\circ}$ The measure of $\angle A B C$ is $39^{\circ}$ .