Math / Geometry

Question4. If $\overrightarrow{B D}$ bisects $<A B C$ . Solve for x and find $m<$ $A B C . \mathrm{m}<A B C=5 x-8$ and the $\mathrm{m} \angle \mathrm{DBC}=57^{\circ}$

Studdy Solution

STEP 1

1. The angle $\angle ABC$ is bisected by $\overrightarrow{BD}$ .
2. The measure of $\angle ABC$ is given by $5x - 8$ .
3. The measure of $\angle DBC$ is $57^\circ$ .

STEP 2

1. Understand the relationship between the angles.
2. Set up an equation for $x$ .
3. Solve for $x$ .
4. Calculate $m\angle ABC$ .

STEP 3

Since $\overrightarrow{BD}$ bisects $\angle ABC$ , it means that $\angle ABD = \angle DBC$ .

STEP 4

Given that $\angle DBC = 57^\circ$ , and since $\angle ABD = \angle DBC$ , we have:
$\angle ABD = 57^\circ$

STEP 5

Since $\angle ABC$ is bisected, it means:
$\angle ABC = \angle ABD + \angle DBC$ $\angle ABC = 57^\circ + 57^\circ$ $\angle ABC = 114^\circ$

STEP 6

We know $m\angle ABC = 5x - 8$ . Set up the equation:
$5x - 8 = 114$

STEP 7

Solve for $x$ :
$5x - 8 = 114$ $5x = 114 + 8$ $5x = 122$ $x = \frac{122}{5}$ $x = 24.4$

STEP 8

Now, find $m\angle ABC$ using $x = 24.4$ :
$m\angle ABC = 5x - 8$ $m\angle ABC = 5(24.4) - 8$ $m\angle ABC = 122 - 8$ $m\angle ABC = 114^\circ$
The value of $x$ is $24.4$ and the measure of $\angle ABC$ is $114^\circ$ .

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Question4. If $\overrightarrow{B D}$ bisects $<A B C$ . Solve for x and find $m<$ $A B C . \mathrm{m}<A B C=5 x-8$ and the $\mathrm{m} \angle \mathrm{DBC}=57^{\circ}$

Studdy Solution

STEP 1

1. The angle $\angle ABC$ is bisected by $\overrightarrow{BD}$ .
2. The measure of $\angle ABC$ is given by $5x - 8$ .
3. The measure of $\angle DBC$ is $57^\circ$ .

STEP 2

1. Understand the relationship between the angles.
2. Set up an equation for $x$ .
3. Solve for $x$ .
4. Calculate $m\angle ABC$ .

STEP 3

Since $\overrightarrow{BD}$ bisects $\angle ABC$ , it means that $\angle ABD = \angle DBC$ .

STEP 4

Given that $\angle DBC = 57^\circ$ , and since $\angle ABD = \angle DBC$ , we have:
$\angle ABD = 57^\circ$

STEP 5

Since $\angle ABC$ is bisected, it means:
$\angle ABC = \angle ABD + \angle DBC$ $\angle ABC = 57^\circ + 57^\circ$ $\angle ABC = 114^\circ$

STEP 6

We know $m\angle ABC = 5x - 8$ . Set up the equation:
$5x - 8 = 114$

STEP 7

Solve for $x$ :
$5x - 8 = 114$ $5x = 114 + 8$ $5x = 122$ $x = \frac{122}{5}$ $x = 24.4$

STEP 8

Now, find $m\angle ABC$ using $x = 24.4$ :
$m\angle ABC = 5x - 8$ $m\angle ABC = 5(24.4) - 8$ $m\angle ABC = 122 - 8$ $m\angle ABC = 114^\circ$
The value of $x$ is $24.4$ and the measure of $\angle ABC$ is $114^\circ$ .

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Question4. If BDundefined\overrightarrow{B D}BD bisects <ABC<A B C<ABC. Solve for x and find m<m<m< ABC.m<ABC=5x−8A B C . \mathrm{m}<A B C=5 x-8ABC.m<ABC=5x−8 and the m∠DBC=57∘\mathrm{m} \angle \mathrm{DBC}=57^{\circ}m∠DBC=57∘

Studdy Solution

STEP 1

1. The angle ∠ABC \angle ABC ∠ABC is bisected by BDundefined \overrightarrow{BD} BD.2. The measure of ∠ABC \angle ABC ∠ABC is given by 5x−8 5x - 8 5x−8.3. The measure of ∠DBC \angle DBC ∠DBC is 57∘ 57^\circ 57∘.

STEP 2

1. Understand the relationship between the angles.2. Set up an equation for x x x.3. Solve for x x x.4. Calculate m∠ABC m\angle ABC m∠ABC.

STEP 3

Since BDundefined \overrightarrow{BD} BD bisects ∠ABC \angle ABC ∠ABC, it means that ∠ABD=∠DBC \angle ABD = \angle DBC ∠ABD=∠DBC.

STEP 4

Given that ∠DBC=57∘ \angle DBC = 57^\circ ∠DBC=57∘, and since ∠ABD=∠DBC \angle ABD = \angle DBC ∠ABD=∠DBC, we have:∠ABD=57∘ \angle ABD = 57^\circ ∠ABD=57∘

STEP 5

Since ∠ABC \angle ABC ∠ABC is bisected, it means:∠ABC=∠ABD+∠DBC \angle ABC = \angle ABD + \angle DBC ∠ABC=∠ABD+∠DBC ∠ABC=57∘+57∘ \angle ABC = 57^\circ + 57^\circ ∠ABC=57∘+57∘ ∠ABC=114∘ \angle ABC = 114^\circ ∠ABC=114∘

STEP 6

We know m∠ABC=5x−8 m\angle ABC = 5x - 8 m∠ABC=5x−8. Set up the equation:5x−8=114 5x - 8 = 114 5x−8=114

STEP 7

Solve for x x x:5x−8=114 5x - 8 = 114 5x−8=114 5x=114+8 5x = 114 + 8 5x=114+8 5x=122 5x = 122 5x=122 x=1225 x = \frac{122}{5} x=5122​ x=24.4 x = 24.4 x=24.4

STEP 8

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Question4. If $\overrightarrow{B D}$ bisects $<A B C$ . Solve for x and find $m<$ $A B C . \mathrm{m}<A B C=5 x-8$ and the $\mathrm{m} \angle \mathrm{DBC}=57^{\circ}$

1. The angle $\angle ABC$ is bisected by $\overrightarrow{BD}$ .
2. The measure of $\angle ABC$ is given by $5x - 8$ .
3. The measure of $\angle DBC$ is $57^\circ$ .

1. Understand the relationship between the angles.
2. Set up an equation for $x$ .
3. Solve for $x$ .
4. Calculate $m\angle ABC$ .

Since $\overrightarrow{BD}$ bisects $\angle ABC$ , it means that $\angle ABD = \angle DBC$ .

Given that $\angle DBC = 57^\circ$ , and since $\angle ABD = \angle DBC$ , we have:
$\angle ABD = 57^\circ$

Since $\angle ABC$ is bisected, it means:
$\angle ABC = \angle ABD + \angle DBC$ $\angle ABC = 57^\circ + 57^\circ$ $\angle ABC = 114^\circ$

We know $m\angle ABC = 5x - 8$ . Set up the equation:
$5x - 8 = 114$

Solve for $x$ :
$5x - 8 = 114$ $5x = 114 + 8$ $5x = 122$ $x = \frac{122}{5}$ $x = 24.4$