QuestionFind $m \angle E H F$ if $\overrightarrow{H G}$ bisects it, with $m \angle E H G=(15 x-19)^{\circ}$ and $m \angle G H F=(9 x+11)^{\circ}$ .

Studdy Solution

STEP 1

Assumptions1. Vector $\overrightarrow{H G}$ bisects $\angle H F$ . . The measure of angle $H G$ is given by $(15 x-19)^\circ$ .
3. The measure of angle $G H F$ is given by $(9 x+11)^\circ$ .
4. We need to find the measure of angle $H F$ .

STEP 2

Since $\overrightarrow{H G}$ bisects $\angle H F$ , we know that $\angle H G$ is equal to $\angle G H F$ . So, we can set the expressions for these angles equal to each other and solve for $x$ .
$(15 x-19)^\circ = (9 x+11)^\circ$

STEP 3

To solve for $x$ , first subtract $9x$ from both sides of the equation.
$(15 x -9 x) -19^\circ =11^\circ$

STEP 4

implify the left side of the equation.
$6x -19^\circ =11^\circ$

STEP 5

Next, add $19^\circ$ to both sides of the equation to isolate $x$ on one side.
$x =11^\circ +19^\circ$

STEP 6

Calculate the right side of the equation.
$6x =30^\circ$

STEP 7

Finally, divide both sides of the equation by6 to solve for $x$ .
$x = \frac{30^\circ}{6}$

STEP 8

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

STEP 9

Now that we have the value of $x$ , we can substitute it back into the expressions for $\angle H G$ and $\angle G H F$ to find the measure of these angles.
$m \angle H G = (15 \cdot5 -19)^\circ$ $m \angle G H F = (9 \cdot5 +11)^\circ$

STEP 10

Calculate the measure of $\angle H G$ and $\angle G H F$ .
$m \angle H G =56^\circ$ $m \angle G H F =56^\circ$

STEP 11

Since $\overrightarrow{H G}$ bisects $\angle H F$ , the measure of $\angle H F$ is twice the measure of $\angle H G$ or $\angle G H F$ . So, we can calculate the measure of $\angle H F$ as follows $m \angle H F = \cdot m \angle H G$

STEP 12

Substitute the measure of $\angle H G$ into the equation.
$m \angle H F =2 \cdot56^\circ$

STEP 13

Calculate the measure of $\angle H F$ .
$m \angle H F =112^\circ$ The measure of $\angle H F$ is $112^\circ$ .

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QuestionFind $m \angle E H F$ if $\overrightarrow{H G}$ bisects it, with $m \angle E H G=(15 x-19)^{\circ}$ and $m \angle G H F=(9 x+11)^{\circ}$ .

Studdy Solution

STEP 1

Assumptions1. Vector $\overrightarrow{H G}$ bisects $\angle H F$ . . The measure of angle $H G$ is given by $(15 x-19)^\circ$ .
3. The measure of angle $G H F$ is given by $(9 x+11)^\circ$ .
4. We need to find the measure of angle $H F$ .

STEP 2

Since $\overrightarrow{H G}$ bisects $\angle H F$ , we know that $\angle H G$ is equal to $\angle G H F$ . So, we can set the expressions for these angles equal to each other and solve for $x$ .
$(15 x-19)^\circ = (9 x+11)^\circ$

STEP 3

To solve for $x$ , first subtract $9x$ from both sides of the equation.
$(15 x -9 x) -19^\circ =11^\circ$

STEP 4

implify the left side of the equation.
$6x -19^\circ =11^\circ$

STEP 5

Next, add $19^\circ$ to both sides of the equation to isolate $x$ on one side.
$x =11^\circ +19^\circ$

STEP 6

Calculate the right side of the equation.
$6x =30^\circ$

STEP 7

Finally, divide both sides of the equation by6 to solve for $x$ .
$x = \frac{30^\circ}{6}$

STEP 8

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

STEP 9

Now that we have the value of $x$ , we can substitute it back into the expressions for $\angle H G$ and $\angle G H F$ to find the measure of these angles.
$m \angle H G = (15 \cdot5 -19)^\circ$ $m \angle G H F = (9 \cdot5 +11)^\circ$

STEP 10

Calculate the measure of $\angle H G$ and $\angle G H F$ .
$m \angle H G =56^\circ$ $m \angle G H F =56^\circ$

STEP 11

Since $\overrightarrow{H G}$ bisects $\angle H F$ , the measure of $\angle H F$ is twice the measure of $\angle H G$ or $\angle G H F$ . So, we can calculate the measure of $\angle H F$ as follows $m \angle H F = \cdot m \angle H G$

STEP 12

Substitute the measure of $\angle H G$ into the equation.
$m \angle H F =2 \cdot56^\circ$

STEP 13

Calculate the measure of $\angle H F$ .
$m \angle H F =112^\circ$ The measure of $\angle H F$ is $112^\circ$ .

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QuestionFind m∠EHFm \angle E H Fm∠EHF if HGundefined\overrightarrow{H G}HG bisects it, with m∠EHG=(15x−19)∘m \angle E H G=(15 x-19)^{\circ}m∠EHG=(15x−19)∘ and m∠GHF=(9x+11)∘m \angle G H F=(9 x+11)^{\circ}m∠GHF=(9x+11)∘.

Studdy Solution

STEP 1

STEP 2

STEP 3

To solve for xxx, first subtract 9x9x9x from both sides of the equation.(15x−9x)−19∘=11∘(15 x -9 x) -19^\circ =11^\circ(15x−9x)−19∘=11∘

STEP 4

implify the left side of the equation.6x−19∘=11∘6x -19^\circ =11^\circ6x−19∘=11∘

STEP 5

Next, add 19∘19^\circ19∘ to both sides of the equation to isolate xxx on one side.x=11∘+19∘x =11^\circ +19^\circx=11∘+19∘

STEP 6

Calculate the right side of the equation.6x=30∘6x =30^\circ6x=30∘

STEP 7

Finally, divide both sides of the equation by6 to solve for xxx.x=30∘6x = \frac{30^\circ}{6}x=630∘​

STEP 8

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

STEP 9

STEP 10

Calculate the measure of ∠HG\angle H G∠HG and ∠GHF\angle G H F∠GHF.m∠HG=56∘m \angle H G =56^\circm∠HG=56∘m∠GHF=56∘m \angle G H F =56^\circm∠GHF=56∘

STEP 11

STEP 12

Substitute the measure of ∠HG\angle H G∠HG into the equation.m∠HF=2⋅56∘m \angle H F =2 \cdot56^\circm∠HF=2⋅56∘

STEP 13

Calculate the measure of ∠HF\angle H F∠HF.m∠HF=112∘m \angle H F =112^\circm∠HF=112∘The measure of ∠HF\angle H F∠HF is 112∘112^\circ112∘.

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QuestionFind $m \angle E H F$ if $\overrightarrow{H G}$ bisects it, with $m \angle E H G=(15 x-19)^{\circ}$ and $m \angle G H F=(9 x+11)^{\circ}$ .

To solve for $x$ , first subtract $9x$ from both sides of the equation.
$(15 x -9 x) -19^\circ =11^\circ$

implify the left side of the equation.
$6x -19^\circ =11^\circ$

Next, add $19^\circ$ to both sides of the equation to isolate $x$ on one side.
$x =11^\circ +19^\circ$

Calculate the right side of the equation.
$6x =30^\circ$

Finally, divide both sides of the equation by6 to solve for $x$ .
$x = \frac{30^\circ}{6}$

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

Calculate the measure of $\angle H G$ and $\angle G H F$ .
$m \angle H G =56^\circ$ $m \angle G H F =56^\circ$

Substitute the measure of $\angle H G$ into the equation.
$m \angle H F =2 \cdot56^\circ$

Calculate the measure of $\angle H F$ .
$m \angle H F =112^\circ$ The measure of $\angle H F$ is $112^\circ$ .