QuestionIn right triangle DOG, with $\angle D = 32^{\circ}$ and $OS$ bisecting $\angle DOG$ , find $\angle OSG$ . Options: a. $58^{\circ}$ b. $77^{\circ}$ c. $103^{\circ}$ d. $122^{\circ}$

Studdy Solution

STEP 1

Assumptions1. We have a right triangle DOG. . Line segment OS bisects angle DOG.
3. The measure of angle D is given as $32^{\circ}$ .
4. The sum of the interior angles of a triangle is $180^{\circ}$ .
5. We are asked to find the measure of angle OSG.

STEP 2

Since OS bisects angle DOG, it means that angle DOS is equal to angle SOG. Let's denote the measure of these angles as $x$ . So, we have $\angle DOS = \angle SOG = x$

STEP 3

Now, we know that the sum of the angles in a triangle is $180^{\circ}$ . So, in triangle DOG, we have $\angle DOG + \angle OGD + \angle GDO =180^{\circ}$

STEP 4

Substitute the given and derived values into the equation. Angle DOG is a right angle, so its measure is $90^{\circ}$ . Angle GDO is given as $32^{\circ}$ , and angle OGD is the sum of angles DOS and SOG, which are both $x$ .
$90^{\circ} +32^{\circ} +2x =180^{\circ}$

STEP 5

implify the equation to find the value of $x$ .
$2x =180^{\circ} -90^{\circ} -32^{\circ}$

STEP 6

Calculate the value on the right side of the equation.
$2x =58^{\circ}$

STEP 7

Divide each side of the equation by2 to solve for $x$ .
$x =58^{\circ} /2$

STEP 8

Calculate the value of $x$ .
$x =29^{\circ}$

STEP 9

Now that we have the value of $x$ , we can find the measure of angle OSG. Angle OSG is the sum of angles DOS and SOG, which are both $x$ .
$\angle OSG =2x$

STEP 10

Substitute the value of $x$ into the equation.
$\angle OSG =2 \\times29^{\circ}$

STEP 11

Calculate the measure of angle OSG.
$\angle OSG = \\times29^{\circ} =58^{\circ}$ So, the measure of angle OSG is $58^{\circ}$ .

Was this helpful?

QuestionIn right triangle DOG, with $\angle D = 32^{\circ}$ and $OS$ bisecting $\angle DOG$ , find $\angle OSG$ . Options: a. $58^{\circ}$ b. $77^{\circ}$ c. $103^{\circ}$ d. $122^{\circ}$

Studdy Solution

STEP 1

Assumptions1. We have a right triangle DOG. . Line segment OS bisects angle DOG.
3. The measure of angle D is given as $32^{\circ}$ .
4. The sum of the interior angles of a triangle is $180^{\circ}$ .
5. We are asked to find the measure of angle OSG.

STEP 2

Since OS bisects angle DOG, it means that angle DOS is equal to angle SOG. Let's denote the measure of these angles as $x$ . So, we have $\angle DOS = \angle SOG = x$

STEP 3

Now, we know that the sum of the angles in a triangle is $180^{\circ}$ . So, in triangle DOG, we have $\angle DOG + \angle OGD + \angle GDO =180^{\circ}$

STEP 4

Substitute the given and derived values into the equation. Angle DOG is a right angle, so its measure is $90^{\circ}$ . Angle GDO is given as $32^{\circ}$ , and angle OGD is the sum of angles DOS and SOG, which are both $x$ .
$90^{\circ} +32^{\circ} +2x =180^{\circ}$

STEP 5

implify the equation to find the value of $x$ .
$2x =180^{\circ} -90^{\circ} -32^{\circ}$

STEP 6

Calculate the value on the right side of the equation.
$2x =58^{\circ}$

STEP 7

Divide each side of the equation by2 to solve for $x$ .
$x =58^{\circ} /2$

STEP 8

Calculate the value of $x$ .
$x =29^{\circ}$

STEP 9

Now that we have the value of $x$ , we can find the measure of angle OSG. Angle OSG is the sum of angles DOS and SOG, which are both $x$ .
$\angle OSG =2x$

STEP 10

Substitute the value of $x$ into the equation.
$\angle OSG =2 \\times29^{\circ}$

STEP 11

Calculate the measure of angle OSG.
$\angle OSG = \\times29^{\circ} =58^{\circ}$ So, the measure of angle OSG is $58^{\circ}$ .

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.

QuestionIn right triangle DOG, with ∠D=32∘\angle D = 32^{\circ}∠D=32∘ and OSOSOS bisecting ∠DOG\angle DOG∠DOG, find ∠OSG\angle OSG∠OSG. Options: a. 58∘58^{\circ}58∘ b. 77∘77^{\circ}77∘ c. 103∘103^{\circ}103∘ d. 122∘122^{\circ}122∘

Studdy Solution

STEP 1

Assumptions1. We have a right triangle DOG. . Line segment OS bisects angle DOG.3. The measure of angle D is given as 32∘32^{\circ}32∘.4. The sum of the interior angles of a triangle is 180∘180^{\circ}180∘.5. We are asked to find the measure of angle OSG.

STEP 2

Since OS bisects angle DOG, it means that angle DOS is equal to angle SOG. Let's denote the measure of these angles as xxx. So, we have∠DOS=∠SOG=x\angle DOS = \angle SOG = x∠DOS=∠SOG=x

STEP 3

Now, we know that the sum of the angles in a triangle is 180∘180^{\circ}180∘. So, in triangle DOG, we have∠DOG+∠OGD+∠GDO=180∘\angle DOG + \angle OGD + \angle GDO =180^{\circ}∠DOG+∠OGD+∠GDO=180∘

STEP 4

STEP 5

implify the equation to find the value of xxx.2x=180∘−90∘−32∘2x =180^{\circ} -90^{\circ} -32^{\circ}2x=180∘−90∘−32∘

STEP 6

Calculate the value on the right side of the equation.2x=58∘2x =58^{\circ}2x=58∘

STEP 7

Divide each side of the equation by2 to solve for xxx.x=58∘/2x =58^{\circ} /2x=58∘/2

STEP 8

Calculate the value of xxx.x=29∘x =29^{\circ}x=29∘

STEP 9

Now that we have the value of xxx, we can find the measure of angle OSG. Angle OSG is the sum of angles DOS and SOG, which are both xxx.∠OSG=2x\angle OSG =2x∠OSG=2x

STEP 10

Substitute the value of xxx into the equation.∠OSG=2times29∘\angle OSG =2 \\times29^{\circ}∠OSG=2times29∘

STEP 11

Calculate the measure of angle OSG.∠OSG=times29∘=58∘\angle OSG = \\times29^{\circ} =58^{\circ}∠OSG=times29∘=58∘So, the measure of angle OSG is 58∘58^{\circ}58∘.

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.

QuestionIn right triangle DOG, with $\angle D = 32^{\circ}$ and $OS$ bisecting $\angle DOG$ , find $\angle OSG$ . Options: a. $58^{\circ}$ b. $77^{\circ}$ c. $103^{\circ}$ d. $122^{\circ}$

Assumptions1. We have a right triangle DOG. . Line segment OS bisects angle DOG.
3. The measure of angle D is given as $32^{\circ}$ .
4. The sum of the interior angles of a triangle is $180^{\circ}$ .
5. We are asked to find the measure of angle OSG.

Since OS bisects angle DOG, it means that angle DOS is equal to angle SOG. Let's denote the measure of these angles as $x$ . So, we have $\angle DOS = \angle SOG = x$

Now, we know that the sum of the angles in a triangle is $180^{\circ}$ . So, in triangle DOG, we have $\angle DOG + \angle OGD + \angle GDO =180^{\circ}$

implify the equation to find the value of $x$ .
$2x =180^{\circ} -90^{\circ} -32^{\circ}$

Calculate the value on the right side of the equation.
$2x =58^{\circ}$

Divide each side of the equation by2 to solve for $x$ .
$x =58^{\circ} /2$

Calculate the value of $x$ .
$x =29^{\circ}$

Now that we have the value of $x$ , we can find the measure of angle OSG. Angle OSG is the sum of angles DOS and SOG, which are both $x$ .
$\angle OSG =2x$

Substitute the value of $x$ into the equation.
$\angle OSG =2 \\times29^{\circ}$

Calculate the measure of angle OSG.
$\angle OSG = \\times29^{\circ} =58^{\circ}$ So, the measure of angle OSG is $58^{\circ}$ .