QuestionAre triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is dilated and maps onto Triangle DEF. . The measure of angle A in Triangle ABC is30 degrees.
3. The measure of angle F in Triangle DEF is38 degrees.
4. The measure of angle D in Triangle DEF is30 degrees.

STEP 2

In order to determine if the triangles are similar, we need to check if the corresponding angles are equal. For two triangles to be similar, their corresponding angles must be equal and their corresponding sides must be in proportion.

STEP 3

We know that the measure of angle A in Triangle ABC is30 degrees and the measure of angle D in Triangle DEF is also30 degrees. So, angle A corresponds to angle D.
$\angle A = \angle D =30^{\circ}$

STEP 4

We also know that the measure of angle F in Triangle DEF is38 degrees. However, we do not know the measure of the corresponding angle in Triangle ABC. To find this, we need to use the fact that the sum of the angles in a triangle is180 degrees.

STEP 5

Let's denote the measure of angle B in Triangle ABC as x and the measure of angle C as y. We can write the equation for the sum of the angles in Triangle ABC.
$\angle A + \angle B + \angle C =180^{\circ}$

STEP 6

Substitute the known values into the equation.
$30^{\circ} + x + y =180^{\circ}$

STEP 7

implify the equation to find the sum of angles B and C.
$x + y =180^{\circ} -30^{\circ} =150^{\circ}$

STEP 8

Since the sum of angles D,, and F in Triangle DEF is also180 degrees, and we know that angle D is30 degrees and angle F is38 degrees, we can find the measure of angle.
$\angle D + \angle + \angle F =180^{\circ}$

STEP 9

Substitute the known values into the equation.
$30^{\circ} + \angle +38^{\circ} =180^{\circ}$

STEP 10

implify the equation to find the measure of angle.
$\angle =180^{\circ} -30^{\circ} -38^{\circ} =112^{\circ}$

STEP 11

Now we can see that the sum of angles B and C in Triangle ABC is150 degrees, and the measure of angle in Triangle DEF is112 degrees. Since these are not equal, the triangles are not similar by AA similarity.
The correct answer is (d) No, the triangles are not proportional.

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QuestionAre triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is dilated and maps onto Triangle DEF. . The measure of angle A in Triangle ABC is30 degrees.
3. The measure of angle F in Triangle DEF is38 degrees.
4. The measure of angle D in Triangle DEF is30 degrees.

STEP 2

In order to determine if the triangles are similar, we need to check if the corresponding angles are equal. For two triangles to be similar, their corresponding angles must be equal and their corresponding sides must be in proportion.

STEP 3

We know that the measure of angle A in Triangle ABC is30 degrees and the measure of angle D in Triangle DEF is also30 degrees. So, angle A corresponds to angle D.
$\angle A = \angle D =30^{\circ}$

STEP 4

We also know that the measure of angle F in Triangle DEF is38 degrees. However, we do not know the measure of the corresponding angle in Triangle ABC. To find this, we need to use the fact that the sum of the angles in a triangle is180 degrees.

STEP 5

Let's denote the measure of angle B in Triangle ABC as x and the measure of angle C as y. We can write the equation for the sum of the angles in Triangle ABC.
$\angle A + \angle B + \angle C =180^{\circ}$

STEP 6

Substitute the known values into the equation.
$30^{\circ} + x + y =180^{\circ}$

STEP 7

implify the equation to find the sum of angles B and C.
$x + y =180^{\circ} -30^{\circ} =150^{\circ}$

STEP 8

Since the sum of angles D,, and F in Triangle DEF is also180 degrees, and we know that angle D is30 degrees and angle F is38 degrees, we can find the measure of angle.
$\angle D + \angle + \angle F =180^{\circ}$

STEP 9

Substitute the known values into the equation.
$30^{\circ} + \angle +38^{\circ} =180^{\circ}$

STEP 10

implify the equation to find the measure of angle.
$\angle =180^{\circ} -30^{\circ} -38^{\circ} =112^{\circ}$

STEP 11

Now we can see that the sum of angles B and C in Triangle ABC is150 degrees, and the measure of angle in Triangle DEF is112 degrees. Since these are not equal, the triangles are not similar by AA similarity.
The correct answer is (d) No, the triangles are not proportional.

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QuestionAre triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF similar given ∠A=30∘\angle A=30^{\circ}∠A=30∘, ∠D=30∘\angle D=30^{\circ}∠D=30∘, ∠F=38∘\angle F=38^{\circ}∠F=38∘? Justify.

Studdy Solution

STEP 1

Assumptions1. Triangle ABC is dilated and maps onto Triangle DEF. . The measure of angle A in Triangle ABC is30 degrees.3. The measure of angle F in Triangle DEF is38 degrees.4. The measure of angle D in Triangle DEF is30 degrees.

STEP 2

In order to determine if the triangles are similar, we need to check if the corresponding angles are equal. For two triangles to be similar, their corresponding angles must be equal and their corresponding sides must be in proportion.

STEP 3

We know that the measure of angle A in Triangle ABC is30 degrees and the measure of angle D in Triangle DEF is also30 degrees. So, angle A corresponds to angle D.∠A=∠D=30∘\angle A = \angle D =30^{\circ}∠A=∠D=30∘

STEP 4

We also know that the measure of angle F in Triangle DEF is38 degrees. However, we do not know the measure of the corresponding angle in Triangle ABC. To find this, we need to use the fact that the sum of the angles in a triangle is180 degrees.

STEP 5

Let's denote the measure of angle B in Triangle ABC as x and the measure of angle C as y. We can write the equation for the sum of the angles in Triangle ABC.∠A+∠B+∠C=180∘\angle A + \angle B + \angle C =180^{\circ}∠A+∠B+∠C=180∘

STEP 6

Substitute the known values into the equation.30∘+x+y=180∘30^{\circ} + x + y =180^{\circ}30∘+x+y=180∘

STEP 7

implify the equation to find the sum of angles B and C.x+y=180∘−30∘=150∘x + y =180^{\circ} -30^{\circ} =150^{\circ}x+y=180∘−30∘=150∘

STEP 8

Since the sum of angles D,, and F in Triangle DEF is also180 degrees, and we know that angle D is30 degrees and angle F is38 degrees, we can find the measure of angle.∠D+∠+∠F=180∘\angle D + \angle + \angle F =180^{\circ}∠D+∠+∠F=180∘

STEP 9

Substitute the known values into the equation.30∘+∠+38∘=180∘30^{\circ} + \angle +38^{\circ} =180^{\circ}30∘+∠+38∘=180∘

STEP 10

implify the equation to find the measure of angle.∠=180∘−30∘−38∘=112∘\angle =180^{\circ} -30^{\circ} -38^{\circ} =112^{\circ}∠=180∘−30∘−38∘=112∘

STEP 11

Now we can see that the sum of angles B and C in Triangle ABC is150 degrees, and the measure of angle in Triangle DEF is112 degrees. Since these are not equal, the triangles are not similar by AA similarity.The correct answer is (d) No, the triangles are not proportional.

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

QuestionAre triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

Assumptions1. Triangle ABC is dilated and maps onto Triangle DEF. . The measure of angle A in Triangle ABC is30 degrees.
3. The measure of angle F in Triangle DEF is38 degrees.
4. The measure of angle D in Triangle DEF is30 degrees.

We know that the measure of angle A in Triangle ABC is30 degrees and the measure of angle D in Triangle DEF is also30 degrees. So, angle A corresponds to angle D.
$\angle A = \angle D =30^{\circ}$

Let's denote the measure of angle B in Triangle ABC as x and the measure of angle C as y. We can write the equation for the sum of the angles in Triangle ABC.
$\angle A + \angle B + \angle C =180^{\circ}$

Substitute the known values into the equation.
$30^{\circ} + x + y =180^{\circ}$

implify the equation to find the sum of angles B and C.
$x + y =180^{\circ} -30^{\circ} =150^{\circ}$

Since the sum of angles D,, and F in Triangle DEF is also180 degrees, and we know that angle D is30 degrees and angle F is38 degrees, we can find the measure of angle.
$\angle D + \angle + \angle F =180^{\circ}$

Substitute the known values into the equation.
$30^{\circ} + \angle +38^{\circ} =180^{\circ}$

implify the equation to find the measure of angle.
$\angle =180^{\circ} -30^{\circ} -38^{\circ} =112^{\circ}$

Now we can see that the sum of angles B and C in Triangle ABC is150 degrees, and the measure of angle in Triangle DEF is112 degrees. Since these are not equal, the triangles are not similar by AA similarity.
The correct answer is (d) No, the triangles are not proportional.