Math / Geometry

QuestionListen
Find the measure of each acute angle in a right triangle where the measure of one acute angle is 8 times the measure of the other acute angle.
The smaller acute angles measures $\square$ ${ }^{\circ}$ and the larger acute angle measures $\square$ ${ }^{\circ}$ .

Studdy Solution

STEP 1

What is this asking? We need to find the measurement of two angles in a right triangle, knowing one is 8 times bigger than the other. Watch out! Remember all the angles in a triangle add up to $180^\circ$ , and a right triangle has a $90^\circ$ angle.

STEP 2

1. Set up the equation
2. Solve for $x$
3. Calculate the larger angle

STEP 3

Let $x$ be the measure of the smaller angle.
Since the larger angle is 8 times the smaller angle, we can express it as $8x$ .

STEP 4

We know that the sum of the angles in any triangle is $180^\circ$ .
This triangle has a right angle ( $90^\circ$ ), a smaller angle ( $x$ ), and a larger angle ( $8x$ ).
So, let's write that down: $x + 8x + 90 = 180$ .

STEP 5

We can combine the $x$ terms: $x + 8x = 9x$ .
Our equation now looks like this: $9x + 90 = 180$ .

STEP 6

To isolate the term with $x$ , let's subtract 90 from both sides of the equation: $9x + 90 - 90 = 180 - 90$ .
This simplifies to $9x = 90$ .

STEP 7

Now, we divide both sides by 9 to find the value of $x$ : $\frac{9x}{9} = \frac{90}{9}$ .
This gives us $x = 10$ .
So the smaller angle is $10^\circ$ !

STEP 8

The larger angle is 8 times the smaller angle ( $x$ ), which we now know is $10^\circ$ .
So, the larger angle is $8 \cdot 10 = 80^\circ$ .

STEP 9

The smaller acute angle measures $10^\circ$ and the larger acute angle measures $80^\circ$ .

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QuestionListen
Find the measure of each acute angle in a right triangle where the measure of one acute angle is 8 times the measure of the other acute angle.
The smaller acute angles measures $\square$ ${ }^{\circ}$ and the larger acute angle measures $\square$ ${ }^{\circ}$ .

Studdy Solution

STEP 1

What is this asking? We need to find the measurement of two angles in a right triangle, knowing one is 8 times bigger than the other. Watch out! Remember all the angles in a triangle add up to $180^\circ$ , and a right triangle has a $90^\circ$ angle.

STEP 2

1. Set up the equation
2. Solve for $x$
3. Calculate the larger angle

STEP 3

Let $x$ be the measure of the smaller angle.
Since the larger angle is 8 times the smaller angle, we can express it as $8x$ .

STEP 4

We know that the sum of the angles in any triangle is $180^\circ$ .
This triangle has a right angle ( $90^\circ$ ), a smaller angle ( $x$ ), and a larger angle ( $8x$ ).
So, let's write that down: $x + 8x + 90 = 180$ .

STEP 5

We can combine the $x$ terms: $x + 8x = 9x$ .
Our equation now looks like this: $9x + 90 = 180$ .

STEP 6

To isolate the term with $x$ , let's subtract 90 from both sides of the equation: $9x + 90 - 90 = 180 - 90$ .
This simplifies to $9x = 90$ .

STEP 7

Now, we divide both sides by 9 to find the value of $x$ : $\frac{9x}{9} = \frac{90}{9}$ .
This gives us $x = 10$ .
So the smaller angle is $10^\circ$ !

STEP 8

The larger angle is 8 times the smaller angle ( $x$ ), which we now know is $10^\circ$ .
So, the larger angle is $8 \cdot 10 = 80^\circ$ .

STEP 9

The smaller acute angle measures $10^\circ$ and the larger acute angle measures $80^\circ$ .

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QuestionListenFind the measure of each acute angle in a right triangle where the measure of one acute angle is 8 times the measure of the other acute angle.The smaller acute angles measures □\square□ ∘{ }^{\circ}∘ and the larger acute angle measures □\square□ ∘{ }^{\circ}∘.

Studdy Solution

STEP 1

What is this asking? We need to find the measurement of two angles in a right triangle, knowing one is 8 times bigger than the other. Watch out! Remember all the angles in a triangle add up to 180∘180^\circ180∘, and a right triangle has a 90∘90^\circ90∘ angle.

STEP 2

1. Set up the equation2. Solve for xxx3. Calculate the larger angle

STEP 3

Let xxx be the measure of the smaller angle.Since the larger angle is **8 times** the smaller angle, we can express it as 8x8x8x.

STEP 4

We know that the sum of the angles in *any* triangle is 180∘180^\circ180∘.This triangle has a right angle (90∘90^\circ90∘), a smaller angle (xxx), and a larger angle (8x8x8x).So, let's write that down: x+8x+90=180x + 8x + 90 = 180x+8x+90=180.

STEP 5

We can combine the xxx terms: x+8x=9xx + 8x = 9xx+8x=9x.Our equation now looks like this: 9x+90=1809x + 90 = 1809x+90=180.

STEP 6

To isolate the term with xxx, let's subtract **90** from both sides of the equation: 9x+90−90=180−909x + 90 - 90 = 180 - 909x+90−90=180−90.This simplifies to 9x=909x = 909x=90.

STEP 7

Now, we divide both sides by **9** to find the value of xxx: 9x9=909\frac{9x}{9} = \frac{90}{9}99x​=990​.This gives us x=10x = 10x=10.So the smaller angle is 10∘10^\circ10∘!

STEP 8

The larger angle is 8 times the smaller angle (xxx), which we now know is 10∘10^\circ10∘.So, the larger angle is 8⋅10=80∘8 \cdot 10 = 80^\circ8⋅10=80∘.

STEP 9

The smaller acute angle measures 10∘10^\circ10∘ and the larger acute angle measures 80∘80^\circ80∘.

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Start learning now

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

QuestionListen
Find the measure of each acute angle in a right triangle where the measure of one acute angle is 8 times the measure of the other acute angle.
The smaller acute angles measures $\square$ ${ }^{\circ}$ and the larger acute angle measures $\square$ ${ }^{\circ}$ .

What is this asking? We need to find the measurement of two angles in a right triangle, knowing one is 8 times bigger than the other. Watch out! Remember all the angles in a triangle add up to $180^\circ$ , and a right triangle has a $90^\circ$ angle.

1. Set up the equation
2. Solve for $x$
3. Calculate the larger angle

Let $x$ be the measure of the smaller angle.
Since the larger angle is 8 times the smaller angle, we can express it as $8x$ .

We know that the sum of the angles in any triangle is $180^\circ$ .
This triangle has a right angle ( $90^\circ$ ), a smaller angle ( $x$ ), and a larger angle ( $8x$ ).
So, let's write that down: $x + 8x + 90 = 180$ .

We can combine the $x$ terms: $x + 8x = 9x$ .
Our equation now looks like this: $9x + 90 = 180$ .

To isolate the term with $x$ , let's subtract 90 from both sides of the equation: $9x + 90 - 90 = 180 - 90$ .
This simplifies to $9x = 90$ .

Now, we divide both sides by 9 to find the value of $x$ : $\frac{9x}{9} = \frac{90}{9}$ .
This gives us $x = 10$ .
So the smaller angle is $10^\circ$ !

The larger angle is 8 times the smaller angle ( $x$ ), which we now know is $10^\circ$ .
So, the larger angle is $8 \cdot 10 = 80^\circ$ .

The smaller acute angle measures $10^\circ$ and the larger acute angle measures $80^\circ$ .