Math / Geometry

Question4. Which equation could be used to find the value of $x$ in the triangle below? A. $12 x+20=180$ B. $12 x=180$ C. $12 x-20=180$ D. $8 x-10=4 x+10$
5. Find the $m \angle U L E$ in the triangle below.
6. In triangle $A B C, m \angle A=40^{\circ}$ and $m \angle B=80^{\circ}$ . Which of the following triangles is simitar to triangle $A B C$ ? A. $B$ . C. D.

Studdy Solution

STEP 1

What is this asking? We need to find an equation that helps us solve for $x$ in a triangle, and then use it to find the measure of one of the angles.
We also need to find a similar triangle to another given triangle. Watch out! Don't forget that the angles of a triangle always add up to $180$ degrees!
Also, similar triangles have the same angle measures, but not necessarily the same side lengths.

STEP 2

1. Find the Equation for $x$
2. Solve for $x$
3. Find the Measure of the Angle
4. Find the Similar Triangle

STEP 3

Alright, so we've got this triangle, and we know its angles are $(2x)^{\circ}$ , $(6x - 10)^{\circ}$ , and $(4x + 10)^{\circ}$ .
The sum of the angles in any triangle is always $180$ degrees.

STEP 4

So, let's add those angles together and set the sum equal to $180$ : $(2x) + (6x - 10) + (4x + 10) = 180$

STEP 5

Now, let's simplify!
Combine the $x$ terms: $2x + 6x + 4x = 12x$ .
And look, the $-10$ and $+10$ add to zero!
So, our equation becomes: $12x = 180$

STEP 6

We've got $12x = 180$ .
To isolate $x$ , we need to divide both sides of the equation by $12$ .
Remember, we're dividing by $12$ because $12$ divided by $12$ is $1$ , giving us $1 \cdot x$ or just $x$ .

STEP 7

So, $x = \frac{180}{12}$ .
Doing the division, we get $x = \textbf{15}$ .
Awesome!

STEP 8

The problem asks for the measure of $\angle ULE$ , which is the same as the angle $(6x - 10)^{\circ}$ .
Since we know $x = \textbf{15}$ , we can substitute that value into the expression: $6 \cdot (\textbf{15}) - 10$

STEP 9

Now, let's calculate: $6 \cdot 15 = 90$ , and $90 - 10 = \textbf{80}$ .
So, $m\angle ULE = \textbf{80}$ degrees!

STEP 10

We're given that in triangle $ABC$ , $m\angle A = 40^{\circ}$ and $m\angle B = 80^{\circ}$ .
Since the angles in a triangle add up to $180^{\circ}$ , we can find $m\angle C$ : $40 + 80 + m\angle C = 180$ $120 + m\angle C = 180$ $m\angle C = 180 - 120$ $m\angle C = 60^{\circ}$

STEP 11

Similar triangles have the same angle measures.
We're looking for a triangle with angles measuring $40^{\circ}$ , $80^{\circ}$ , and $60^{\circ}$ .
We'll need to check the given options to see which one matches these angles.
This part is left to the student to check against the provided image, as the image is not provided to me.

STEP 12

The equation to find $x$ is $12x = 180$ .
The value of $x$ is $15$ .
The measure of $\angle ULE$ is $80$ degrees.
The similar triangle will have angles measuring $40^{\circ}$ , $80^{\circ}$ , and $60^{\circ}$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the Equation for $x$
2. Solve for $x$
3. Find the Measure of the Angle
4. Find the Similar Triangle

STEP 3

Alright, so we've got this triangle, and we know its angles are $(2x)^{\circ}$ , $(6x - 10)^{\circ}$ , and $(4x + 10)^{\circ}$ .
The sum of the angles in any triangle is always $180$ degrees.

STEP 4

So, let's add those angles together and set the sum equal to $180$ : $(2x) + (6x - 10) + (4x + 10) = 180$

STEP 5

Now, let's simplify!
Combine the $x$ terms: $2x + 6x + 4x = 12x$ .
And look, the $-10$ and $+10$ add to zero!
So, our equation becomes: $12x = 180$

STEP 6

We've got $12x = 180$ .
To isolate $x$ , we need to divide both sides of the equation by $12$ .
Remember, we're dividing by $12$ because $12$ divided by $12$ is $1$ , giving us $1 \cdot x$ or just $x$ .

STEP 7

So, $x = \frac{180}{12}$ .
Doing the division, we get $x = \textbf{15}$ .
Awesome!

STEP 8

The problem asks for the measure of $\angle ULE$ , which is the same as the angle $(6x - 10)^{\circ}$ .
Since we know $x = \textbf{15}$ , we can substitute that value into the expression: $6 \cdot (\textbf{15}) - 10$

STEP 9

Now, let's calculate: $6 \cdot 15 = 90$ , and $90 - 10 = \textbf{80}$ .
So, $m\angle ULE = \textbf{80}$ degrees!

STEP 10

We're given that in triangle $ABC$ , $m\angle A = 40^{\circ}$ and $m\angle B = 80^{\circ}$ .
Since the angles in a triangle add up to $180^{\circ}$ , we can find $m\angle C$ : $40 + 80 + m\angle C = 180$ $120 + m\angle C = 180$ $m\angle C = 180 - 120$ $m\angle C = 60^{\circ}$

STEP 11

STEP 12

The equation to find $x$ is $12x = 180$ .
The value of $x$ is $15$ .
The measure of $\angle ULE$ is $80$ degrees.
The similar triangle will have angles measuring $40^{\circ}$ , $80^{\circ}$ , and $60^{\circ}$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the Equation for xxx2. Solve for xxx3. Find the Measure of the Angle4. Find the Similar Triangle

STEP 3

Alright, so we've got this triangle, and we know its angles are (2x)∘(2x)^{\circ}(2x)∘, (6x−10)∘(6x - 10)^{\circ}(6x−10)∘, and (4x+10)∘(4x + 10)^{\circ}(4x+10)∘.The sum of the angles in *any* triangle is always **180180180 degrees**.

STEP 4

So, let's **add** those angles together and **set** the sum equal to 180180180: (2x)+(6x−10)+(4x+10)=180(2x) + (6x - 10) + (4x + 10) = 180(2x)+(6x−10)+(4x+10)=180

STEP 5

Now, let's **simplify**!Combine the xxx terms: 2x+6x+4x=12x2x + 6x + 4x = 12x2x+6x+4x=12x.And look, the −10-10−10 and +10+10+10 add to zero!So, our equation becomes: 12x=18012x = 18012x=180

STEP 6

We've got 12x=18012x = 18012x=180.To **isolate** xxx, we need to **divide** both sides of the equation by **121212**.Remember, we're dividing by 121212 because 121212 divided by 121212 is 111, giving us 1⋅x1 \cdot x1⋅x or just xxx.

STEP 7

So, x=18012x = \frac{180}{12}x=12180​.Doing the division, we get x=15x = \textbf{15}x=15.Awesome!

STEP 8

The problem asks for the measure of ∠ULE\angle ULE∠ULE, which is the same as the angle (6x−10)∘(6x - 10)^{\circ}(6x−10)∘.Since we know x=15x = \textbf{15}x=15, we can **substitute** that value into the expression: 6⋅(15)−106 \cdot (\textbf{15}) - 106⋅(15)−10

STEP 9

Now, let's **calculate**: 6⋅15=906 \cdot 15 = 906⋅15=90, and 90−10=8090 - 10 = \textbf{80}90−10=80.So, m∠ULE=80m\angle ULE = \textbf{80}m∠ULE=80 degrees!

STEP 10

STEP 11

STEP 12

The equation to find xxx is 12x=18012x = 18012x=180.The value of xxx is 151515.The measure of ∠ULE\angle ULE∠ULE is 808080 degrees.The similar triangle will have angles measuring 40∘40^{\circ}40∘, 80∘80^{\circ}80∘, and 60∘60^{\circ}60∘.

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1. Find the Equation for $x$
2. Solve for $x$
3. Find the Measure of the Angle
4. Find the Similar Triangle

Alright, so we've got this triangle, and we know its angles are $(2x)^{\circ}$ , $(6x - 10)^{\circ}$ , and $(4x + 10)^{\circ}$ .
The sum of the angles in any triangle is always $180$ degrees.

So, let's add those angles together and set the sum equal to $180$ : $(2x) + (6x - 10) + (4x + 10) = 180$

Now, let's simplify!
Combine the $x$ terms: $2x + 6x + 4x = 12x$ .
And look, the $-10$ and $+10$ add to zero!
So, our equation becomes: $12x = 180$

We've got $12x = 180$ .
To isolate $x$ , we need to divide both sides of the equation by $12$ .
Remember, we're dividing by $12$ because $12$ divided by $12$ is $1$ , giving us $1 \cdot x$ or just $x$ .

So, $x = \frac{180}{12}$ .
Doing the division, we get $x = \textbf{15}$ .
Awesome!

The problem asks for the measure of $\angle ULE$ , which is the same as the angle $(6x - 10)^{\circ}$ .
Since we know $x = \textbf{15}$ , we can substitute that value into the expression: $6 \cdot (\textbf{15}) - 10$

Now, let's calculate: $6 \cdot 15 = 90$ , and $90 - 10 = \textbf{80}$ .
So, $m\angle ULE = \textbf{80}$ degrees!

The equation to find $x$ is $12x = 180$ .
The value of $x$ is $15$ .
The measure of $\angle ULE$ is $80$ degrees.
The similar triangle will have angles measuring $40^{\circ}$ , $80^{\circ}$ , and $60^{\circ}$ .