Math / Geometry

QuestionIntroduction to parallel lines cut by a transversal 1) List all of the acute angles in the diagram: $\qquad$ 2) List all of the obtuse angles in the diagram: $\qquad$ 3) List all of the vertical angles in the diagram: $\qquad$ 4) List all of supplementary angles in the diagram. $\qquad$ Challenge: If the measure of angle 2 in the diagram above is $45^{\circ}$ , find the measure of all of the rest of the missing angles!

Studdy Solution

STEP 1

What is this asking? We're looking at two parallel lines crossed by another line, and we need to find different types of angles, and then figure out all the angle sizes if we know one of them! Watch out! Don't mix up acute and obtuse angles, and remember, supplementary angles add up to $180^\circ$ !

STEP 2

1. Angle Types
2. Angle Sizes

STEP 3

Let's define acute angles!
Acute angles are the little guys, less than $90^\circ$ !
Looking at our diagram, angles $\angle 2$ , $\angle 4$ , $\angle 6$ , and $\angle 8$ are our acute angles.

STEP 4

Now for obtuse angles!
These are the bigger angles, greater than $90^\circ$ .
In our diagram, angles $\angle 1$ , $\angle 3$ , $\angle 5$ , and $\angle 7$ are obtuse.

STEP 5

Vertical angles are like mirror images!
They're across from each other when two lines cross.
So, $\angle 1$ and $\angle 4$ are vertical angles.
Also, $\angle 2$ and $\angle 3$ , $\angle 5$ and $\angle 8$ , and $\angle 6$ and $\angle 7$ are vertical angle pairs!

STEP 6

Supplementary angles add up to $180^\circ$ !
They make a straight line together.
Lots of these here! $\angle 1$ and $\angle 2$ are supplementary, as are $\angle 3$ and $\angle 4$ , $\angle 5$ and $\angle 6$ , and $\angle 7$ and $\angle 8$ .
Also, angles next to each other on the same line are supplementary, like $\angle 1$ and $\angle 3$ , $\angle 2$ and $\angle 4$ , etc.

STEP 7

We know $\angle 2$ is $45^\circ$ .
Since $\angle 2$ and $\angle 4$ are vertical angles, $\angle 4$ is also $45^\circ$ !

STEP 8

$\angle 2$ and $\angle 1$ are supplementary, meaning they add up to $180^\circ$ .
So, $\angle 1 = 180^\circ - \angle 2 = 180^\circ - 45^\circ = 135^\circ$ .

STEP 9

Since $\angle 1$ and $\angle 3$ are vertical angles, $\angle 3$ is also $135^\circ$ !

STEP 10

Now, because the lines are parallel, corresponding angles are equal. $\angle 2$ and $\angle 6$ are corresponding angles, so $\angle 6 = 45^\circ$ .

STEP 11

$\angle 6$ and $\angle 8$ are vertical angles, so $\angle 8 = 45^\circ$ too!

STEP 12

$\angle 5$ and $\angle 6$ are supplementary, so $\angle 5 = 180^\circ - \angle 6 = 180^\circ - 45^\circ = 135^\circ$ .

STEP 13

Finally, $\angle 5$ and $\angle 7$ are vertical angles, meaning $\angle 7 = 135^\circ$ !

STEP 14

Acute angles: $\angle 2$ , $\angle 4$ , $\angle 6$ , and $\angle 8$ . Obtuse angles: $\angle 1$ , $\angle 3$ , $\angle 5$ , and $\angle 7$ . Vertical angles: $\angle 1$ and $\angle 4$ ; $\angle 2$ and $\angle 3$ ; $\angle 5$ and $\angle 8$ ; $\angle 6$ and $\angle 7$ . Supplementary angles: $\angle 1$ and $\angle 2$ ; $\angle 3$ and $\angle 4$ ; $\angle 5$ and $\angle 6$ ; $\angle 7$ and $\angle 8$ ; and many more pairs! Angle sizes: $\angle 1 = 135^\circ$ , $\angle 2 = 45^\circ$ , $\angle 3 = 135^\circ$ , $\angle 4 = 45^\circ$ , $\angle 5 = 135^\circ$ , $\angle 6 = 45^\circ$ , $\angle 7 = 135^\circ$ , $\angle 8 = 45^\circ$ .

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Studdy Solution

STEP 1

What is this asking? We're looking at two parallel lines crossed by another line, and we need to find different types of angles, and then figure out all the angle sizes if we know one of them! Watch out! Don't mix up acute and obtuse angles, and remember, supplementary angles add up to $180^\circ$ !

STEP 2

1. Angle Types
2. Angle Sizes

STEP 3

Let's define acute angles!
Acute angles are the little guys, less than $90^\circ$ !
Looking at our diagram, angles $\angle 2$ , $\angle 4$ , $\angle 6$ , and $\angle 8$ are our acute angles.

STEP 4

Now for obtuse angles!
These are the bigger angles, greater than $90^\circ$ .
In our diagram, angles $\angle 1$ , $\angle 3$ , $\angle 5$ , and $\angle 7$ are obtuse.

STEP 5

Vertical angles are like mirror images!
They're across from each other when two lines cross.
So, $\angle 1$ and $\angle 4$ are vertical angles.
Also, $\angle 2$ and $\angle 3$ , $\angle 5$ and $\angle 8$ , and $\angle 6$ and $\angle 7$ are vertical angle pairs!

STEP 6

STEP 7

We know $\angle 2$ is $45^\circ$ .
Since $\angle 2$ and $\angle 4$ are vertical angles, $\angle 4$ is also $45^\circ$ !

STEP 8

$\angle 2$ and $\angle 1$ are supplementary, meaning they add up to $180^\circ$ .
So, $\angle 1 = 180^\circ - \angle 2 = 180^\circ - 45^\circ = 135^\circ$ .

STEP 9

Since $\angle 1$ and $\angle 3$ are vertical angles, $\angle 3$ is also $135^\circ$ !

STEP 10

Now, because the lines are parallel, corresponding angles are equal. $\angle 2$ and $\angle 6$ are corresponding angles, so $\angle 6 = 45^\circ$ .

STEP 11

$\angle 6$ and $\angle 8$ are vertical angles, so $\angle 8 = 45^\circ$ too!

STEP 12

$\angle 5$ and $\angle 6$ are supplementary, so $\angle 5 = 180^\circ - \angle 6 = 180^\circ - 45^\circ = 135^\circ$ .

STEP 13

Finally, $\angle 5$ and $\angle 7$ are vertical angles, meaning $\angle 7 = 135^\circ$ !

STEP 14

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Studdy Solution

STEP 1

STEP 2

1. Angle Types2. Angle Sizes

STEP 3

Let's **define** acute angles!Acute angles are the little guys, less than 90∘90^\circ90∘!Looking at our diagram, angles ∠2\angle 2∠2, ∠4\angle 4∠4, ∠6\angle 6∠6, and ∠8\angle 8∠8 are our **acute angles**.

STEP 4

Now for **obtuse** angles!These are the bigger angles, greater than 90∘90^\circ90∘.In our diagram, angles ∠1\angle 1∠1, ∠3\angle 3∠3, ∠5\angle 5∠5, and ∠7\angle 7∠7 are **obtuse**.

STEP 5

STEP 6

STEP 7

We know ∠2\angle 2∠2 is 45∘45^\circ45∘.Since ∠2\angle 2∠2 and ∠4\angle 4∠4 are **vertical angles**, ∠4\angle 4∠4 is also 45∘45^\circ45∘!

STEP 8

∠2\angle 2∠2 and ∠1\angle 1∠1 are **supplementary**, meaning they add up to 180∘180^\circ180∘.So, ∠1=180∘−∠2=180∘−45∘=135∘\angle 1 = 180^\circ - \angle 2 = 180^\circ - 45^\circ = 135^\circ∠1=180∘−∠2=180∘−45∘=135∘.

STEP 9

Since ∠1\angle 1∠1 and ∠3\angle 3∠3 are **vertical angles**, ∠3\angle 3∠3 is also 135∘135^\circ135∘!

STEP 10

Now, because the lines are **parallel**, corresponding angles are equal. ∠2\angle 2∠2 and ∠6\angle 6∠6 are **corresponding angles**, so ∠6=45∘\angle 6 = 45^\circ∠6=45∘.

STEP 11

∠6\angle 6∠6 and ∠8\angle 8∠8 are **vertical angles**, so ∠8=45∘\angle 8 = 45^\circ∠8=45∘ too!

STEP 12

∠5\angle 5∠5 and ∠6\angle 6∠6 are **supplementary**, so ∠5=180∘−∠6=180∘−45∘=135∘\angle 5 = 180^\circ - \angle 6 = 180^\circ - 45^\circ = 135^\circ∠5=180∘−∠6=180∘−45∘=135∘.

STEP 13

Finally, ∠5\angle 5∠5 and ∠7\angle 7∠7 are **vertical angles**, meaning ∠7=135∘\angle 7 = 135^\circ∠7=135∘!

STEP 14

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1. Angle Types
2. Angle Sizes

Let's define acute angles!
Acute angles are the little guys, less than $90^\circ$ !
Looking at our diagram, angles $\angle 2$ , $\angle 4$ , $\angle 6$ , and $\angle 8$ are our acute angles.

Now for obtuse angles!
These are the bigger angles, greater than $90^\circ$ .
In our diagram, angles $\angle 1$ , $\angle 3$ , $\angle 5$ , and $\angle 7$ are obtuse.

We know $\angle 2$ is $45^\circ$ .
Since $\angle 2$ and $\angle 4$ are vertical angles, $\angle 4$ is also $45^\circ$ !

$\angle 2$ and $\angle 1$ are supplementary, meaning they add up to $180^\circ$ .
So, $\angle 1 = 180^\circ - \angle 2 = 180^\circ - 45^\circ = 135^\circ$ .

Since $\angle 1$ and $\angle 3$ are vertical angles, $\angle 3$ is also $135^\circ$ !

Now, because the lines are parallel, corresponding angles are equal. $\angle 2$ and $\angle 6$ are corresponding angles, so $\angle 6 = 45^\circ$ .

$\angle 6$ and $\angle 8$ are vertical angles, so $\angle 8 = 45^\circ$ too!

$\angle 5$ and $\angle 6$ are supplementary, so $\angle 5 = 180^\circ - \angle 6 = 180^\circ - 45^\circ = 135^\circ$ .

Finally, $\angle 5$ and $\angle 7$ are vertical angles, meaning $\angle 7 = 135^\circ$ !