Math / Geometry

Question1
In the picture below, line PQ is parallel to line RS, and the lines are cut by a transversal, line TU. The transversal is not perpendicular to the parallel lines.
Aora: Fegtre is not dram to scale
Which of the following are congruent angles? A. $\angle X \cong \angle G$ B. $\angle \mathrm{X} \cong \angle \mathrm{Y}$ c. $\angle \mathrm{X} \cong \angle \mathrm{E}$ D. $\angle X \cong \angle F$

Studdy Solution

STEP 1

What is this asking? Which angles are the same when a line crosses two parallel lines? Watch out! Don't mix up the different types of angles: corresponding, alternate interior, and vertical!

STEP 2

1. Angle Relationships
2. Find Congruent Angles

STEP 3

Alright, let's break this down!
We've got two parallel lines, PQ and RS, and they're crossed by another line, TU, called a transversal.
This creates a bunch of angles, and we need to figure out which ones are the same, or as we mathematicians like to say, congruent.

STEP 4

When a transversal crosses parallel lines, it forms several special angle pairs.
Let's review them! Corresponding angles are in matching corners. Alternate interior angles are on opposite sides of the transversal and inside the parallel lines.
And vertical angles are across from each other when two lines intersect.

STEP 5

Let's look at $\angle X$ . $\angle Y$ is a vertical angle to $\angle X$ , so they're congruent.
That means $\angle X \cong \angle Y$ .
Boom! We found one pair.

STEP 6

Now, $\angle G$ is next to $\angle X$ on line PQ.
These are adjacent angles, and they add up to $180^\circ$ since they form a straight line.
Since the transversal isn't perpendicular to the parallel lines, $\angle X$ and $\angle G$ can't be congruent.

STEP 7

$\angle E$ is in the same position on the other parallel line as $\angle X$ is on line PQ.
These are corresponding angles, and since the lines are parallel, corresponding angles are congruent!
So, $\angle X \cong \angle E$ .
Another pair down!

STEP 8

Finally, $\angle F$ is a vertical angle to $\angle E$ .
Since $\angle E$ is congruent to $\angle X$ , and $\angle F$ is congruent to $\angle E$ , then $\angle F$ is also congruent to $\angle X$ !
So, $\angle X \cong \angle F$ .

STEP 9

$\angle X$ is congruent to $\angle Y$ , $\angle E$ , and $\angle F$ .
The correct answers are B, C, and D.

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Studdy Solution

STEP 1

What is this asking? Which angles are the same when a line crosses two parallel lines? Watch out! Don't mix up the different types of angles: corresponding, alternate interior, and vertical!

STEP 2

1. Angle Relationships
2. Find Congruent Angles

STEP 3

Alright, let's break this down!
We've got two parallel lines, PQ and RS, and they're crossed by another line, TU, called a transversal.
This creates a bunch of angles, and we need to figure out which ones are the same, or as we mathematicians like to say, congruent.

STEP 4

STEP 5

Let's look at $\angle X$ . $\angle Y$ is a vertical angle to $\angle X$ , so they're congruent.
That means $\angle X \cong \angle Y$ .
Boom! We found one pair.

STEP 6

Now, $\angle G$ is next to $\angle X$ on line PQ.
These are adjacent angles, and they add up to $180^\circ$ since they form a straight line.
Since the transversal isn't perpendicular to the parallel lines, $\angle X$ and $\angle G$ can't be congruent.

STEP 7

$\angle E$ is in the same position on the other parallel line as $\angle X$ is on line PQ.
These are corresponding angles, and since the lines are parallel, corresponding angles are congruent!
So, $\angle X \cong \angle E$ .
Another pair down!

STEP 8

Finally, $\angle F$ is a vertical angle to $\angle E$ .
Since $\angle E$ is congruent to $\angle X$ , and $\angle F$ is congruent to $\angle E$ , then $\angle F$ is also congruent to $\angle X$ !
So, $\angle X \cong \angle F$ .

STEP 9

$\angle X$ is congruent to $\angle Y$ , $\angle E$ , and $\angle F$ .
The correct answers are B, C, and D.

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Studdy Solution

STEP 1

What is this asking? Which angles are the same when a line crosses two parallel lines? Watch out! Don't mix up the different types of angles: **corresponding**, **alternate interior**, and **vertical**!

STEP 2

1. Angle Relationships2. Find Congruent Angles

STEP 3

Alright, let's break this down!We've got two parallel lines, PQ and RS, and they're crossed by another line, TU, called a transversal.This creates a bunch of angles, and we need to figure out which ones are the same, or as we mathematicians like to say, *congruent*.

STEP 4

STEP 5

Let's look at ∠X\angle X∠X. ∠Y\angle Y∠Y is a **vertical angle** to ∠X\angle X∠X, so they're congruent.That means ∠X≅∠Y\angle X \cong \angle Y∠X≅∠Y.Boom! We found one pair.

STEP 6

Now, ∠G\angle G∠G is next to ∠X\angle X∠X on line PQ.These are **adjacent angles**, and they add up to 180∘180^\circ180∘ since they form a straight line.Since the transversal isn't perpendicular to the parallel lines, ∠X\angle X∠X and ∠G\angle G∠G can't be congruent.

STEP 7

∠E\angle E∠E is in the same position on the other parallel line as ∠X\angle X∠X is on line PQ.These are **corresponding angles**, and since the lines are parallel, corresponding angles are congruent!So, ∠X≅∠E\angle X \cong \angle E∠X≅∠E.Another pair down!

STEP 8

Finally, ∠F\angle F∠F is a **vertical angle** to ∠E\angle E∠E.Since ∠E\angle E∠E is congruent to ∠X\angle X∠X, and ∠F\angle F∠F is congruent to ∠E\angle E∠E, then ∠F\angle F∠F is also congruent to ∠X\angle X∠X!So, ∠X≅∠F\angle X \cong \angle F∠X≅∠F.

STEP 9

∠X\angle X∠X is congruent to ∠Y\angle Y∠Y, ∠E\angle E∠E, and ∠F\angle F∠F.The correct answers are B, C, and D.

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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.

What is this asking? Which angles are the same when a line crosses two parallel lines? Watch out! Don't mix up the different types of angles: corresponding, alternate interior, and vertical!

1. Angle Relationships
2. Find Congruent Angles

Alright, let's break this down!
We've got two parallel lines, PQ and RS, and they're crossed by another line, TU, called a transversal.
This creates a bunch of angles, and we need to figure out which ones are the same, or as we mathematicians like to say, congruent.

Let's look at $\angle X$ . $\angle Y$ is a vertical angle to $\angle X$ , so they're congruent.
That means $\angle X \cong \angle Y$ .
Boom! We found one pair.

Now, $\angle G$ is next to $\angle X$ on line PQ.
These are adjacent angles, and they add up to $180^\circ$ since they form a straight line.
Since the transversal isn't perpendicular to the parallel lines, $\angle X$ and $\angle G$ can't be congruent.

$\angle E$ is in the same position on the other parallel line as $\angle X$ is on line PQ.
These are corresponding angles, and since the lines are parallel, corresponding angles are congruent!
So, $\angle X \cong \angle E$ .
Another pair down!

Finally, $\angle F$ is a vertical angle to $\angle E$ .
Since $\angle E$ is congruent to $\angle X$ , and $\angle F$ is congruent to $\angle E$ , then $\angle F$ is also congruent to $\angle X$ !
So, $\angle X \cong \angle F$ .

$\angle X$ is congruent to $\angle Y$ , $\angle E$ , and $\angle F$ .
The correct answers are B, C, and D.