Math / Geometry

Question16. In the figure below, lines $A$ and $B$ are parallel.
Find the $m \angle 5$ if $m \angle 1=75^{\circ}$ and $m \angle 3=40^{\circ}$ . A. $34^{\circ}, 117^{\circ}$ and $29^{\circ}$ B. $31^{\circ}, 106^{\circ}$ and $43^{\circ}$ C. $37^{\circ}, 128^{\circ}$ and $15^{\circ}$ D. $29^{\circ}, 84^{\circ}$ and $67^{\circ}$

Studdy Solution

STEP 1

What is this asking? We need to find the measure of angle 5, given two other angles formed by a line crossing two parallel lines. Watch out! Don't mix up corresponding and alternate interior angles!
Also, remember parallel lines create special angle relationships.

STEP 2

1. Find angle 4
2. Find angle 2
3. Find angle 5

STEP 3

Angles 1, 2, 3, and 4 make a straight line, so they add up to $180^\circ$ .
We know $\angle 1 = 75^\circ$ and $\angle 3 = 40^\circ$ , so $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 180^\circ$ .

STEP 4

Since lines A and B are parallel, angles 1 and 2 are consecutive interior angles, meaning they add up to $180^\circ$ .
So, $\angle 1 + \angle 2 = 180^\circ$ , and since $\angle 1 = 75^\circ$ , we have $75^\circ + \angle 2 = 180^\circ$ .

STEP 5

Subtracting $75^\circ$ from both sides gives us $\angle 2 = 180^\circ - 75^\circ = 105^\circ$ .
So, $\angle 2 = \mathbf{105^\circ}$ .

STEP 6

Similarly, since lines A and B are parallel, angles 3 and 4 are also consecutive interior angles, meaning $\angle 3 + \angle 4 = 180^\circ$ .
Since $\angle 3 = 40^\circ$ , we have $40^\circ + \angle 4 = 180^\circ$ .

STEP 7

Subtracting $40^\circ$ from both sides gives us $\angle 4 = 180^\circ - 40^\circ = 140^\circ$ .
So, $\angle 4 = \mathbf{140^\circ}$ .

STEP 8

We already found $\angle 2$ in the previous step!
Remember, we found that $\angle 2 = \mathbf{105^\circ}$ .

STEP 9

Angles 2, 4, and 5 form a straight line, so they add up to $180^\circ$ .
That means $\angle 2 + \angle 4 + \angle 5 = 180^\circ$ .

STEP 10

We know $\angle 2 = 105^\circ$ and $\angle 4 = 140^\circ$ , so we can plug those values into our equation: $105^\circ + 140^\circ + \angle 5 = 180^\circ$ .

STEP 11

Adding the angles we know gives us $245^\circ + \angle 5 = 180^\circ$ .
Oops, that's not right!
Angles 4 and 2 are on one side of the transversal, and add up to $105^\circ + 140^\circ = 245^\circ$ .
Angle 5 is on the other side of the transversal.
Let's try another approach.

STEP 12

Angle 1 and angle 5 are corresponding angles, which means they are congruent.
Since $\angle 1 = 75^\circ$ , then $\angle 5$ must also be $75^\circ$ .
So, $\angle 5 = \mathbf{75^\circ}$ .

STEP 13

The measure of angle 5 is $75^\circ$ .
None of the answer choices are correct!

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Studdy Solution

STEP 1

What is this asking? We need to find the measure of angle 5, given two other angles formed by a line crossing two parallel lines. Watch out! Don't mix up corresponding and alternate interior angles!
Also, remember parallel lines create special angle relationships.

STEP 2

1. Find angle 4
2. Find angle 2
3. Find angle 5

STEP 3

Angles 1, 2, 3, and 4 make a straight line, so they add up to $180^\circ$ .
We know $\angle 1 = 75^\circ$ and $\angle 3 = 40^\circ$ , so $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 180^\circ$ .

STEP 4

Since lines A and B are parallel, angles 1 and 2 are consecutive interior angles, meaning they add up to $180^\circ$ .
So, $\angle 1 + \angle 2 = 180^\circ$ , and since $\angle 1 = 75^\circ$ , we have $75^\circ + \angle 2 = 180^\circ$ .

STEP 5

Subtracting $75^\circ$ from both sides gives us $\angle 2 = 180^\circ - 75^\circ = 105^\circ$ .
So, $\angle 2 = \mathbf{105^\circ}$ .

STEP 6

Similarly, since lines A and B are parallel, angles 3 and 4 are also consecutive interior angles, meaning $\angle 3 + \angle 4 = 180^\circ$ .
Since $\angle 3 = 40^\circ$ , we have $40^\circ + \angle 4 = 180^\circ$ .

STEP 7

Subtracting $40^\circ$ from both sides gives us $\angle 4 = 180^\circ - 40^\circ = 140^\circ$ .
So, $\angle 4 = \mathbf{140^\circ}$ .

STEP 8

We already found $\angle 2$ in the previous step!
Remember, we found that $\angle 2 = \mathbf{105^\circ}$ .

STEP 9

Angles 2, 4, and 5 form a straight line, so they add up to $180^\circ$ .
That means $\angle 2 + \angle 4 + \angle 5 = 180^\circ$ .

STEP 10

We know $\angle 2 = 105^\circ$ and $\angle 4 = 140^\circ$ , so we can plug those values into our equation: $105^\circ + 140^\circ + \angle 5 = 180^\circ$ .

STEP 11

Adding the angles we know gives us $245^\circ + \angle 5 = 180^\circ$ .
Oops, that's not right!
Angles 4 and 2 are on one side of the transversal, and add up to $105^\circ + 140^\circ = 245^\circ$ .
Angle 5 is on the other side of the transversal.
Let's try another approach.

STEP 12

Angle 1 and angle 5 are corresponding angles, which means they are congruent.
Since $\angle 1 = 75^\circ$ , then $\angle 5$ must also be $75^\circ$ .
So, $\angle 5 = \mathbf{75^\circ}$ .

STEP 13

The measure of angle 5 is $75^\circ$ .
None of the answer choices are correct!

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Studdy Solution

STEP 1

What is this asking? We need to find the measure of angle 5, given two other angles formed by a line crossing two parallel lines. Watch out! Don't mix up corresponding and alternate interior angles!Also, remember parallel lines create special angle relationships.

STEP 2

1. Find angle 42. Find angle 23. Find angle 5

STEP 3

Angles 1, 2, 3, and 4 make a straight line, so they add up to 180∘180^\circ180∘.We know ∠1=75∘\angle 1 = 75^\circ∠1=75∘ and ∠3=40∘\angle 3 = 40^\circ∠3=40∘, so ∠1+∠2+∠3+∠4=180∘\angle 1 + \angle 2 + \angle 3 + \angle 4 = 180^\circ∠1+∠2+∠3+∠4=180∘.

STEP 4

STEP 5

Subtracting 75∘75^\circ75∘ from both sides gives us ∠2=180∘−75∘=105∘\angle 2 = 180^\circ - 75^\circ = 105^\circ∠2=180∘−75∘=105∘.So, ∠2=105∘\angle 2 = \mathbf{105^\circ}∠2=105∘.

STEP 6

Similarly, since lines A and B are parallel, angles 3 and 4 are also **consecutive interior angles**, meaning ∠3+∠4=180∘\angle 3 + \angle 4 = 180^\circ∠3+∠4=180∘.Since ∠3=40∘\angle 3 = 40^\circ∠3=40∘, we have 40∘+∠4=180∘40^\circ + \angle 4 = 180^\circ40∘+∠4=180∘.

STEP 7

Subtracting 40∘40^\circ40∘ from both sides gives us ∠4=180∘−40∘=140∘\angle 4 = 180^\circ - 40^\circ = 140^\circ∠4=180∘−40∘=140∘.So, ∠4=140∘\angle 4 = \mathbf{140^\circ}∠4=140∘.

STEP 8

We already found ∠2\angle 2∠2 in the previous step!Remember, we found that ∠2=105∘\angle 2 = \mathbf{105^\circ}∠2=105∘.

STEP 9

Angles 2, 4, and 5 form a straight line, so they add up to 180∘180^\circ180∘.That means ∠2+∠4+∠5=180∘\angle 2 + \angle 4 + \angle 5 = 180^\circ∠2+∠4+∠5=180∘.

STEP 10

We know ∠2=105∘\angle 2 = 105^\circ∠2=105∘ and ∠4=140∘\angle 4 = 140^\circ∠4=140∘, so we can plug those values into our equation: 105∘+140∘+∠5=180∘105^\circ + 140^\circ + \angle 5 = 180^\circ105∘+140∘+∠5=180∘.

STEP 11

STEP 12

Angle 1 and angle 5 are **corresponding angles**, which means they are congruent.Since ∠1=75∘\angle 1 = 75^\circ∠1=75∘, then ∠5\angle 5∠5 must also be 75∘75^\circ75∘.So, ∠5=75∘\angle 5 = \mathbf{75^\circ}∠5=75∘.

STEP 13

The measure of angle 5 is 75∘75^\circ75∘.None of the answer choices are correct!

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What is this asking? We need to find the measure of angle 5, given two other angles formed by a line crossing two parallel lines. Watch out! Don't mix up corresponding and alternate interior angles!
Also, remember parallel lines create special angle relationships.

1. Find angle 4
2. Find angle 2
3. Find angle 5

Angles 1, 2, 3, and 4 make a straight line, so they add up to $180^\circ$ .
We know $\angle 1 = 75^\circ$ and $\angle 3 = 40^\circ$ , so $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 180^\circ$ .

Subtracting $75^\circ$ from both sides gives us $\angle 2 = 180^\circ - 75^\circ = 105^\circ$ .
So, $\angle 2 = \mathbf{105^\circ}$ .

Similarly, since lines A and B are parallel, angles 3 and 4 are also consecutive interior angles, meaning $\angle 3 + \angle 4 = 180^\circ$ .
Since $\angle 3 = 40^\circ$ , we have $40^\circ + \angle 4 = 180^\circ$ .

Subtracting $40^\circ$ from both sides gives us $\angle 4 = 180^\circ - 40^\circ = 140^\circ$ .
So, $\angle 4 = \mathbf{140^\circ}$ .

We already found $\angle 2$ in the previous step!
Remember, we found that $\angle 2 = \mathbf{105^\circ}$ .

Angles 2, 4, and 5 form a straight line, so they add up to $180^\circ$ .
That means $\angle 2 + \angle 4 + \angle 5 = 180^\circ$ .

We know $\angle 2 = 105^\circ$ and $\angle 4 = 140^\circ$ , so we can plug those values into our equation: $105^\circ + 140^\circ + \angle 5 = 180^\circ$ .

Angle 1 and angle 5 are corresponding angles, which means they are congruent.
Since $\angle 1 = 75^\circ$ , then $\angle 5$ must also be $75^\circ$ .
So, $\angle 5 = \mathbf{75^\circ}$ .

The measure of angle 5 is $75^\circ$ .
None of the answer choices are correct!