Math / Geometry

QuestionIn the figure below, lines $l$ and $k$ are parallel. Suppose that $m \angle 2=50^{\circ}$ and $m \angle 4=30^{\circ}$ .
Complete the statements below.
We see that $\angle 1$ and $\angle 4$ are Choose one
And since lines $l$ and $k$ are parallel, $\angle 1$ and $\angle 4$ are Choose one
So, $m \angle 1=$ $\square$ ${ }^{\circ}$ 。
We see that $\angle 2$ and $\angle 5$ are Choose one
And since lines $l$ and $k$ are parallel, $\angle 2$ and $\angle 5$ are Choose one
So, $m \angle 5=$ $\square$ ${ }^{\circ}$ 。
By the angle addition property, $m \angle 5+m \angle 4+m \angle 3=$ $\square$ Note that $m \angle 5=50^{\circ}$ and $m \angle 4=30^{\circ}$ , so $m \angle 3=$ $\square$ ㅇ.。
Therefore, $m \angle 1+m \angle 2+m \angle 3=$ $\square$ $]^{\circ}$ .
The relationship between $\angle 1, \angle 2$ , and $\angle 3$ is an example of the following rule. The sum of the interior angle measures of a triangle is $\square$ ○

Studdy Solution

STEP 1

1. Lines $l$ and $k$ are parallel.
2. Angles are formed by a transversal intersecting the parallel lines.
3. We are given $m \angle 2 = 50^\circ$ and $m \angle 4 = 30^\circ$ .
4. We need to identify relationships between angles and calculate specific angle measures.

STEP 2

1. Determine the relationship between $\angle 1$ and $\angle 4$ .
2. Determine the relationship between $\angle 2$ and $\angle 5$ .
3. Use the angle addition property to find $m \angle 3$ .
4. Calculate the sum of $m \angle 1 + m \angle 2 + m \angle 3$ .
5. Identify the rule related to the sum of the interior angles of a triangle.

STEP 3

Identify the relationship between $\angle 1$ and $\angle 4$ . Since they are on the same side of the transversal and one is an exterior angle and the other is an interior angle, they are supplementary angles.

STEP 4

Since lines $l$ and $k$ are parallel, $\angle 1$ and $\angle 4$ are corresponding angles. Therefore, they are equal.

STEP 5

So, $m \angle 1 = m \angle 4 = 30^\circ$ .

STEP 6

Identify the relationship between $\angle 2$ and $\angle 5$ . These angles are alternate interior angles.

STEP 7

Since lines $l$ and $k$ are parallel, $\angle 2$ and $\angle 5$ are equal.

STEP 8

So, $m \angle 5 = m \angle 2 = 50^\circ$ .

STEP 9

By the angle addition property, $m \angle 5 + m \angle 4 + m \angle 3 = 180^\circ$ .

STEP 10

Note that $m \angle 5 = 50^\circ$ and $m \angle 4 = 30^\circ$ , so:
$m \angle 3 = 180^\circ - (50^\circ + 30^\circ) = 100^\circ$ .

STEP 11

Therefore, $m \angle 1 + m \angle 2 + m \angle 3 = 30^\circ + 50^\circ + 100^\circ = 180^\circ$ .

STEP 12

The relationship between $\angle 1, \angle 2$ , and $\angle 3$ is an example of the following rule: The sum of the interior angle measures of a triangle is $180^\circ$ .

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Studdy Solution

STEP 1

1. Lines $l$ and $k$ are parallel.
2. Angles are formed by a transversal intersecting the parallel lines.
3. We are given $m \angle 2 = 50^\circ$ and $m \angle 4 = 30^\circ$ .
4. We need to identify relationships between angles and calculate specific angle measures.

STEP 2

STEP 3

Identify the relationship between $\angle 1$ and $\angle 4$ . Since they are on the same side of the transversal and one is an exterior angle and the other is an interior angle, they are supplementary angles.

STEP 4

Since lines $l$ and $k$ are parallel, $\angle 1$ and $\angle 4$ are corresponding angles. Therefore, they are equal.

STEP 5

So, $m \angle 1 = m \angle 4 = 30^\circ$ .

STEP 6

Identify the relationship between $\angle 2$ and $\angle 5$ . These angles are alternate interior angles.

STEP 7

Since lines $l$ and $k$ are parallel, $\angle 2$ and $\angle 5$ are equal.

STEP 8

So, $m \angle 5 = m \angle 2 = 50^\circ$ .

STEP 9

By the angle addition property, $m \angle 5 + m \angle 4 + m \angle 3 = 180^\circ$ .

STEP 10

Note that $m \angle 5 = 50^\circ$ and $m \angle 4 = 30^\circ$ , so:
$m \angle 3 = 180^\circ - (50^\circ + 30^\circ) = 100^\circ$ .

STEP 11

Therefore, $m \angle 1 + m \angle 2 + m \angle 3 = 30^\circ + 50^\circ + 100^\circ = 180^\circ$ .

STEP 12

The relationship between $\angle 1, \angle 2$ , and $\angle 3$ is an example of the following rule: The sum of the interior angle measures of a triangle is $180^\circ$ .

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Studdy Solution

STEP 1

STEP 2

STEP 3

Identify the relationship between ∠1\angle 1∠1 and ∠4\angle 4∠4. Since they are on the same side of the transversal and one is an exterior angle and the other is an interior angle, they are supplementary angles.

STEP 4

Since lines l l l and k k k are parallel, ∠1\angle 1∠1 and ∠4\angle 4∠4 are corresponding angles. Therefore, they are equal.

STEP 5

So, m∠1=m∠4=30∘ m \angle 1 = m \angle 4 = 30^\circ m∠1=m∠4=30∘.

STEP 6

Identify the relationship between ∠2\angle 2∠2 and ∠5\angle 5∠5. These angles are alternate interior angles.

STEP 7

Since lines l l l and k kk are parallel, ∠2\angle 2∠2 and ∠5\angle 5∠5 are equal.

STEP 8

So, m∠5=m∠2=50∘ m \angle 5 = m \angle 2 = 50^\circ m∠5=m∠2=50∘.

STEP 9

By the angle addition property, m∠5+m∠4+m∠3=180∘ m \angle 5 + m \angle 4 + m \angle 3 = 180^\circ m∠5+m∠4+m∠3=180∘.

STEP 10

Note that m∠5=50∘ m \angle 5 = 50^\circ m∠5=50∘ and m∠4=30∘ m \angle 4 = 30^\circ m∠4=30∘, so:m∠3=180∘−(50∘+30∘)=100∘ m \angle 3 = 180^\circ - (50^\circ + 30^\circ) = 100^\circ m∠3=180∘−(50∘+30∘)=100∘.

STEP 11

Therefore, m∠1+m∠2+m∠3=30∘+50∘+100∘=180∘ m \angle 1 + m \angle 2 + m \angle 3 = 30^\circ + 50^\circ + 100^\circ = 180^\circ m∠1+m∠2+m∠3=30∘+50∘+100∘=180∘.

STEP 12

The relationship between ∠1,∠2\angle 1, \angle 2∠1,∠2, and ∠3\angle 3∠3 is an example of the following rule: The sum of the interior angle measures of a triangle is 180∘ 180^\circ 180∘.

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Identify the relationship between $\angle 1$ and $\angle 4$ . Since they are on the same side of the transversal and one is an exterior angle and the other is an interior angle, they are supplementary angles.

Since lines $l$ and $k$ are parallel, $\angle 1$ and $\angle 4$ are corresponding angles. Therefore, they are equal.

So, $m \angle 1 = m \angle 4 = 30^\circ$ .

Identify the relationship between $\angle 2$ and $\angle 5$ . These angles are alternate interior angles.

Since lines $l$ and $k$ are parallel, $\angle 2$ and $\angle 5$ are equal.

So, $m \angle 5 = m \angle 2 = 50^\circ$ .

By the angle addition property, $m \angle 5 + m \angle 4 + m \angle 3 = 180^\circ$ .

Note that $m \angle 5 = 50^\circ$ and $m \angle 4 = 30^\circ$ , so:
$m \angle 3 = 180^\circ - (50^\circ + 30^\circ) = 100^\circ$ .

Therefore, $m \angle 1 + m \angle 2 + m \angle 3 = 30^\circ + 50^\circ + 100^\circ = 180^\circ$ .

The relationship between $\angle 1, \angle 2$ , and $\angle 3$ is an example of the following rule: The sum of the interior angle measures of a triangle is $180^\circ$ .