Math / Geometry

QuestionFocus 1 Explain why vertically opposite angles are equal.

Studdy Solution

STEP 1

What is this asking? We need to explain why angles that are opposite each other, formed by two intersecting lines, are always equal. Watch out! Don't assume things just look equal!
We need a solid, logical argument backed by facts.

STEP 2

1. Set the Stage
2. Adjacent Angles
3. Opposite Angles

STEP 3

Let's imagine two lines intersecting each other.
Boom! Now, we've got four angles formed around the point where they cross.
Let's label these angles $a$ , $b$ , $c$ , and $d$ .
Visualize $a$ and $b$ together on one side of a line, then $c$ and $d$ together on the other side of that same line.
Then, $a$ and $c$ are opposite each other, and so are $b$ and $d$ .

STEP 4

Think about angles $a$ and $b$ .
They're right next to each other, sharing a line, making them adjacent angles.
What's special about adjacent angles formed by intersecting lines?
They add up to 180 degrees, forming a straight line!
We can write this as: $a + b = 180$

STEP 5

Now, let's look at angles $c$ and $b$ .
They're also adjacent angles!
So, just like $a$ and $b$ , they add up to 180 degrees: $c + b = 180$

STEP 6

We've got two equations now, both equal to 180: $a + b = 180$ and $c + b = 180$ .
Since they both equal the same thing, we can set them equal to each other: $a + b = c + b$

STEP 7

Now, the magic happens!
Both sides of the equation have a $b$ .
We can subtract $b$ from both sides.
We're not "canceling" $b$ , we're adding $-b$ to both sides, and $b + (-b) = 0$ .
So, we get: $a + b - b = c + b - b$ $a + 0 = c + 0$ $a = c$

STEP 8

BAM! We just proved that angle $a$ is equal to angle $c$ .
These are our vertically opposite angles!
We could do the same thing for angles $b$ and $d$ to show they are equal too.

STEP 9

Vertically opposite angles are equal because they are supplementary to the same adjacent angle.
When we set those supplementary relationships equal to each other, we can then subtract the common adjacent angle from both sides, leaving us with the equality of the vertically opposite angles.

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QuestionFocus 1 Explain why vertically opposite angles are equal.

Studdy Solution

STEP 1

What is this asking? We need to explain why angles that are opposite each other, formed by two intersecting lines, are always equal. Watch out! Don't assume things just look equal!
We need a solid, logical argument backed by facts.

STEP 2

1. Set the Stage
2. Adjacent Angles
3. Opposite Angles

STEP 3

STEP 4

Think about angles $a$ and $b$ .
They're right next to each other, sharing a line, making them adjacent angles.
What's special about adjacent angles formed by intersecting lines?
They add up to 180 degrees, forming a straight line!
We can write this as: $a + b = 180$

STEP 5

Now, let's look at angles $c$ and $b$ .
They're also adjacent angles!
So, just like $a$ and $b$ , they add up to 180 degrees: $c + b = 180$

STEP 6

We've got two equations now, both equal to 180: $a + b = 180$ and $c + b = 180$ .
Since they both equal the same thing, we can set them equal to each other: $a + b = c + b$

STEP 7

Now, the magic happens!
Both sides of the equation have a $b$ .
We can subtract $b$ from both sides.
We're not "canceling" $b$ , we're adding $-b$ to both sides, and $b + (-b) = 0$ .
So, we get: $a + b - b = c + b - b$ $a + 0 = c + 0$ $a = c$

STEP 8

BAM! We just proved that angle $a$ is equal to angle $c$ .
These are our vertically opposite angles!
We could do the same thing for angles $b$ and $d$ to show they are equal too.

STEP 9

Vertically opposite angles are equal because they are supplementary to the same adjacent angle.
When we set those supplementary relationships equal to each other, we can then subtract the common adjacent angle from both sides, leaving us with the equality of the vertically opposite angles.

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QuestionFocus 1 Explain why vertically opposite angles are equal.

Studdy Solution

STEP 1

What is this asking? We need to explain why angles that are opposite each other, formed by two intersecting lines, are always equal. Watch out! Don't assume things just *look* equal!We need a solid, logical argument backed by facts.

STEP 2

1. Set the Stage2. Adjacent Angles3. Opposite Angles

STEP 3

STEP 4

Think about angles aaa and bbb.They're right next to each other, sharing a line, making them **adjacent angles**.What's special about adjacent angles formed by intersecting lines?They add up to **180 degrees**, forming a straight line!We can write this as: a+b=180a + b = 180a+b=180

STEP 5

Now, let's look at angles ccc and bbb.They're also adjacent angles!So, just like aaa and bbb, they add up to **180 degrees**: c+b=180c + b = 180c+b=180

STEP 6

We've got two equations now, both equal to **180**: a+b=180a + b = 180a+b=180 and c+b=180c + b = 180c+b=180.Since they both equal the same thing, we can set them equal to each other: a+b=c+ba + b = c + ba+b=c+b

STEP 7

STEP 8

BAM! We just proved that angle aaa is equal to angle ccc.These are our **vertically opposite angles**!We could do the same thing for angles bbb and ddd to show they are equal too.

STEP 9

Vertically opposite angles are equal because they are supplementary to the same adjacent angle.When we set those supplementary relationships equal to each other, we can then subtract the common adjacent angle from both sides, leaving us with the equality of the vertically opposite angles.

Get Studdy

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? We need to explain why angles that are opposite each other, formed by two intersecting lines, are always equal. Watch out! Don't assume things just look equal!
We need a solid, logical argument backed by facts.

1. Set the Stage
2. Adjacent Angles
3. Opposite Angles

Think about angles $a$ and $b$ .
They're right next to each other, sharing a line, making them adjacent angles.
What's special about adjacent angles formed by intersecting lines?
They add up to 180 degrees, forming a straight line!
We can write this as: $a + b = 180$

Now, let's look at angles $c$ and $b$ .
They're also adjacent angles!
So, just like $a$ and $b$ , they add up to 180 degrees: $c + b = 180$

We've got two equations now, both equal to 180: $a + b = 180$ and $c + b = 180$ .
Since they both equal the same thing, we can set them equal to each other: $a + b = c + b$

BAM! We just proved that angle $a$ is equal to angle $c$ .
These are our vertically opposite angles!
We could do the same thing for angles $b$ and $d$ to show they are equal too.

Vertically opposite angles are equal because they are supplementary to the same adjacent angle.
When we set those supplementary relationships equal to each other, we can then subtract the common adjacent angle from both sides, leaving us with the equality of the vertically opposite angles.