7th Grade Mathematics worksheets - Edform - Free Printable

Let’s solve each problem step by step. We’ll go one at a time, carefully checking our work.

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Problem 1:

We are given two angles on a straight line:
One angle is labeled (3x – 3)°, and the other is 147°.
They form a linear pair → they add up to 180°.

So we write the equation:

> (3x – 3) + 147 = 180

Simplify:

> 3x + 144 = 180

Subtract 144 from both sides:

> 3x = 36

Divide by 3:

> x = 12

✔ Check: Plug x=12 into (3x–3): 3(12)–3 = 36–3 = 33°. Then 33° + 147° = 180° → correct.

Type of Angles: Linear Pair (or Supplementary Angles)
Key Information: Angles on a straight line sum to 180°
Equation: (3x – 3) + 147 = 180
Solution: x = 12

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Problem 2:

Angle XYZ = 60°, and it’s split into two parts: 5x° and 2x°.

So:

> 5x + 2x = 60
> 7x = 60
> x = 60 ÷ 7 ≈ 8.571... but let’s keep as fraction: x = 60/7

Wait — that seems messy. Let me double-check the diagram description.

Actually, looking again: The full angle ∠XYZ is 60°, made of two adjacent angles: 5x° and 2x°. So yes, they add to 60°.

But maybe I misread? Wait — in some diagrams, sometimes the total is not 60°, but here it says “m∠XYZ = 60°”, so yes.

But 60 divided by 7 is not a whole number. That’s okay — unless there’s a mistake.

Wait — perhaps the 60° is NOT the total? Let me re-read.

The label says: “m∠XYZ = 60°” and inside the angle are two parts: 5x° and 2x°. So yes, they should add to 60°.

But let’s check if maybe it’s vertical or something else? No — it’s clearly an angle split into two parts.

Alternatively, maybe the 60° is one part? But no — the notation “m∠XYZ = 60°” means the whole angle is 60°.

Hmm. Maybe it's a typo? Or maybe I need to proceed.

Wait — actually, looking back at the original image description (even though I can’t see it), the user wrote:

“② ... m∠XYZ = 60°” with rays YX, YW, YZ, and angles marked 5x° and 2x° between them.

Assuming W is between X and Z, then yes: 5x + 2x = 60 → 7x=60 → x=60/7.

But that’s unusual for a worksheet. Maybe it’s supposed to be 70°? Or maybe I misread.

Wait — another possibility: Perhaps the 60° is NOT the total angle, but one of the angles? But the text says “m∠XYZ = 60°”, which typically means the entire angle from X to Z via Y is 60°.

Alternatively, maybe the diagram shows that 5x and 2x are adjacent and together make 60° — so we have to go with that.

But let’s hold on — maybe I made a mistake in interpretation.

Wait — perhaps the angle labeled 60° is actually the measure of one of the smaller angles? But the way it’s written: “m∠XYZ = 60°” usually refers to the whole angle.

Looking at standard problems like this, often the total is given, and you add the parts.

But 60 isn't divisible by 7. Hmm.

Wait — maybe it’s 70°? Because 5x + 2x = 7x, and 7*10=70. That would make sense.

But the problem says 60°. Unless it’s a trick.

Alternatively, maybe the 60° is the measure of the larger angle, and 5x and 2x are parts — still same thing.

I think we have to go with what’s written.

So:

Equation: 5x + 2x = 60 → 7x = 60 → x = 60/7 ≈ 8.57

But since this is a school worksheet, likely expects integer answer. So perhaps I misread the diagram.

Wait — another thought: Maybe ∠XYZ is 60°, but the two angles shown are NOT adding to it? For example, maybe one is outside?

But based on typical layout, and since it’s problem #2 after a linear pair, likely it’s angle addition.

Perhaps the 60° is the measure of the angle between YW and YZ or something? But the label says ∠XYZ.

Let me assume for now that it’s correct and proceed.

But wait — let’s look ahead. Problem 4 has 90°, problem 5 has 71°, etc. All integers. So probably 60° is meant to be the total, and 5x+2x=60.

Maybe it’s acceptable as fraction.

But let’s calculate exactly:

x = 60/7

As mixed number: 8 4/7

But perhaps the problem intends for us to leave as improper fraction or decimal? Unlikely.

Wait — maybe I misread the labels. What if the 60° is not the total, but one of the angles?

For example, suppose ∠XYZ is composed of 5x and 2x, but the 60° is labeling one of them? But the text says “m∠XYZ = 60°”, which should mean the whole angle.

Unless in the diagram, point W is such that ∠XYW = 5x, ∠WYZ = 2x, and ∠XYZ = 60°, so yes, 5x+2x=60.

I think we have to go with x = 60/7.

But let’s move on and come back.

Actually, upon second thought — in many textbooks, when they say "m∠ABC = value", and show two rays inside, it means the whole angle is that value.

So I'll proceed.

Type of Angles: Adjacent Angles (forming a larger angle)
Key Information: The sum of the parts equals the whole angle
Equation: 5x + 2x = 60
Solution: x = 60/7 or approximately 8.57

But this feels off. Let me check online or recall similar problems.

Wait — perhaps the 60° is the measure of the angle between the two outer rays, and the two inner angles are 5x and 2x, so yes.

Another idea: Maybe it's vertical angles or something else? No, the diagram shows three rays from Y: to X, W, Z, with W between X and Z.

So I think it's correct.

But let's do Problem 3 first and come back.

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Problem 3:

Two lines intersecting. One angle is 45°, another is (4x - 12)°, and they are vertical angles? Or adjacent?

From description: "two lines crossing, one angle 45°, another (4x-12)°"

In intersecting lines, vertical angles are equal.

If 45° and (4x-12)° are vertical angles, then:

> 4x - 12 = 45

Add 12 to both sides:

> 4x = 57

x = 57/4 = 14.25

Again, not integer. Hmm.

If they are adjacent and supplementary, then:

> 4x - 12 + 45 = 180

> 4x + 33 = 180

> 4x = 147

> x = 147/4 = 36.75 — worse.

But in the diagram, likely they are vertical angles because they are opposite each other.

In standard worksheets, when two lines cross, and two opposite angles are labeled, they are vertical and equal.

So probably 4x - 12 = 45

Then x = 57/4 = 14.25

Still not nice.

Perhaps the 45° is adjacent to (4x-12)°, and they form a linear pair.

Let me think.

In the description: "two lines intersecting, angles marked 45° and (4x-12)°"

Typically, if they are on a straight line, they are supplementary.

But in intersecting lines, adjacent angles are supplementary, vertical are equal.

Without seeing the diagram, it's ambiguous.

But in problem 1, it was clear linear pair.

In problem 3, likely the two angles are vertical, so equal.

But 4x-12 = 45 gives x=14.25

Or perhaps the 45° is not related directly.

Another possibility: The angle labeled 45° and the angle labeled (4x-12)° are adjacent and form a right angle or something? But not specified.

Let's look at problem 4.

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Problem 4:

Right angle symbol, so 90°. Split into two parts: (5x + 1)° and 49°.

So:

> (5x + 1) + 49 = 90

> 5x + 50 = 90

> 5x = 40

> x = 8

Nice! Integer.

Check: 5*8 +1 = 41, 41+49=90 → correct.

Type of Angles: Complementary Angles (sum to 90°)
Key Information: Right angle measures 90°, so parts add to 90°
Equation: (5x + 1) + 49 = 90
Solution: x = 8

Good.

Now back to problem 2 and 3.

For problem 2, if we assume the total angle is 60°, and parts are 5x and 2x, then x=60/7≈8.57, but perhaps it's 70°? Let me see if there's a common mistake.

Maybe "m∠XYZ = 60°" is a red herring, and the 60° is one of the angles? But the text says "m∠XYZ = 60°", which is the whole angle.

Another idea: Perhaps the 60° is the measure of the angle between YX and YW or something, but the label is on ∠XYZ.

I think for consistency, since problem 4 worked out nicely, perhaps in problem 2, the total is 70°, but it's written as 60°.

Or maybe it's 63°? 7*9=63.

But it's written as 60°.

Perhaps in the diagram, the 60° is not the total, but the measure of one of the angles.

Let's read the user's input again: "② ... m∠XYZ = 60°" with angles 5x° and 2x° inside.

Perhaps ∠XYZ is 60°, and it's composed of 5x and 2x, so 7x=60.

I think we have to go with that.

But let's do problem 5 and 6.

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Problem 5:

Three lines intersecting at a point. Angles around a point sum to 360°.

Given: one angle is 71°, another is (2x + 7)°, and there are more angles.

From description: "three lines intersecting, angles marked 71°, (2x+7)°, and others."

Typically, in such diagrams, they might give two angles and imply symmetry or something.

But here, likely the 71° and (2x+7)° are vertical angles or adjacent.

In the text: "⑤ ... 71° and (2x+7)°"

And it's three lines, so six angles, but usually they label opposite ones.

Probably 71° and (2x+7)° are vertical angles, so equal.

So:

> 2x + 7 = 71

> 2x = 64

> x = 32

Check: 2*32 +7 = 64+7=71 → yes.

Perfect.

Type of Angles: Vertical Angles
Key Information: Vertical angles are equal
Equation: 2x + 7 = 71
Solution: x = 32

Great.

Now problem 6.

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Problem 6:

Two lines intersecting. Angles marked 40° and (5x - 6)°.

Likely vertical angles or adjacent.

If vertical, then 5x - 6 = 40

> 5x = 46

> x = 9.2 — not integer.

If adjacent and supplementary:

> 5x - 6 + 40 = 180

> 5x + 34 = 180

> 5x = 146

> x = 29.2 — worse.

But in the diagram, probably they are vertical angles.

Perhaps the 40° and (5x-6)° are not directly related, but in context, likely vertical.

Another possibility: The angle labeled 40° and the angle labeled (5x-6)° are adjacent and form a straight line with another angle, but not specified.

In standard problems, when two angles are labeled on intersecting lines and no other info, they are often vertical.

But 5x-6=40 gives x=9.2.

Perhaps it's 5x - 6 = 40, and we accept decimal.

But let's see if there's a better fit.

Maybe the 40° is one angle, and (5x-6)° is its supplement or something.

Suppose they are adjacent on a straight line, then:

> 40 + (5x - 6) = 180

> 5x + 34 = 180

> 5x = 146

> x = 29.2 — still not good.

Perhaps the diagram shows that 40° and (5x-6)° are vertical, so equal.

I think we have to go with that.

But let's list all solutions so far.

Perhaps for problem 2, the total angle is 70°, not 60°. Let me assume that for a moment.

If in problem 2, m∠XYZ = 70°, then 5x + 2x = 70, 7x=70, x=10. Nice.

Similarly, for problem 3, if 4x-12 = 45, x=14.25, but if they are supplementary, 4x-12 + 45 = 180, 4x=147, x=36.75.

But in problem 3, perhaps the 45° and (4x-12)° are not vertical; maybe they are adjacent and the third angle is given or something.

Another idea for problem 3: When two lines intersect, the vertical angles are equal, and adjacent are supplementary.

If the 45° and (4x-12)° are on the same side, they might be supplementary if they form a linear pair.

But in the diagram, likely they are vertical.

Perhaps the 45° is one angle, and (4x-12)° is the vertically opposite, so equal.

I think for consistency, since problems 1,4,5 gave nice answers, probably problem 2 has a typo, and it's 70°.

Similarly, problem 6 might be 5x - 6 = 40, x=9.2, or perhaps 5x - 6 = 44 or something.

Let's calculate what would make it nice.

For problem 6, if 5x - 6 = 40, x=9.2

If 5x - 6 = 44, then 5x=50, x=10.

Or if 5x - 6 = 39, 5x=45, x=9.

But it's given as 40°.

Perhaps in problem 3, if 4x - 12 = 48, then 4x=60, x=15, but it's 45°.

Another thought for problem 3: Maybe the 45° and (4x-12)° are not the vertical pair; perhaps there is another angle.

But in the description, only those two are mentioned.

Perhaps they are adjacent and their sum is 180° minus something, but unlikely.

Let's look back at the user's input for problem 3: "③ ... 45° and (4x - 12)°"

And for problem 6: "⑥ ... 40° and (5x - 6)°"

In many worksheets, for intersecting lines, if two angles are labeled and they are opposite, they are equal.

So for problem 3: 4x - 12 = 45 → x = 57/4 = 14.25

For problem 6: 5x - 6 = 40 → x = 46/5 = 9.2

But that seems odd.

Perhaps for problem 3, the 45° and (4x-12)° are adjacent and form a straight line with the vertex, but that doesn't make sense.

Another idea: In problem 3, the two angles are on a straight line, so supplementary.

So 4x - 12 + 45 = 180

4x + 33 = 180

4x = 147

x = 36.75 — still bad.

Perhaps the 45° is not related to the (4x-12)° directly, but in the diagram, they are vertical.

I recall that in some diagrams, the angle labeled might be the acute or obtuse, but here it's specified.

Let's consider that for problem 2, if we take m∠XYZ = 60°, and 5x + 2x = 60, x=60/7, but perhaps the problem is to find the measure, not x, but it asks for x.

The instruction is "solve for the missing value", and in the box, it's "Solution: Type answer", and for others, it's x, so likely x.

Perhaps in problem 2, the 60° is the measure of one of the angles, not the total.

For example, suppose ∠XYW = 5x, ∠WYZ = 2x, and m∠XYZ = 60°, which is the same as ∠XYW + ∠WYZ = 5x + 2x = 7x = 60, same thing.

Unless m∠XYZ is not the sum, but that doesn't make sense.

Another possibility: Perhaps "m∠XYZ = 60°" means that the angle at Y between X and Z is 60°, and the ray YW is such that it creates angles 5x and 2x, but perhaps 5x and 2x are not both inside; but that would be unusual.

I think I have to proceed with the calculations as per standard interpretation.

So let's summarize:

Problem 1: x = 12

Problem 2: 5x + 2x = 60 → 7x = 60 → x = 60/7

Problem 3: Assuming vertical angles, 4x - 12 = 45 → 4x = 57 → x = 57/4

Problem 4: (5x+1) + 49 = 90 → 5x = 40 → x = 8

Problem 5: 2x + 7 = 71 → 2x = 64 → x = 32

Problem 6: Assuming vertical angles, 5x - 6 = 40 → 5x = 46 → x = 46/5 = 9.2

But this is messy. Perhaps for problem 3 and 6, they are supplementary.

Let's try for problem 3: if 4x - 12 and 45 are adjacent and supplementary, then 4x - 12 + 45 = 180 → 4x = 147 → x = 36.75

Worse.

Perhaps in problem 3, the 45° and (4x-12)° are not the only angles; but the diagram might show that they are vertical.

Another idea for problem 3: When two lines intersect, the vertical angles are equal, and the adjacent are supplementary. If the 45° is one angle, then the vertical angle is also 45°, and the adjacent are 135° each.

If (4x-12)° is one of the adjacent angles, then 4x-12 = 135 → 4x = 147 → x = 36.75 again.

Same thing.

Perhaps (4x-12)° is the vertical angle to 45°, so equal.

I think we have to accept the fractions.

But let's check problem 6 similarly.

Perhaps for problem 6, the 40° and (5x-6)° are adjacent and their sum is 180° if they are on a straight line, but in intersecting lines, adjacent angles are supplementary only if they form a linear pair, which they do if they are next to each other on a straight line.

In two intersecting lines, each pair of adjacent angles are supplementary.

So if 40° and (5x-6)° are adjacent, then 40 + (5x-6) = 180

5x + 34 = 180

5x = 146

x = 29.2

Still bad.

If they are vertical, 5x-6 = 40, x=9.2

Perhaps the 40° is not the angle, but part of it.

I recall that in some diagrams, the angle labeled might be the difference or something, but unlikely.

Another thought for problem 2: Perhaps "m∠XYZ = 60°" is the measure of the angle, and 5x and 2x are the measures of the two parts, but maybe 5x is for one part, 2x for another, and they add to 60, so x=60/7.

Perhaps the problem is to find the measure of the angles, but the solution box is for x.

Let's look at the response format; it says "Solution: Type answer", and for others, it's x, so likely x.

Perhaps for problem 2, it's 5x and 2x, but the total is not 60; maybe 60 is one of them.

Suppose that m∠XYZ = 60°, but that is the measure of the angle between YX and YZ, and the ray YW is such that ∠XYW = 5x, ∠WYZ = 2x, so 5x + 2x = 60, same thing.

I think I have to go with the calculations.

But let's assume that in problem 2, the total is 70°, as it's a common number, and 7*10=70.

Similarly, for problem 3, if 4x-12 = 48, then x=15, but it's 45°.

For problem 6, if 5x-6 = 44, x=10, or 5x-6 = 39, x=9.

But it's given as 40°.

Perhaps in problem 6, the 40° and (5x-6)° are not the vertical pair; maybe there is another angle.

In the description for problem 6: "⑥ ... 40° and (5x - 6)°" and "two lines intersecting", so likely they are vertical or adjacent.

Perhaps the 40° is one angle, and (5x-6)° is the vertically opposite, so equal.

I think for the sake of this, I'll use the exact values.

But let's double-check problem 1,4,5 which are clean.

Problem 1: x=12

Problem 4: x=8

Problem 5: x=32

For problem 2: let's calculate x = 60/7 = 8 4/7

For problem 3: x = 57/4 = 14 1/4

For problem 6: x = 46/5 = 9 1/5

But perhaps the worksheet expects decimals or fractions.

Maybe in problem 3, the 45° and (4x-12)° are supplementary because they are on a straight line.

In the diagram, if the two lines intersect, and the 45° and (4x-12)° are on the same straight line, then they are supplementary.

For example, if they are adjacent angles forming a linear pair.

In that case, for problem 3: 4x - 12 + 45 = 180

4x + 33 = 180

4x = 147

x = 36.75

Still not good.

Perhaps the 45° is not related; but that doesn't make sense.

Another idea for problem 3: The angle labeled 45° and the angle labeled (4x-12)° are vertical angles, so equal, and the 45° is given, so 4x-12 = 45, x=14.25.

I think I have to proceed.

Perhaps for problem 2, "m∠XYZ = 60°" is a mistake, and it's 70°.

Let me search my memory: in many textbooks, a common problem is 5x + 2x = 70, x=10.

Similarly, for problem 6, 5x - 6 = 44, x=10, or 5x - 6 = 39, x=9.

But let's look at the numbers.

In problem 4, 5x+1 +49=90, 5x=40, x=8.

In problem 5, 2x+7=71, 2x=64, x=32.

In problem 1, 3x-3 +147=180, 3x=36, x=12.

All multiples of 4 or 8 or 12.

For problem 2, if 7x=70, x=10.

For problem 3, if 4x-12 = 48, x=15, but it's 45°.

If 4x-12 = 44, 4x=56, x=14.

Or if 4x-12 = 40, 4x=52, x=13.

None match 45.

Perhaps the 45° is the measure of the angle, and (4x-12)° is the supplement, so 4x-12 = 180 - 45 = 135, then 4x = 147, x=36.75.

Same.

I think for the purpose of this, I'll use the exact values, but since the user might expect nice numbers, perhaps there's a different interpretation.

Let's try for problem 2: perhaps "m∠XYZ = 60°" means that the angle is 60°, and 5x and 2x are the measures, but maybe 5x is for the whole, but that doesn't make sense.

Another idea: In problem 2, the ray YW is such that ∠XYW = 5x, ∠WYZ = 2x, and m∠XYZ = 60°, which is correct, but perhaps the 60° is not the sum, but the measure of one of them, but the label is on ∠XYZ, which is the whole.

I recall that in some notations, m∠XYZ means the angle at Y between X and Z, so it should be the sum.

Perhaps the diagram shows that 5x and 2x are not both inside; but that would be specified.

I think I have to box the answers as calculated.

So let's list:

1. x = 12

2. x = 60/7

3. x = 57/4

4. x = 8

5. x = 32

6. x = 46/5

But for the final answer, since the user may want the values, and perhaps in the context, they are to be left as fractions.

Perhaps for problem 3 and 6, they are supplementary.

Let's assume for problem 3 that the 45° and (4x-12)° are adjacent and supplementary, so 4x-12 + 45 = 180, 4x = 147, x = 36.75

For problem 6, 40 + (5x-6) = 180, 5x = 146, x = 29.2

Still bad.

Perhaps in problem 6, the 40° and (5x-6)° are vertical, so 5x-6 = 40, x=9.2

I think I'll go with the vertical angle assumption for 3 and 6.

So:

Final Answer for each:

1. x = 12

2. x = 60/7

3. x = 57/4

4. x = 8

5. x = 32

6. x = 46/5

But to write as mixed numbers or decimals? Probably fractions.

Since the worksheet may expect simplified fractions.

For problem 2: 60/7

Problem 3: 57/4

Problem 6: 46/5

Now for the types and key information.

Let's fill in for each.

Problem 1:
Type of Angles: Linear Pair (or Supplementary Angles)
Key Information: Angles on a straight line sum to 180 degrees
Equation: (3x - 3) + 147 = 180
Solution: x = 12

Problem 2:
Type of Angles: Adjacent Angles (angle addition)
Key Information: The sum of the parts equals the whole angle
Equation: 5x + 2x = 60
Solution: x = 60/7

Problem 3:
Type of Angles: Vertical Angles
Key Information: Vertical angles are equal
Equation: 4x - 12 = 45
Solution: x = 57/4

Problem 4:
Type of Angles: Complementary Angles
Key Information: Angles in a right angle sum to 90 degrees
Equation: (5x + 1) + 49 = 90
Solution: x = 8

Problem 5:
Type of Angles: Vertical Angles
Key Information: Vertical angles are equal
Equation: 2x + 7 = 71
Solution: x = 32

Problem 6:
Type of Angles: Vertical Angles
Key Information: Vertical angles are equal
Equation: 5x - 6 = 40
Solution: x = 46/5

Now for the final answer, since the user said "provide only the final correct answer", but there are six problems, so probably list all solutions.

The instruction: "Final Answer: [Provide only the final correct answer]"

But for multiple problems, likely list the values of x for each.

Perhaps in order.

So Final Answer:
1. 12
2. 60/7
3. 57/4
4. 8
5. 32
6. 46/5

To make it neat, perhaps as decimals, but fractions are exact.

Since the worksheet may have intended nice numbers, but based on calculation, this is it.

Perhaps for problem 2, if we interpret that m∠XYZ = 60° is the measure, and 5x and 2x are the measures, but maybe 5x is for the larger part, but still sum to 60.

I think it's correct.

So I'll go with that.

Final Answer:
1. 12
2. \frac{60}{7}
3. \frac{57}{4}
4. 8
5. 32
6. \frac{46}{5}

But to write in boxed or something.

Since the user said "provide only the final correct answer", and for multiple, perhaps list them.

In the context, perhaps they want the values.

So I'll write:

Final Answer:
For problem 1: 12
For problem 2: \frac{60}{7}
For problem 3: \frac{57}{4}
For problem 4: 8
For problem 5: 32
For problem 6: \frac{46}{5}

But to save space, perhaps comma-separated.

Since the assistant is to provide the final answer after reasoning, and the reasoning is done, so:

Final Answer: 12, \frac{60}{7}, \frac{57}{4}, 8, 32, \frac{46}{5}

But let's confirm problem 3 and 6 once more.

Upon second thought, in problem 3, if two lines intersect, and one angle is 45°, then the vertical angle is 45°, and the adjacent are 135°.

If (4x-12)° is labeled as one of the angles, and if it's the vertical angle to 45°, then 4x-12 = 45, x=14.25.

If it's an adjacent angle, then 4x-12 = 135, x=36.75.

In the diagram, likely it's the vertical angle, as it's commonly done.

Similarly for problem 6.

Perhaps in problem 6, the 40° and (5x-6)° are not the vertical pair; maybe the 40° is one, and (5x-6)° is the adjacent, but then sum to 180.

But 40 + 5x -6 = 180, 5x = 146, x=29.2.

I think vertical is more likely.

So I'll stick with that.

Final Answer: 12, \frac{60}{7}, \frac{57}{4}, 8, 32, \frac{46}{5}

To write as improper fractions or mixed, but improper is fine.

Perhaps reduce, but they are already reduced.

So for the box.

Since the user may expect the answers in order, and for the solution boxes.

So I'll provide the final answer as the values of x for each problem in sequence.

Final Answer: 12, \frac{60}{7}, \frac{57}{4}, 8, 32, \frac{46}{5}