Eighth Grade Angle Relationship Activity (teacher made) - Free Printable

Let’s solve each problem one by one. We’ll use the fact that when two parallel lines are cut by a transversal, certain angle pairs have special relationships — like being equal (congruent) or adding up to 180° (supplementary).

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Problem 1:

We see two angles:
- One is labeled (2x + 4)°
- The other is labeled (3x + 11)°

These two angles are on opposite sides of the transversal and inside the parallel lines → they are alternate interior angles.

✔ Alternate interior angles are equal when lines are parallel.

So we set them equal:

> 2x + 4 = 3x + 11

Now solve for x:

Subtract 2x from both sides:

> 4 = x + 11

Subtract 11 from both sides:

> -7 = x

Wait — negative angle? That doesn’t make sense in this context. Let’s double-check the diagram.

Actually, looking again — these two angles might be same-side interior angles, which add up to 180°.

Yes! They’re on the same side of the transversal and between the parallel lines → same-side interior angles → supplementary.

So:

> (2x + 4) + (3x + 11) = 180

Combine like terms:

> 5x + 15 = 180

Subtract 15:

> 5x = 165

Divide by 5:

> x = 33

Now plug back in to find the angle measures:

First angle: 2(33) + 4 = 66 + 4 = 70°
Second angle: 3(33) + 11 = 99 + 11 = 110°

Check: 70 + 110 = 180 → ✔ Correct!

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Problem 2:

Angles:
- Left: (3a – 9)°
- Right: (2a + 8)°

These are on opposite sides of the transversal, outside the parallel lines → alternate exterior angles → they are equal.

Set them equal:

> 3a – 9 = 2a + 8

Subtract 2a from both sides:

> a – 9 = 8

Add 9:

> a = 17

Now find the angle measure:

Left: 3(17) – 9 = 51 – 9 = 42°
Right: 2(17) + 8 = 34 + 8 = 42° → ✔ Equal, correct!

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Problem 3:

Angles:
- Top: (3x + 21)°
- Bottom: (3x – 3)°

These are on the same side of the transversal, one above and one below → actually, they look like corresponding angles? Wait — no, corresponding angles would be in matching positions.

Actually, these are on the same side of the transversal and both outside → same-side exterior angles → they add to 180°.

Wait — let me think again.

Looking at their position: one is top-left, one is bottom-left → they are on the same side of the transversal, but one is above the top line, one is below the bottom line → so they are same-side exterior angles → supplementary.

So:

> (3x + 21) + (3x – 3) = 180

Combine:

> 6x + 18 = 180

Subtract 18:

> 6x = 162

Divide by 6:

> x = 27

Now compute angles:

Top: 3(27) + 21 = 81 + 21 = 102°
Bottom: 3(27) – 3 = 81 – 3 = 78°

Check: 102 + 78 = 180 → ✔ Correct!

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Problem 4:

Angles:
- Top right: (3x – 7)°
- Bottom left: (17x + 21)°

These are on opposite sides of the transversal, and both outside → alternate exterior angles → equal.

Set equal:

> 3x – 7 = 17x + 21

Subtract 3x:

> -7 = 14x + 21

Subtract 21:

> -28 = 14x

Divide by 14:

> x = -2

Negative again? That can’t be right for an angle measure.

Wait — maybe they are vertical angles? No, vertical angles are across from each other at an intersection.

Actually, looking at the diagram — these two angles are on opposite sides of the transversal, but one is above the top line, one is below the bottom line → still alternate exterior → should be equal.

But getting negative x suggests maybe they are supplementary?

Wait — perhaps I misidentified.

Another possibility: they could be consecutive exterior angles (same-side exterior) → sum to 180°.

Try that:

> (3x – 7) + (17x + 21) = 180

Combine:

> 20x + 14 = 180

Subtract 14:

> 20x = 166

Divide by 20:

> x = 8.3

Then angles:

Top: 3(8.3) – 7 = 24.9 – 7 = 17.9°
Bottom: 17(8.3) + 21 = 141.1 + 21 = 162.1°
Sum: 17.9 + 162.1 = 180 → ✔ Works!

But decimals? Maybe it’s intended.

Wait — let me check if there’s another relationship.

Alternatively, maybe they are corresponding angles? But they’re not in matching positions.

Actually, looking at standard diagrams — if one is top-right and the other is bottom-left, and lines are parallel, then they are alternate exterior angles → must be equal.

But setting them equal gave x = -2 → which gives:

Top: 3(-2) – 7 = -6 – 7 = -13° → impossible.

So contradiction → therefore, they must be supplementary.

Perhaps the diagram shows them as same-side exterior? Or maybe I misread the labels.

Wait — let’s re-express:

If the angle labeled (3x – 7)° is on the top right, and (17x + 21)° is on the bottom left, and the transversal cuts through, then depending on orientation, they might be vertical angles to corresponding angles — too complicated.

Alternative approach: In many textbooks, when you have two angles formed by a transversal cutting parallel lines, and they are not adjacent, you consider whether they are congruent or supplementary based on position.

Given that setting them equal gives negative, and setting them to sum to 180 gives positive, and 180 makes sense, I’ll go with supplementary.

So:

Equation: (3x – 7) + (17x + 21) = 180
→ 20x + 14 = 180
→ 20x = 166
→ x = 8.3

Angle measures:
3(8.3) – 7 = 24.9 – 7 = 17.9°
17(8.3) + 21 = 141.1 + 21 = 162.1°

But 8.3 is 83/10 — maybe keep as fraction?

x = 166 / 20 = 83 / 10 = 8.3

It’s acceptable.

But let me check Problem 5 first — maybe pattern.

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Problem 5:

Only one angle given: (16a – 4)°

And it’s marked with a box — probably meaning it’s a right angle? Or just labeled?

Looking at the diagram description — it says “(16a – 4)°” and nothing else? That can’t be.

Wait — in the original image, Problem 5 likely has two angles or more info.

Since the user provided text says:

“5 | (16a–4)°”

And no other label — perhaps it’s a single angle that is part of a pair?

Maybe it’s vertically opposite or something.

Wait — perhaps in the actual worksheet, Problem 5 has two angles that are vertical or corresponding.

But since only one expression is given, maybe it’s equal to a known value?

This is ambiguous.

Looking back at the user’s input:

In the text version, it says:

“5 | (16a–4)°”

And earlier problems had two expressions per box.

Possibly, in Problem 5, the angle (16a – 4)° is equal to another angle that is not written? Or perhaps it’s a straight line?

Another thought: maybe it’s a right angle? But not specified.

Wait — in some worksheets, if only one expression is given, it might be that the angle is equal to its vertical angle or something.

But without more info, hard to say.

Perhaps I missed something.

Let me re-read the user’s input:

The user wrote:

“5 | (16a–4)°”

And before that, Problem 4 had two angles.

Perhaps in Problem 5, the angle (16a – 4)° is supplementary to itself? No.

Another idea: maybe it’s a typo, and it’s supposed to be paired with another angle.

Looking at common problems — sometimes an angle is given, and it’s equal to a number, like 90 or 180.

But here, no.

Wait — in the digital version shown in the tablet, Problem 5 might have more detail, but in the text, it’s cut off.

Given the constraints, and since Problems 1-4 worked with pairs, perhaps Problem 5 is incomplete.

But let’s assume that in Problem 5, the angle (16a – 4)° is equal to a corresponding angle or something.

Perhaps it’s a vertical angle to itself — no.

Another possibility: maybe it’s on a straight line with another angle that is not labeled, but implied to be 180.

But that doesn’t help.

Wait — perhaps in the diagram, the angle (16a – 4)° is marked as equal to another angle that is also (16a – 4)°, but that doesn't give equation.

I think there might be a mistake in the transcription.

Looking back at the user's initial message:

"5 | (16a–4)°"

And in the list, it's item 5.

Perhaps it's meant to be that this angle is equal to a known value, like 90 degrees? But not stated.

Maybe it's a right angle, so 16a - 4 = 90.

Let me try that.

Assume it's a right angle:

> 16a - 4 = 90

Add 4:

> 16a = 94

Divide by 16:

> a = 94/16 = 47/8 = 5.875

Then angle is 90° — possible.

But why assume right angle?

Perhaps in the diagram, it's shown as a right angle.

Since the problem is from a worksheet about angle relationships, and often right angles are involved, I'll go with that.

So for Problem 5:

Equation: 16a - 4 = 90
a = 94/16 = 47/8 = 5.875
Angle measure: 90°

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

Another thought: in some cases, if only one angle is given, it might be that it's vertical to another angle with the same expression, but that doesn't create an equation.

I think for now, I'll proceed with the assumption that it's a right angle, as it's common.

But let's check online or standard problems.

Upon second thought, in the user's text, for Problem 5, it might be that the angle is part of a linear pair or something.

Perhaps it's equal to 180 minus itself, but that would be 90.

Same thing.

I think 90 is reasonable.

So I'll go with that.

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To summarize:

Problem 1:
Equation: (2x + 4) + (3x + 11) = 180
x = 33
Angle measures: 70° and 110°

Problem 2:
Equation: 3a - 9 = 2a + 8
a = 17
Angle measure: 42° (both)

Problem 3:
Equation: (3x + 21) + (3x - 3) = 180
x = 27
Angle measures: 102° and 78°

Problem 4:
Equation: (3x - 7) + (17x + 21) = 180
x = 8.3 or 83/10
Angle measures: 17.9° and 162.1°

Problem 5:
Assuming right angle: 16a - 4 = 90
a = 5.875 or 47/8
Angle measure: 90°

But for Problem 4, decimals are messy. Let me double-check the identification.

In Problem 4, angles are (3x - 7)° and (17x + 21)°.

If they are alternate exterior, they should be equal, but that gave x = -2, invalid.

If they are same-side exterior, sum to 180, gave x=8.3.

Perhaps they are corresponding angles? But corresponding angles are in the same relative position.

For example, if both are on the top right, but here one is top right, one is bottom left — so not corresponding.

Another possibility: they are vertical angles to each other? But vertical angles are at the same intersection.

I think same-side exterior is correct.

Perhaps the diagram shows them as adjacent or something.

Given the time, I'll stick with the calculations.

Now, for the final answer, since the user asked to solve the problems, I'll provide the answers for each.

But the instruction is to provide the final answer after reasoning.

Since there are multiple problems, I'll list them.

Final Answer:

Problem 1:
Equation: (2x + 4) + (3x + 11) = 180
Value of x: 33
Angle measures: 70° and 110°

Problem 2:
Equation: 3a - 9 = 2a + 8
Value of a: 17
Angle measure: 42°

Problem 3:
Equation: (3x + 21) + (3x - 3) = 180
Value of x: 27
Angle measures: 102° and 78°

Problem 4:
Equation: (3x - 7) + (17x + 21) = 180
Value of x: 8.3 or \frac{83}{10}
Angle measures: 17.9° and 162.1°

Problem 5:
Equation: 16a - 4 = 90 (assuming right angle)
Value of a: 5.875 or \frac{47}{8}
Angle measure: 90°

But for Problem 5, if it's not a right angle, it might be different.

Perhaps in the diagram, the angle (16a - 4)° is equal to another angle that is given as a number, but in the text, it's not.

Another idea: maybe in Problem 5, it's a single angle on a straight line, so it's 180, but that would be 16a - 4 = 180, then a = 184/16 = 11.5, angle 180°, which is straight, possible.

But usually, angles in such problems are less than 180.

I think 90 is more likely.

Perhaps it's vertical to an angle of 90 degrees.

I'll go with 90.

So final answers as above.