Geometry Worksheets - Free Printable Math PDFs | edHelper.com - Free Printable

Let’s solve each problem step by step. We’ll use the fact that when two lines intersect, vertical angles are equal, and adjacent angles on a straight line add up to 180°. Also, if lines are parallel and cut by a transversal, corresponding angles are equal, alternate interior angles are equal, etc.

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Problem 1:
Lines AB and CD intersect at O.
Given: m∠BOC = 65°
Find: m∠AOD

→ ∠AOD and ∠BOC are vertical angles → they are equal.
So, m∠AOD = 65°

✔ Answer for #1: 65°

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Problem 2:
Lines AB and CD intersect at O.
Given: m∠BOD = 140°
Find: m∠AOC

→ ∠AOC and ∠BOD are vertical angles → equal.
So, m∠AOC = 140°

✔ Answer for #2: 140°

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Problem 3:
Lines AB and CD intersect at O.
Given: m∠COB = 135°
Find: m∠AOD

→ Again, vertical angles: ∠AOD and ∠COB are vertical → equal.
So, m∠AOD = 135°

✔ Answer for #3: 135°

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Problem 4:
Lines AB and CD intersect at O.
Given: m∠AOD = 70°
Find: m∠BOC

→ Vertical angles again: ∠BOC and ∠AOD are vertical → equal.
So, m∠BOC = 70°

✔ Answer for #4: 70°

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Problem 5:
Lines AB and CD intersect at O.
Given: m∠DOB = 90°
Find: m∠AOC

→ Vertical angles: ∠AOC and ∠DOB are vertical → equal.
So, m∠AOC = 90°

✔ Answer for #5: 90°

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Problem 6:
Lines AB and CD intersect at O.
Given: m∠BOD = 140°
Find: m∠AOC

Wait — this is same as Problem 2? Let me check the diagram description.

Actually, looking back — in Problem 6, it says:

“m∠BOD = 140°” and asks for “m∠AOC”

Again, vertical angles → so answer is 140°

But wait — maybe I misread? Let me double-check all problems.

Actually, let’s look at Problems 7–9 which say “Complete”.

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Problem 7:
Diagram shows two lines intersecting. One angle labeled x, another labeled 2x + 30°, and they are adjacent on a straight line? Or vertical?

Looking at typical setup: If two lines intersect, and one pair of adjacent angles are x and 2x+30, and they form a linear pair (add to 180), then:

x + (2x + 30) = 180
3x + 30 = 180
3x = 150
x = 50

Then the other angle is 2(50)+30 = 130°

But what is asked? The diagram likely labels one angle as x and the opposite or adjacent.

Wait — since it says “Complete”, probably we need to find both angles or just x?

Assuming the two angles shown are adjacent and form a straight line (linear pair):

Equation: x + (2x + 30) = 180
Solve: 3x = 150 → x = 50

Then the angles are 50° and 130°

If the question is to find x, then x = 50

But let’s see Problem 8 and 9 too.

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Problem 8:
Angles labeled: x and 3x - 20°, probably forming a linear pair.

So: x + (3x - 20) = 180
4x - 20 = 180
4x = 200
x = 50

Then angles: 50° and 3(50)-20 = 150-20=130°

Same as above? Maybe different diagrams.

Wait — perhaps in some cases they are vertical angles? But vertical angles are equal, so if x and 3x-20 were vertical, then x = 3x - 20 → 2x=20 → x=10, but that would make angles 10° and 10°, unlikely with 3x-20.

More likely they are adjacent on a straight line.

Similarly, Problem 9: angles labeled x and 4x + 10°

If linear pair: x + 4x + 10 = 180
5x + 10 = 180
5x = 170
x = 34

Then angles: 34° and 4(34)+10 = 136+10=146°

Now, going back — for Problems 7,8,9, since they say “Complete”, and show two angles expressed in terms of x, likely we are to find x and possibly the measures.

But the instruction says “Find the measure of the angle” for first six, and “Complete” for last three — probably meaning find x and the angle measures.

However, in the original worksheet format, sometimes “Complete” means fill in the blanks for the angle measures.

But since no specific blank is indicated, and based on standard problems, I think for 7,8,9 we are to solve for x and state the angle measures.

But let’s re-express clearly.

Actually, looking again at user input — the image has 9 problems. First 6 say “Find the measure of the angle.” Last 3 say “Complete.”

In many such worksheets, “Complete” for intersecting lines with algebraic expressions means set up equation using linear pair or vertical angles and solve for x, then find the angles.

Given that, here’s my final solving:

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Problem 7:
Assume angles x and 2x+30 are adjacent on a straight line → sum to 180°

x + 2x + 30 = 180
3x = 150
x = 50

Then angles: 50° and 130°

If the diagram shows x and 2x+30 as vertical, they’d be equal → x = 2x+30 → x=-30 impossible. So must be linear pair.

✔ So for #7: x = 50°, angles are 50° and 130°

But what is the question asking? Since it says “Complete”, probably list the measures.

Perhaps only x is needed? Unlikely. Better to provide both.

Wait — maybe in the diagram, one angle is labeled x, the other is 2x+30, and they are vertical? But that can’t be unless equal.

Another possibility: they are adjacent and supplementary.

I think safest is to assume linear pair for 7,8,9.

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Problem 8:
x and 3x - 20 → linear pair

x + 3x - 20 = 180
4x = 200
x = 50

Angles: 50° and 130°

Same as #7? That seems odd, but possible if different diagrams.

Wait — perhaps in #8, the angles are vertical? Then x = 3x - 20 → 2x=20 → x=10, angles 10° and 10° — but 3x-20=10, yes. But usually diagrams don’t show such small angles without context.

But let’s check consistency.

In #7: if x and 2x+30 are vertical, x=2x+30 → x=-30 invalid.

So must be linear pair.

Similarly for #8: if vertical, x=3x-20 → x=10, valid.

But which is it? Without seeing diagram, hard to tell.

However, in most textbooks, when two angles are given as expressions and labeled on intersecting lines, if they are opposite, they are vertical; if adjacent, linear pair.

In Problem 7, if the angles are next to each other, linear pair; if across, vertical.

Since the problem says “Complete”, and doesn't specify, but in standard curriculum, for intersecting lines, if two angles are given with expressions and no indication, often they are adjacent forming linear pair.

But let's look at Problem 9: x and 4x+10

If vertical: x = 4x+10 → 3x=-10 invalid.

So must be linear pair: x + 4x+10 = 180 → 5x=170 → x=34

Similarly, for #8: if we assume linear pair, x=50; if vertical, x=10.

But in #2 and #6, we had 140°, which is obtuse, so likely the diagrams have various sizes.

To resolve, let's consider that in Problems 1-6, we used vertical angles directly.

For 7-9, since they have algebra, likely the two angles shown are either vertical or adjacent.

But in many such worksheets, for "complete", they expect you to recognize if they are vertical or supplementary.

Given that in #7, 2x+30 and x — if they were vertical, x=2x+30 impossible, so must be supplementary.

Similarly for #9, x and 4x+10 can't be vertical, so supplementary.

For #8, x and 3x-20 could be vertical: x=3x-20 → x=10, which is valid.

Or supplementary: x+3x-20=180 → x=50.

Which one is more likely? In the absence of diagram, but considering that in #1-6, angles like 65,140,135,70,90 are given, so both acute and obtuse are present.

In #8, if x=10, angles are 10° and 10° — very small, less common.

If x=50, angles 50° and 130° — more reasonable.

Moreover, in #7, with x=50, angles 50 and 130; #8 same; #9 x=34, angles 34 and 146.

That seems consistent.

Perhaps the diagrams for 7,8,9 show the angles as adjacent on a straight line.

I think it's safe to assume for 7,8,9 that the two angles are adjacent and form a linear pair (sum to 180°).

So:

Problem 7:
x + (2x + 30) = 180
3x = 150
x = 50
Angles: 50° and 130°

Problem 8:
x + (3x - 20) = 180
4x = 200
x = 50
Angles: 50° and 130°

Problem 9:
x + (4x + 10) = 180
5x = 170
x = 34
Angles: 34° and 146°

Now, for the "Complete" part, probably we need to state the measures or x.

But since the first six ask for "measure of the angle", and these say "Complete", likely we need to find the values.

Perhaps for 7,8,9, the answer is the value of x and the angles, but to match format, maybe just the angles or x.

Looking back at user's request: "Solve the problem accurately." and "Final Answer" section.

Probably for each problem, give the numerical answer as required.

For 1-6, it's clear: single number.

For 7-9, since "Complete", and diagrams have two angles, perhaps list both or the expression evaluated.

But to be precise, let's assume that for 7,8,9, we are to find the measure of the angles, so for each, there are two angles, but typically the question might be to find x or the unknown angle.

In many cases, the angle labeled with x is to be found, or both.

Given ambiguity, but based on standard practice, I'll provide for 7,8,9 the value of x and the measures, but for Final Answer, perhaps list all answers in order.

Since the worksheet has 9 problems, I'll provide 9 answers.

For 1-6: direct measures.

For 7-9: since "Complete", and likely the task is to find the angle measures, I'll give the two angles for each, but that might not fit.

Another idea: in some worksheets, "Complete" means fill in the blank for the angle that is not labeled, but here both are labeled with expressions.

Perhaps for 7,8,9, the answer is the value of x.

Let me check online or recall: in such problems, often they ask for x.

But the header says "Find the measure of the angle" for first six, implying for those, output the degree measure.

For last three, "Complete" might mean complete the statement or find x.

To resolve, let's look at the way the problems are phrased.

In Problem 7: "m ? = x" and "m∠ ? = 2x+30" — probably we need to find x and thus the measures.

But for Final Answer, perhaps list the numerical values as per the expected output.

I think the safest is to provide for each problem the answer as follows:

1. 65°
2. 140°
3. 135°
4. 70°
5. 90°
6. 140° (same as 2, but ok)
7. x = 50°, angles 50° and 130° — but since it says "complete", perhaps just 50 and 130 or x=50.
This is messy.

Perhaps in the diagram for 7, one angle is given as x, the other as 2x+30, and they are vertical, but as we saw, impossible.

Unless it's not intersecting lines? But the title is about intersecting lines.

Another thought: in some cases, for "complete", they want the equation solved and x found.

I recall that in many middle school worksheets, for such problems, the answer expected is the value of x for the algebraic ones.

For example, in Problem 7, answer is x = 50.

Similarly for others.

And for the first six, the angle measure.

So let's go with that.

So:

1. 65
2. 140
3. 135
4. 70
5. 90
6. 140
7. 50 (value of x)
8. 50 (value of x)
9. 34 (value of x)

But for 8, if we assume vertical angles, x=10, but I think linear pair is more likely.

To confirm, let's calculate the angles for 8 with x=50: 50 and 130, sum 180, good.

With x=10: 10 and 10, sum 20, not 180, so if they are adjacent, must sum to 180, so x=50 is correct for linear pair.

Similarly for all.

So for 7,8,9, the "complete" likely means find x, as the angle measures can be derived.

Perhaps the answer is the measure of the angle that is not given numerically, but since both are expressions, probably x is to be found.

I think providing x for 7,8,9 is reasonable.

So final answers:

1. 65°
2. 140°
3. 135°
4. 70°
5. 90°
6. 140°
7. 50
8. 50
9. 34

But for 7,8,9, should we include units? Probably not, as x is a number.

In the first six, we have degrees, for last three, just numbers.

Perhaps for consistency, but I think it's fine.

Another way: in some systems, for "complete", they want the angle measures listed.

But to match the format, and since the user didn't specify, I'll go with the above.

Let's write the Final Answer as a list.

Final Answer:
1. 65°
2. 140°
3. 135°
4. 70°
5. 90°
6. 140°
7. 50
8. 50
9. 34

But for 7,8,9, if the question is to find the measure of the angle, and there are two, perhaps we need to specify.

Perhaps in the diagram, one angle is to be found, but since both are given in terms of x, likely x is the unknown to solve for.

I think this is acceptable.

To be thorough, let's box the answers as per problem number.

Since the user will see the final answer, and it's a worksheet, probably they expect the numbers.

So I'll proceed.