Geometry worksheet with problems to find the value of x in triangles and angles, featuring diagrams and algebraic expressions.

Let’s solve each problem step by step. We’ll use basic triangle rules:
- The sum of angles in any triangle is 180°.
- An exterior angle equals the sum of the two opposite interior angles.
- Vertical angles are equal.
- Supplementary angles add to 180°.
- In right triangles, one angle is 90°.

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Problem 13:
We have two triangles sharing a vertical angle (the “X” shape).
Left triangle: angles x°, x°, and some unknown angle.
Right triangle: angles 43°, 57°, and the same unknown angle (vertical angle).

First, find the third angle in the right triangle:
180 - 43 - 57 = 80° → this is the vertical angle.

Now left triangle has angles: x°, x°, and 80°.
So: x + x + 80 = 180 → 2x = 100 → x = 50

✔ Check: 50 + 50 + 80 = 180 ✔️

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Problem 14:
Big triangle on left: right triangle with 65° and 90° → third angle = 180 - 65 - 90 = 25°
This 25° is part of the big bottom line.
On the right, small triangle has angles 50°, 80°, so third angle = 180 - 50 - 80 = 50°
But wait — that 50° is adjacent to x° along the straight line? Actually, look again.

Actually, the diagram shows two triangles sharing a side. Let’s label:

The big triangle on left: angles 65°, 90°, and let’s call the bottom-left angle A.
A = 180 - 65 - 90 = 25°

Now, the whole bottom line is straight. On the right, there’s a triangle with angles 50°, 80°, and the angle at the shared vertex is x°.

Wait — actually, the angle labeled x° is inside the small triangle on the right. But we can also see that the big triangle’s bottom-right angle is made up of x° and something else?

Better approach: Look at the large triangle formed by combining both? Or use exterior angle.

Actually, notice: the angle outside the small triangle (on the right) is an exterior angle for the left triangle? Not quite.

Alternative: The angle between the two triangles at the bottom is common.

Let me redraw mentally:

There’s a horizontal base. Left triangle: right-angled at bottom-left, top angle 65°, so bottom-right angle of left triangle is 25°.

Then from that point, another triangle goes up to the right with angles 50° at top, 80° at bottom-right, and x° at the shared vertex (bottom-left of small triangle).

So at the shared vertex (where the two triangles meet on the base), the total angle on the straight line is 180°.

The left triangle contributes 25° at that vertex. The small triangle contributes x° at that same vertex. But they are adjacent? Actually, no — the 25° is inside the left triangle, and x° is inside the small triangle, and together they form the angle between the two slanted lines? I think I’m overcomplicating.

Wait — perhaps the key is that the angle marked x° is actually the difference or sum.

Another way: Consider the big triangle that includes both shapes? Not really.

Look at the small triangle on the right: it has angles 50°, 80°, and x°? But 50+80=130, so x would be 50? But that doesn’t match the diagram because x is not the third angle of that triangle — wait, yes it is!

In problem 14, the small triangle on the right has three angles: one is 50° (top), one is 80° (bottom-right), and the third is x° (at the left vertex of that small triangle). So:

x + 50 + 80 = 180 → x = 50

But then why is there a left triangle? Maybe to distract? Or maybe x is not that angle.

Looking back at the image description: "14) x = _____" with a figure showing a right triangle on left (65°, 90°), connected to a smaller triangle on right with 50° and 80°, and x° is at the junction.

Actually, upon standard interpretation, x° is the angle inside the small triangle at the shared vertex. And since the small triangle has angles 50°, 80°, and x°, then x = 180 - 50 - 80 = 50.

But let’s verify with the left triangle. The left triangle has angles 65°, 90°, and 25° at the bottom-right. That 25° is adjacent to the x° angle? If they are on a straight line, then 25 + x + something = 180? No.

Perhaps the 25° and the x° are parts of the same angle? I think I need to assume that x is simply the third angle of the small triangle.

Maybe the diagram shows that the small triangle's left angle is x, and it's separate.

I recall that in such problems, often x is found using the fact that the exterior angle equals sum of remote interiors.

Notice: the 80° angle in the small triangle might be an exterior angle for the left triangle? Let's see.

The left triangle has angles 65°, 90°, 25°. The 25° is at the bottom-right. Now, if we extend the base, the exterior angle at that vertex would be 180 - 25 = 155°, but that doesn't help.

Another idea: the angle between the two triangles at the top is vertical or something.

Perhaps the 50° and 65° are related.

Let’s calculate the angle at the very top where the two triangles meet. The left triangle has 65° at top-left, the small triangle has 50° at top-right, but they are not adjacent.

I think I made a mistake. Let me search for a standard solution approach.

Upon second thought, in many textbooks, for problem 14, the x is the angle in the small triangle, and it's 50 degrees. But let's double-check with the given numbers.

Small triangle: angles 50°, 80°, and x. Sum must be 180, so x = 50. Yes.

The left triangle is probably there to provide context or for another purpose, but for finding x, we only need the small triangle.

So x = 50.

But let's confirm with the answer later. For now, proceed.

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Problem 15:
Triangle with angles 44°, and two base angles. One base angle is marked with arc, the other is x°, and they are on a straight line with the base extended? No.

The diagram shows a triangle with top angle 44°, and the two base angles are equal? No, one is marked with a single arc, the other with x°, and the base is extended to form an exterior angle x°? Wait.

Actually, looking at the description: "15) x = _____" with a triangle, top angle 44°, left base angle has an arc (meaning it's equal to something?), and the right base angle is adjacent to x° on a straight line.

Standard interpretation: the triangle has angles: top 44°, left base angle let's say y, right base angle z. Then y + z + 44 = 180.

But the right base angle z and x° are supplementary because they form a straight line. So z + x = 180.

Also, if the left base angle is marked with an arc, it might mean it's equal to the right base angle? But in the diagram, only one arc on left, and x on the extension.

Typically, in such diagrams, if no mark, we don't assume equality. But here, the left base angle has an arc, which usually means it's equal to another angle, but there's no other arc. Perhaps it's just indicating the angle.

I think the intention is that the triangle is isosceles with the two base angles equal. Because otherwise, we have two variables.

Assume the two base angles are equal. Then each is (180 - 44)/2 = 136/2 = 68°.

Then, the exterior angle x° is adjacent to one base angle, so x = 180 - 68 = 112°.

If not isosceles, we can't solve. But since it's a standard problem, likely isosceles.

So x = 112.

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Problem 16:
Right triangle with one angle x°, another angle is part of a straight line with 122°.

The 122° is an exterior angle. Specifically, at the top-right vertex, the interior angle of the triangle is supplementary to 122°, so interior angle = 180 - 122 = 58°.

The triangle is right-angled at bottom-right, so angles are: x° (bottom-left), 90° (bottom-right), and 58° (top).

Sum: x + 90 + 58 = 180 → x + 148 = 180 → x = 32.

✔ Check: 32 + 90 + 58 = 180 ✔️

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Problem 17:
Triangle with angles: (6x-7)°, (103-x)°, and 2x°.

Sum of angles in triangle is 180°.

So: (6x - 7) + (103 - x) + 2x = 180

Simplify: 6x - 7 + 103 - x + 2x = 180 → (6x - x + 2x) + (-7 + 103) = 180 → 7x + 96 = 180

7x = 180 - 96 = 84 → x = 12

✔ Check: 6*12 -7 = 72-7=65; 103-12=91; 2*12=24; 65+91+24=180 ✔️

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Problem 18:
Triangle with one angle x°, and the other two angles are not given, but there is an exterior angle or something? The diagram shows a triangle with one angle marked x°, and arrows indicating it's part of a larger figure, but no other angles given.

Wait, re-examining: "18) x = _____" with a triangle, and x° is at the top, and there are arcs on the two base angles, meaning they are equal? Probably isosceles triangle.

If it's isosceles with two base angles equal, and no other info, we can't find x. But likely, the arrows indicate that the two sides are equal, so base angles equal.

But still, without a number, we can't solve. Unless... perhaps the x is the vertex angle, and we need more.

I think I missed something. In the user's image description, it might be that the triangle has angles that sum to 180, but only x is given. That can't be.

Perhaps the diagram shows that the two base angles are equal, and x is the vertex, but no value. This is incomplete.

Wait, looking back at the original request, for problem 18, it's "x = _____" with a triangle and x° at the top, and the two base angles have arcs, suggesting they are equal. But without a numerical value, we can't determine x.

Unless... perhaps in the context, it's implied that the triangle is equilateral or something, but no.

Another possibility: the arrows on the sides indicate that the two sides are equal, so isosceles, but still need more info.

I think there might be a mistake in my assumption. Let me think differently.

Perhaps the x° is an exterior angle. The diagram shows a triangle with one angle x°, and the adjacent angle on the straight line is not given.

I recall that in some problems, if two angles are marked with arcs, they are equal, and if no numbers, perhaps it's a trick.

But for the sake of progress, let's assume that the two base angles are equal, and the sum is 180, but we have one equation with two variables.

Unless the x is the only unknown, but it's not.

Perhaps the diagram has the two base angles marked as equal, and the vertex is x, but we need another condition.

I think I need to skip and come back, or assume a standard case.

Upon second thought, in many worksheets, for such a diagram, if no numbers are given except x, and it's isosceles, perhaps x is to be expressed, but the problem asks for a number.

Another idea: perhaps the arrows indicate that the triangle is equilateral, so all angles 60°, so x=60. But that's assuming too much.

Let's look at problem 19 and 20 for clues, but better to move on and return.

For now, let's say if it's isosceles with base angles equal, and no other info, we can't solve. But perhaps in the diagram, the base angles are given implicitly.

I think there's a misinterpretation. Let me describe what I see in my mind: a triangle with apex angle x°, and the two base angles are equal, and perhaps the sum is 180, but we need a number.

Perhaps the x is not the apex, but one of the base angles. The problem says "x°" at the top, so likely apex.

I found a better way: in some versions, the diagram for 18 has the two base angles marked with arcs, and the apex is x, and it's given that the triangle is isosceles, but still.

Perhaps the answer is 60 if equilateral, but let's calculate later.

Let's do problem 19 first.

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Problem 19:
Triangle with angles: 56° at top, and the two base angles are marked with double ticks, meaning they are equal. Also, the base is divided into two segments with double ticks, suggesting the two parts are equal, so perhaps the triangle is isosceles with AB = AC, but here the base is BC, and it's divided, so maybe D is midpoint, and AD is median, but also angle bisector or altitude?

The diagram shows: triangle ABC, angle at A is 56°, and BD = DC (since double ticks on base segments), and also the two sides AB and AC have double ticks? No, the description says "||" on the two sides from A to the base points, but typically, if the two legs have the same tick, they are equal.

In the text: "56°" at top, and "||" on the two sides from the top vertex to the base, and "||" on the two segments of the base. So likely, the two sides are equal, so isosceles with AB = AC, and also the base is divided equally, so D is midpoint, and since AB=AC, AD is also altitude and angle bisector.

So, angle at A is 56°, so each base angle is (180 - 56)/2 = 124/2 = 62°.

Now, x° is at the bottom-right, which is one of the base angles, so x = 62.

But the diagram shows x° at the end of the base, so yes, it's the base angle.

So x = 62.

✔ Check: 56 + 62 + 62 = 180 ✔️

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Problem 20:
Two triangles sharing a side. Left triangle: angles 50°, 62°, and the third angle is at the shared vertex. Right triangle: angles 53°, 80°, and x° at the shared vertex.

First, find the third angle of the left triangle: 180 - 50 - 62 = 68°.

This 68° is at the shared vertex for the left triangle.

For the right triangle, angles are 53°, 80°, and x° at the shared vertex.

But the shared vertex has two angles: one from each triangle, and they are adjacent, forming a straight line or what?

In the diagram, the two triangles are on opposite sides of the shared side, so at the shared vertex, the two angles are on a straight line? Or are they vertical?

Typically, in such "bowtie" diagrams, the two angles at the intersection are vertical angles, so equal.

Here, the left triangle has an angle of 68° at the shared vertex, and the right triangle has x° at the same vertex, and since they are vertical angles, x = 68.

Is that correct? Let's see.

The shared vertex is where the two triangles meet, and the angles are vertically opposite, so yes, they should be equal.

So x = 68.

Verify with right triangle: if x=68, then angles 53°, 80°, 68° sum to 53+80=133, +68=201 > 180, impossible.

Mistake.

If the two triangles share a common side, and are on the same plane, at the shared vertex, the angles are not necessarily vertical.

In this case, the shared vertex is a point where four rays meet: two from each triangle.

Specifically, for the left triangle, vertices: let's say P, Q, R, with P top, Q left, R right. Angles at P 50°, at Q 62°, so at R 68°.

Right triangle: shares side PR or QR? Typically, they share the side between the two triangles.

Assume they share the side from the top to the bottom, but in the diagram, it's like two triangles sharing a common vertex at the bottom.

From the description: "20) x = _____" with two triangles: left has 50° at top, 62° at bottom-left, right has 53° at top, 80° at bottom-right, and x° at the bottom-center where they meet.

So at the bottom-center point, there are two angles: one from left triangle, one from right triangle, and they are adjacent, forming a straight line with the base.

The base is a straight line, so the sum of the two angles at the bottom-center should be 180° if they are on a straight line, but they are parts of different triangles.

Actually, the two triangles are placed such that their bases are on the same straight line, and they share a common vertex at the top? No.

Standard configuration: two triangles sharing a common side, but here it's likely that they are separate but connected at a point.

Perhaps the x° is the angle between them.

Another way: the angle x° is at the intersection, and it's the same as the vertical angle.

Let's calculate the angles at the shared vertex.

For the left triangle, the angle at the shared vertex (let's call it V) is 180 - 50 - 62 = 68°.

For the right triangle, the angle at V is x°.

But these two angles are adjacent and together form the angle around point V, but in the plane, if the two triangles are on opposite sides of the shared side, then the two angles at V are supplementary if they are on a straight line, but typically in such diagrams, the two angles at the shared vertex are vertical angles, so equal.

But earlier calculation showed that if x=68, then right triangle angles 53+80+68=201>180, impossible.

So perhaps the shared vertex is not the same for both in that way.

Let's think: in the left triangle, angles are at three vertices: top 50°, bottom-left 62°, so bottom-right (shared) is 68°.

In the right triangle, angles are at top 53°, bottom-right 80°, so bottom-left (shared) is 180 - 53 - 80 = 47°.

Then, at the shared vertex, we have two angles: 68° from left triangle and 47° from right triangle, but they are on the same side or opposite?

If the two triangles are on the same side of the base, then at the shared vertex, the total angle is 68° + 47° = 115°, but that's not x.

Perhaps x is the angle between the two triangles, which would be the difference or something.

I recall that in such problems, x is often the vertical angle or the exterior angle.

Another idea: the two triangles share a common side, and x is the angle at the top of the combined figure, but not specified.

Perhaps the x° is the angle at the shared vertex for the right triangle, and we need to find it, but we have all angles for the right triangle except x, so x = 180 - 53 - 80 = 47°.

Then why is the left triangle given? To distract or for verification.

In many problems, the left triangle is used to find a related angle, but here if x is in the right triangle, and we can calculate it directly, then x = 47.

Let me check if that makes sense with the left triangle.

If x=47, and in left triangle, the shared angle is 68°, then at the shared vertex, the two angles are 68° and 47°, which are adjacent, and if they are on a straight line, 68+47=115 < 180, so not straight.

Perhaps the base is straight, so the sum of the two base angles at the shared point should be 180° if they are on a straight line, but 68 + 47 = 115 ≠ 180, so not.

Unless the shared vertex is not on the base.

I think I have it: in the diagram, the two triangles are arranged such that they share a common vertex at the bottom, and the base is not straight, but the angle x is the angle between the two triangles at that vertex.

But typically, for problem 20, x is the angle at the shared vertex for the right triangle, and it's 47 degrees.

And the left triangle is given to possibly find another angle, but for x, it's directly calculable from the right triangle.

So x = 180 - 53 - 80 = 47.

✔ Check: 53 + 80 + 47 = 180 ✔️

So I'll go with that.

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Problem 18 revisited:
After thinking, in many sources, for a triangle with two base angles marked equal (arcs), and no other numbers, but in this case, perhaps it's assumed that the triangle is equilateral, but that's not stated.

Perhaps the x is the only angle given, and we need to realize that the sum is 180, but with two unknowns.

Another possibility: the arrows on the sides indicate that the two sides are equal, so isosceles, and the base angles are equal, but still.

Perhaps in the diagram, the base angles are not marked, but the sides are, and x is the vertex angle, but no value.

I recall that in some problems, if no numbers are given, x might be 60, but let's look for a different approach.

Perhaps the x° is an exterior angle. The diagram shows a triangle with x° at the top, and the base is extended, but no.

Let's assume that the two base angles are equal, and let each be y, then x + 2y = 180, but we have one equation.

Unless there's additional information.

Perhaps the "arrows" indicate that the triangle is equilateral, so x=60.

Or perhaps in the context of the worksheet, it's 60.

But to be precise, let's say if it's isosceles with base angles equal, and no other info, we can't determine, but for the sake of completing, and since other problems have answers, perhaps for 18, it's 60.

I found a better way: in the user's image, for problem 18, it might be that the two base angles are marked with arcs, and the vertex is x, and it's given that the triangle is isosceles, but still.

Perhaps the answer is 60, as a default.

Let's calculate later.

For now, let's list what we have.

Also, for problem 14, I had x=50, but let's verify with the left triangle.

In problem 14, left triangle: 65°, 90°, so bottom-right angle 25°.

Small triangle on right: if it has angles 50°, 80°, and x°, and if x is at the shared vertex, then the angle at the shared vertex for the small triangle is x, and for the left triangle is 25°, and if they are adjacent on the straight line, then 25 + x + the angle between = 180, but there's no angle between.

Actually, the two triangles share the vertex, and the angle between the two slanted lines is the sum or difference.

Perhaps the 25° and the x° are parts of the same angle at the bottom.

Another standard solution: the angle x is the difference between the exterior angle and something.

Let's use the fact that the exterior angle of the left triangle at the bottom-right is 180 - 25 = 155°, and this is equal to the sum of the two remote interior angles of the small triangle or something.

I think I need to accept that for problem 14, x = 50, as per the small triangle alone.

Similarly for others.

For problem 18, let's assume it's isosceles with base angles equal, and perhaps the vertex angle is x, but no value, so maybe it's 60.

Perhaps the diagram has the two base angles marked, and x is one of them, but the problem says "x°" at the top.

Let's look online or recall: in some worksheets, for a triangle with two arcs on base angles, and no numbers, x is not defined, but here it must be.

Another idea: perhaps the "arrows" on the sides indicate that the two sides are equal, so isosceles, and the base angles are equal, and the vertex angle is x, but we need another condition.

I think for the sake of time, I'll set x = 60 for problem 18, as a guess.

But let's do the last part.

Part II: Find the measure of each angle.

Diagram: three lines intersecting, forming various angles.

Given: angle 3 is 90°, angle 1 is 68°, and we need to find angles 1 to 6, but angle 1 and 3 are given? No, the problem says "find the measure of each angle" for 21 to 26, which are ∠1 to ∠6, and in the diagram, ∠3 is marked 90°, ∠1 is marked 68°, so probably those are given, and we need to find the others.

The diagram has:

- A horizontal line.

- A line crossing it at an angle, with angles labeled 6,5 on left, 2,1 on right.

- Another line crossing, with angles 3,4 at top, and 2,1 at bottom, but angle 1 is at the intersection with the horizontal.

Specifically, at the right intersection, angle 1 is 68°, and it's between the horizontal and the slanted line.

Angle 3 is 90°, at the top intersection.

Let me define the points.

Let me call the horizontal line L1.

Line L2 crosses L1 at point A, with angles: above L1, left is ∠6, right is ∠5; below L1, left is ∠ something, but typically, at intersection, vertical angles.

At point A (right intersection), the angles are: ∠1 = 68° (between L1 and L2, say acute angle), then the adjacent angle on the straight line is 180 - 68 = 112°, which might be ∠2 or something.

The labeling: "3 90°" at top, "4" below it, "2 1" at the right intersection, with "1" being 68°.

So likely, at the top intersection (point B), angles are ∠3 = 90°, ∠4 adjacent.

At the right intersection (point A), angles are ∠1 = 68°, ∠2 adjacent.

Also, there is a line connecting or something, but probably two lines intersecting the horizontal.

From the description: "3 90°" and "4" are at the top, so likely at the intersection of two lines at the top.

Then "2 1" at the bottom right, with "1" = 68°.

And "6 5" at the bottom left.

So probably, there are two lines: one from top-left to bottom-right, and another from top-right to bottom-left, intersecting the horizontal line.

But angle 3 is 90°, which might be at the top intersection of the two slanted lines.

Assume that the two slanted lines intersect at a point above, forming angle 3 = 90°.

Then they cross the horizontal line at two points.

At the right crossing point, angle between the slanted line and horizontal is ∠1 = 68°.

At the left crossing point, angles ∠6 and ∠5.

Also, at the top, angle between the two slanted lines is 90°.

Now, to find all angles.

First, at the right intersection (point A): the slanted line and horizontal line intersect. Given ∠1 = 68°, which is likely the acute angle between them. Then the adjacent angle on the straight line is 180 - 68 = 112°, which is probably ∠2, since it's labeled next to it.

In the diagram, "2 1" with 1=68°, so likely ∠2 and ∠1 are adjacent angles at that intersection, so ∠2 = 180 - 68 = 112°.

Vertical angles: the angle opposite to ∠1 is also 68°, and opposite to ∠2 is 112°.

But in the labeling, probably ∠2 is the obtuse angle, so ∠2 = 112°.

Similarly, at the left intersection (point C), angles ∠6 and ∠5. We don't know yet.

Now, the two slanted lines intersect at point B above, with angle ∠3 = 90°.

The line from B to A (right) and B to C (left).

At point B, the angle between the two lines is 90°, so ∠3 = 90°.

Then, the adjacent angles at B are 90° each, since straight line, but at the intersection of two lines, vertical angles are equal, and adjacent are supplementary.

So if ∠3 = 90°, then the vertically opposite angle is also 90°, and the other two angles are 90° each, since 180-90=90, so actually all angles at B are 90°, so the two lines are perpendicular.

Is that possible? If angle at B is 90°, and it's the angle between the two lines, then yes, they are perpendicular, so all four angles at B are 90°.

So ∠3 = 90°, ∠4 = 90° (adjacent), and the other two are also 90°.

But in the diagram, ∠4 is labeled, so probably ∠4 = 90°.

Now, we need to find angles at the intersections with the horizontal.

Consider the triangle formed by the two slanted lines and the horizontal line.

Points: B (top intersection), A (right on horizontal), C (left on horizontal).

So triangle ABC, with BA and CA being the slanted lines, and AC the horizontal.

At B, angle is 90°.

At A, the angle in the triangle is the angle between BA and the horizontal. But at point A, the angle between the slanted line and horizontal is given as ∠1 = 68°, but is this the angle inside the triangle or outside?

In the diagram, ∠1 is likely the angle between the slanted line and the horizontal, on the side towards the triangle or away.

Typically, if the triangle is above the horizontal, then at A, the angle of the triangle is the angle between BA and AC, which is the same as the angle between the slanted line and the horizontal, so if ∠1 = 68°, and it's acute, likely it's the angle inside the triangle.

Similarly at C.

So in triangle ABC, angle at B is 90°, angle at A is 68°, so angle at C is 180 - 90 - 68 = 22°.

Now, at point C (left intersection), the angle between the slanted line and the horizontal is 22°.

Now, the angles at C: the horizontal line, and the slanted line.

The angle inside the triangle is 22°, so the adjacent angle on the straight line is 180 - 22 = 158°.

In the labeling, at left intersection, angles are ∠6 and ∠5. Probably ∠5 is the acute angle or the one inside.

Typically, ∠5 might be the angle between the slanted line and the horizontal on the upper side, which is 22°, and ∠6 is the adjacent angle, 158°.

But let's see the positions.

Also, we have angle 4 at the top, which we said is 90°.

Now, to confirm, let's list all.

First, at point B (top intersection of two slanted lines): since they are perpendicular, all angles are 90°. So ∠3 = 90°, ∠4 = 90° (assuming ∠4 is adjacent to ∠3).

In the diagram, "3 90°" and "4" below it, so likely ∠4 is the angle below, which is also 90°.

Now, at point A (right on horizontal): angles between the slanted line and horizontal.

∠1 = 68° (given), which is probably the angle in the lower right or upper right. Since the triangle is above, likely ∠1 is the angle above the horizontal, inside the triangle, so 68°.

Then the adjacent angle on the straight line is 180 - 68 = 112°, which is ∠2, as labeled.

Vertical angles: the angle below the horizontal at A is also 68° (opposite to ∠1), and the other is 112°.

But in the labeling, probably ∠2 is the obtuse angle, so ∠2 = 112°.

Similarly, at point C (left on horizontal): angle inside the triangle is 22°, as calculated.

So the angle between the slanted line and horizontal, on the upper side, is 22°.

In the diagram, at left, "6 5", so likely ∠5 is the acute angle, 22°, and ∠6 is the obtuse angle, 158°.

Now, we need to assign which is which.

Probably, ∠5 is the angle adjacent to the triangle, so 22°, and ∠6 is the other.

To confirm, the sum around each point should be 360°, but at each intersection, it's two lines, so four angles summing to 360°, with vertical angles equal.

At point A: angles are 68°, 112°, 68°, 112°.

Similarly at C: 22°, 158°, 22°, 158°.

At B: all 90°.

Now, the angles to find are ∠1 to ∠6.

Given: ∠1 = 68°, ∠3 = 90°.

We have ∠2 = 112° (adjacent to ∠1 at A).

∠4 = 90° (at B, adjacent to ∠3).

∠5 and ∠6 at C.

As above, if ∠5 is the acute angle, 22°, ∠6 is 158°.

But let's see the labeling: "6 5" at left, with 6 on left, 5 on right, so probably ∠6 is the left angle, ∠5 is the right angle at that intersection.

At point C, the horizontal line, the slanted line coming down to the right (since it goes to B above and A right).

So at C, the angle between the horizontal and the slanted line, on the side towards A, is the angle inside the triangle, 22°.

If we consider the directions, the angle to the right of the slanted line might be ∠5, etc.

Typically, in such diagrams, ∠5 is the angle between the horizontal and the slanted line on the upper side, which is 22°, and ∠6 is the adjacent angle on the left, 158°.

Since the horizontal line is straight, and the slanted line, the angle on the left side of the slanted line might be ∠6.

To avoid confusion, let's assume that at each intersection, the angles are labeled in order.

For point C: the two lines divide the plane into four angles. The angle above the horizontal and to the left of the slanted line might be ∠6, and above and to the right might be ∠5, but since the slanted line is going up to the right, the angle between the horizontal and the slanted line on the upper side is acute, 22°, and it is on the right side of the slanted line if we face the direction.

Perhaps it's easier to state:

At left intersection (C):

- The angle between the horizontal and the slanted line, measured from horizontal to slanted line upwards, is 22°. This is likely ∠5, as it's closer to the triangle.

- Then the adjacent angle on the straight line is 158°, which is ∠6.

Similarly, at right intersection (A):

- ∠1 = 68° (between horizontal and slanted line, upwards)

- ∠2 = 112° (adjacent, downwards or something, but in the diagram, probably ∠2 is the other angle at that vertex).

In the labeling, "2 1" with 1=68°, so likely ∠1 and ∠2 are adjacent, so ∠2 = 180 - 68 = 112°.

At top, "3 90°" and "4", so ∠3 and 4 are adjacent, so ∠4 = 90°.

At left, "6 5", so ∠6 and ∠5 are adjacent, so if ∠5 is the acute one, 22°, then ∠6 = 158°.

Now, to confirm, in the triangle, angles are at B 90°, at A 68°, at C 22°, sum 180°, good.

So summary for Part II:

21) ∠1 = 68° (given)

22) ∠2 = 112° (supplementary to ∠1)

23) ∠3 = 90° (given)

24) ∠4 = 90° (since perpendicular, or adjacent to ∠3)

25) ∠5 = 22° (angle at C in triangle)

26) ∠6 = 158° (supplementary to ∠5)

Now, back to problem 18.

For problem 18, after research in my mind, I recall that in some versions, the triangle has two base angles equal, and the vertex angle is x, but no value, so perhaps it's 60, or perhaps there's a typo.

Perhaps the "arrows" indicate that the two sides are equal, and the base angles are equal, and x is the vertex, but we need to realize that the sum is 180, but with two variables.

Another possibility: in the diagram, the two base angles are marked with arcs, and x is at the top, and it's given that the triangle is isosceles, but still.

Perhaps for problem 18, x is 60, as a standard answer.

Let's assume that.

Or perhaps it's 70 or something.

Let's calculate if we had more info.

I think for the sake of completing, I'll set x = 60 for problem 18.

But let's look at problem 14 again.

For problem 14, let's solve properly.

Left triangle: right-angled, 65° at top, so bottom-right angle = 180 - 90 - 65 = 25°.

This 25° is at the bottom-right of left triangle.

Now, the small triangle on the right has angles: at top 50°, at bottom-right 80°, and at bottom-left x°.

The bottom-left of small triangle is the same point as the bottom-right of left triangle? No, they are different points.

In the diagram, the two triangles share a common side or vertex.

Typically, they share the vertex at the bottom where the 25° is, but