Determine the unknown angles in the given figures using triangle angle sum properties.

Let’s solve both problems step by step.

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Problem 5:

We have two triangles that share a common vertex (where angles w° and x° are). The key idea is:

- In any triangle, the sum of interior angles = 180°.
- Angles on a straight line add up to 180°.
- Vertically opposite angles are equal — so w° = x°? Wait, no — actually, in this figure, w° and x° are vertically opposite? Let me check.

Looking at the diagram description: Two triangles intersecting at a point. So the angles labeled w° and x° are vertically opposite, meaning they are equal. But wait — let’s not assume yet. Let’s calculate each triangle separately first.

Left triangle has angles: 42°, 85°, and w°.

So:

w° = 180° - 42° - 85°
= 180° - 127°
= 53°

Right triangle has angles: 88°, y°, and x°.

But notice: w° and x° are vertically opposite angles → so x° = w° = 53°

Now plug into right triangle:

x° + 88° + y° = 180°
53° + 88° + y° = 180°
141° + y° = 180°
y° = 180° - 141° = 39°

✔ So for Problem 5:
- w = 53
- x = 53
- y = 39

Wait — but let me double-check: Are w and x really vertically opposite? Yes, because the two lines cross, forming an “X” shape. So the angle across from w is x — so yes, vertically opposite → equal.

Also, we can verify the right triangle: 53 + 88 + 39 = 180 → 53+88=141, 141+39=180 ✔️

Left triangle: 42 + 85 + 53 = 180 → 42+85=127, 127+53=180 ✔️

Perfect.

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Problem 6:

This is a big triangle with a smaller triangle inside it. We’re given:

Big triangle angles: one is 73°, another is 75°, and the third is split into parts involving x°, y°, z°, and 26°.

Actually, looking carefully: There’s a small triangle inside with angles: 26°, y°, and z°. And the big triangle has angles: x°, 73°, and 75°? Wait — no.

Wait — the big triangle’s three corners are:

- Left corner: x°
- Top right: 73°
- Bottom right: 75°

So if those are the three angles of the big triangle, then:

x° + 73° + 75° = 180°
x° + 148° = 180°
x° = 180° - 148° = 32°

Okay, so x = 32.

Now, look at the small inner triangle. It has angles: 26°, y°, and z°.

But also, note that angle z° and the angle next to it (in the big triangle) form a straight line? Or maybe not.

Wait — actually, the small triangle shares a side with the big triangle. Let’s think differently.

The big triangle has angles: x° (left), 73° (top right), 75° (bottom right). We found x = 32°.

Now, inside, there’s a line drawn from the bottom side to the top side, creating a small triangle with angles 26°, y°, z°.

Notice: The angle adjacent to z° (on the same straight line) is part of the big triangle’s top angle? Not exactly.

Alternative approach: Use exterior angle or just focus on the small triangle.

In the small triangle: angles are 26°, y°, z° → so:

26 + y + z = 180 → y + z = 154° ...(1)

Now, look at the quadrilateral or other relationships.

Actually, consider the triangle that includes angle y° and the 73° angle.

Wait — perhaps better: Look at the triangle formed by the left part.

Another way: The angle at the top of the big triangle is 73°. That 73° is split into two parts: one is z°, and the other is... what?

Actually, looking at the diagram description: The small triangle is inside, sharing the base? Hmm.

Wait — here's a better method:

Consider the triangle that has angles: x°, 26°, and the angle above the 26° which is part of the big triangle.

Actually, let’s label points mentally.

Assume the big triangle is ABC:

- A = left vertex = x°
- B = top right = 73°
- C = bottom right = 75°

Then, there’s a point D on AC, and E on AB, such that DE is drawn, forming triangle DEC or something? Actually, the problem says: inside, there’s a triangle with angles 26°, y°, z°.

From the description: "y°" and "z°" are adjacent along a line, and 26° is below.

Perhaps: The small triangle has vertices at: the intersection point on the left side, the bottom side, and somewhere else.

Wait — simpler: Since we know x = 32°, now look at the triangle that contains the 26° angle.

That triangle has angles: 26°, and two others. One of them is y°, and the third is... let’s call it angle P.

But also, angle P and z° are on a straight line? Or maybe not.

Wait — here’s the key: The angle z° and the angle next to it (which is part of the big triangle’s top angle) add up to 73°? Not necessarily.

Alternative correct approach:

Look at the triangle that has the 26° angle. Its three angles are: 26°, y°, and the angle at the top-left of that small triangle.

But that top-left angle is actually supplementary to z°? No.

Wait — I think I see it.

In the big triangle, total angles sum to 180°, we have x=32°, 73°, 75° — good.

Now, the line drawn inside creates a small triangle with angles 26°, y°, z°.

Additionally, the angle adjacent to z° (along the top side) plus z° equals 73°? Maybe.

Actually, let’s consider the polygon or use the fact that the sum around a point is 360°, but that might be overcomplicating.

Better: Consider the triangle that includes the 73° angle.

The 73° angle is at the top right. From that vertex, a line goes down to the bottom side, creating two angles: one is z°, and the other is... let’s say angle Q.

So z° + angle Q = 73°? Only if the line is going to the opposite side — but in this case, it seems the line is connecting two sides.

Wait — perhaps the small triangle is formed by drawing a line from the bottom side to the left side.

Let me try this:

Denote the big triangle as having vertices:

- Left: A (angle x°)
- Top: B (angle 73°)
- Right: C (angle 75°)

Now, suppose we draw a line from a point D on AC to a point E on AB, such that triangle ADE is formed? But the problem mentions a 26° angle inside.

Actually, from the description: "26°" is at the bottom, near the right side.

Perhaps: There is a small triangle at the bottom right with angles 26°, and two others.

Wait — here's a standard trick:

The small triangle has angles: 26°, y°, z°.

Also, the angle y° is an exterior angle to another triangle? Or perhaps y° is equal to the sum of two remote interior angles.

Let’s think about the triangle that has the 26° angle and the 75° angle.

Actually, consider the triangle formed by the bottom-right corner.

At the bottom-right corner of the big triangle, the angle is 75°. Inside, there’s a 26° angle, so the remaining part of that 75° angle is 75° - 26° = 49°? Is that right?

Yes! Because the 26° angle is part of the 75° angle at the bottom-right vertex.

So, at vertex C (bottom right), the big angle is 75°, and it’s split into two parts: one is 26° (inside the small triangle), and the other is, say, angle R = 75° - 26° = 49°.

Now, this angle R = 49° is an angle in the quadrilateral or in another triangle.

Specifically, now consider the triangle that has angles: x° (at left), angle R = 49° (at bottom right), and the angle at the top which is part of the 73°.

Wait — actually, the triangle formed by points A, C, and the intersection point.

Perhaps better: Now we have a triangle with angles: x° = 32°, 49°, and the third angle — let’s call it S.

Then S = 180° - 32° - 49° = 99°.

But where is this angle S? It should be at the top, but the top angle is 73°, so that doesn’t match.

I think I made a mistake.

Let me start over for Problem 6.

Given:

Big triangle angles:
- Left: x°
- Top right: 73°
- Bottom right: 75°

So x + 73 + 75 = 180 → x = 32°, as before.

Now, inside, there is a small triangle with angles: 26°, y°, z°.

Also, from the diagram, it appears that the 26° angle is at the bottom, and y° and z° are at the other two vertices of the small triangle.

Moreover, the angle z° is adjacent to the 73° angle, and together they make up the full angle at that vertex? Not necessarily.

Another idea: The line that forms the small triangle creates a new triangle with the left part.

Consider the triangle that has angles: x° = 32°, y°, and the angle that is supplementary to z° or something.

Wait — here's the correct insight:

The small triangle has angles 26°, y°, z°.

The angle y° is an exterior angle to the triangle that includes the 26° and the part of the big triangle.

Actually, recall that in such figures, often y° is equal to the sum of the two non-adjacent interior angles.

Let’s look at the triangle that has the 26° angle and the 75° angle.

At the bottom-right vertex, the big angle is 75°. The small triangle takes 26° of it, so the remaining angle is 75° - 26° = 49°. This 49° is an angle in the larger triangle that also includes x° and y°.

Specifically, consider the triangle formed by the left vertex (x°), the bottom-right vertex (with the 49° part), and the vertex where y° is.

So that triangle has angles: x° = 32°, 49°, and y°.

Is that possible? Let's see: 32 + 49 + y = 180 → y = 180 - 81 = 99°.

Then, in the small triangle, we have angles: 26°, y° = 99°, and z°.

So 26 + 99 + z = 180 → 125 + z = 180 → z = 55°.

Now, check if this makes sense with the top angle.

The top angle of the big triangle is 73°. At that vertex, the angle is split into z° and another angle. If z° = 55°, then the other part is 73° - 55° = 18°.

Now, is there a triangle that includes that 18°? For example, the triangle at the top might have angles: 18°, and two others.

But we don't need to, because all angles should add up.

Let me verify the entire figure.

Big triangle: 32° + 73° + 75° = 180° ✔️

Small triangle: 26° + 99° + 55° = 180° ✔️

Now, the angle at the top: 73° is composed of z° = 55° and another angle, say T = 18°.

Similarly, at the bottom-right: 75° is composed of 26° and 49°.

Now, consider the triangle that has the 49° angle, the 18° angle, and the angle at the intersection.

The sum around the intersection point should be 360°, but perhaps it's consistent.

The triangle formed by the top, the intersection, and the bottom-right: angles would be 18°, 49°, and the angle at intersection.

18 + 49 = 67, so the third angle is 113°.

But also, at the intersection point, we have angles from different triangles.

Since we have consistency in the triangles we've checked, and no contradiction, this should be correct.

So for Problem 6:
- x = 32
- y = 99
- z = 55

But let me confirm y°: Is y° really 99°?

In the triangle with angles x=32°, the 49° (which is 75°-26°), and y°, yes, 32+49=81, 180-81=99.

And small triangle: 26+99+z=180 → z=55.

And top angle: z=55, so the remaining is 73-55=18, which is fine.

Another way: The angle y° is an exterior angle to the small triangle? No, y° is inside the small triangle.

Actually, y° is at the vertex where the line meets the left side.

I think it's correct.

So final answers:

Problem 5: w=53, x=53, y=39

Problem 6: x=32, y=99, z=55

Now, to box the final answer as per instruction.

Final Answer:
For problem 5: w = 53, x = 53, y = 39
For problem 6: x = 32, y = 99, z = 55