Solving Problems Using Angle Relationships #2 | Interactive ... - Free Printable

Let’s solve each problem one by one, using basic angle rules you’ve learned — like angles on a straight line add to 180°, vertical angles are equal, and angles around a point or in triangles follow specific patterns.

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Problem 1: Find k

We see three rays from a point. The bottom two angles are labeled 26° and (k + 4)°, and together they make up the larger angle that is split into those two parts. But wait — actually, looking closely, it seems like the total angle between the outermost rays is made of two adjacent angles: one is 26°, and the other is (k + 4)°, and there’s no indication they’re part of a straight line or triangle. Actually, re-examining — this looks like an angle bisector or just two adjacent angles adding to a bigger one? Wait — no, actually, the diagram shows two angles sharing a common ray: one is 26°, the other is (k+4)°, and they are adjacent, forming a larger angle. But we don’t have the total. Hmm.

Wait — maybe I misread. Let me think again.

Actually, in many such diagrams, if two angles are shown next to each other with no other info, sometimes they are meant to be equal? No, that’s not safe.

Wait — perhaps the figure is showing that the angle marked “k” is the difference? Or maybe it's a typo in my reading.

Hold on — let’s look at the labels again.

In Problem 1: There are three rays from a vertex. The smallest angle (between the bottom two rays) is 26°. The middle angle (between the middle and top ray) is labeled (k + 4)°. And the entire angle from bottom to top ray is not given. But then what is “k”? The label “k” is pointing to the middle angle? Or is “k” the variable for the expression?

Actually, looking at the way it’s drawn: the angle between the lower two rays is 26°, and the angle between the upper two rays is (k + 4)°, and these two angles are adjacent and together form the whole angle from bottom to top ray. But we still don’t have a total.

Wait — unless... is there a right angle or something? No.

Perhaps I need to assume that the two angles are equal? That would be unusual without marking.

Wait — maybe the diagram is indicating that the angle labeled “k” is the same as the (k+4)°? That doesn’t make sense.

Alternatively — perhaps “k” is the measure of the middle angle, and it’s labeled as (k + 4)°? That would mean k = k + 4, which is impossible.

I think I’m overcomplicating. Let me try a different approach.

Looking back at standard problems like this: often, when you have two adjacent angles and one is expressed in terms of a variable, and no other info, it might be that they are complementary or supplementary — but here, no straight line.

Wait — perhaps the key is that the two angles together form a known total? But nothing is given.

Unless… oh! Maybe the angle marked with the arc and labeled “k” is actually the sum? No.

Let me check online or recall similar problems.

Actually, upon second thought — in some textbooks, when you have two angles sharing a ray, and one is given numerically and the other algebraically, and no other info, it might be that they are equal — but that’s only if marked with same arcs.

Here, both angles have single arcs — meaning they are congruent!

Yes! In geometry diagrams, if two angles have the same number of arcs (like one arc each), it means they are equal in measure.

So in Problem 1: the angle labeled 26° and the angle labeled (k + 4)° both have one arc — so they are equal.

Therefore:

k + 4 = 26

Subtract 4 from both sides:

k = 22

✔ That makes sense.

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Problem 2: Find r

We have two adjacent angles: one is 58°, the other is r°, and together they form a right angle (because there’s a square corner symbol).

Right angle = 90°

So:

58 + r = 90

r = 90 - 58 = 32

✔ r = 32

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Problem 3: Find t

We have a straight line. On it, two angles are shown: one is 145°, the other is t°, and they are adjacent, forming a straight line.

Angles on a straight line add to 180°.

So:

145 + t = 180

t = 180 - 145 = 35

✔ t = 35

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Problem 4: Find a

Two lines intersect, forming vertical angles.

One pair of vertical angles is labeled (5a)° and 140°.

Vertical angles are equal.

So:

5a = 140

Divide both sides by 5:

a = 28

✔ a = 28

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Problem 5: Find d

We have several angles around a point? Or on a straight line?

Looking: there’s a horizontal line going left, then a vertical line going down, making a right angle (90°). Then from the vertex, another ray goes down-right, creating angles.

Specifically: from the vertical line, one angle is 25°, and another is (5d)°, and together with the right angle, they might form a straight line?

Wait — let’s trace:

- Horizontal line to the left.
- Vertical line downward — so angle between them is 90° (marked with square).
- From the same vertex, another ray goes down and right, making two angles with the vertical: one is 25°, the other is (5d)°.
- Also, the horizontal line continues to the right? Not shown, but probably implied.

Actually, the three angles at the vertex: the 90° (between left-horizontal and down-vertical), then from down-vertical to the new ray is 25°, and from new ray to... where?

Wait — perhaps the total angle from left-horizontal to the new ray is composed of 90° + 25° + (5d)°? But that doesn't help.

Another idea: maybe the angles around the point on one side add to 180°.

Notice: the horizontal line is straight. So from left to right is 180°.

From the left end, we go down vertically (90°), then from vertical to the slanted ray is 25°, and from slanted ray to the right horizontal should be the rest.

But the right horizontal isn't drawn, but we can assume the full straight line.

So, starting from left horizontal, turning down to vertical: 90°.

Then from vertical to slanted ray: 25°.

Then from slanted ray to right horizontal: let's call it x.

Total: 90 + 25 + x = 180 → x = 65°.

But in the diagram, the angle between the slanted ray and the vertical is labeled as (5d)°? Wait no — looking back:

The diagram says: from the vertical, one angle is 25°, and another is (5d)°, and they are adjacent, and together with the 90°, they might be on a straight line.

Actually, re-examining: the three angles at the vertex on the lower side:

- Between left-horizontal and vertical: 90°
- Between vertical and first slanted ray: 25°
- Between first slanted ray and second slanted ray: (5d)°
- And then to the right-horizontal? But only two slanted rays are shown.

Wait — perhaps the (5d)° and 25° are on opposite sides? No.

Another interpretation: the angle between the vertical and the slanted ray is split into two parts: 25° and (5d)°, and together they make the angle from vertical to the slanted ray.

But then what is the total?

Perhaps the key is that the angle between the left-horizontal and the slanted ray is 90° + 25° = 115°, and the remaining to 180° is 65°, but that's not labeled.

Wait — look at the label: "d" is pointing to the angle that is (5d)°, and also there's a 25° angle adjacent to it, and both are between the vertical and the slanted ray? That doesn't make sense.

Perhaps the 25° and (5d)° are adjacent angles that together form the angle between the vertical and the horizontal on the right.

Assume that from the vertical line to the right-horizontal line is 90° (since it's a straight line from left to right, and vertical is perpendicular).

So, if from vertical to slanted ray is 25°, and from slanted ray to right-horizontal is (5d)°, then:

25 + 5d = 90

Because they make up the 90° from vertical to right-horizontal.

Is that correct? Let's see: left-horizontal to vertical is 90°, vertical to right-horizontal is another 90°, total 180°.

If the slanted ray is in the lower-right quadrant, then yes, the angle from vertical to right-horizontal is 90°, and it's split into 25° and (5d)°.

So:

25 + 5d = 90

5d = 65

d = 13

✔ d = 13

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Problem 6: Find w

Two lines intersect. One angle is 135°, and the adjacent angle is (45 + 3w)°.

Adjacent angles on a straight line add to 180°.

So:

(45 + 3w) + 135 = 180

Combine like terms:

45 + 135 + 3w = 180

180 + 3w = 180

Subtract 180 from both sides:

3w = 0

w = 0

Wait, that can't be right. Did I misidentify?

If two lines intersect, adjacent angles are supplementary.

So if one angle is 135°, the adjacent angle should be 180 - 135 = 45°.

And that adjacent angle is labeled as (45 + 3w)°.

So:

45 + 3w = 45

Then 3w = 0, w = 0.

But is that possible? Mathematically yes, but let's verify.

If w=0, then the angle is 45 + 0 = 45°, and 45° + 135° = 180°, which is correct for adjacent angles on a straight line.

Also, vertical angles would be equal: the angle opposite 135° is also 135°, and opposite 45° is 45°.

So it checks out.

But w=0 seems odd, but mathematically correct.

Perhaps the diagram has the (45 + 3w)° as the vertical angle to 135°? But no, because vertical angles are equal, so if it were vertical, 45 + 3w = 135, then 3w=90, w=30.

But in the diagram, how are they positioned?

Typically, in such diagrams, if two angles are adjacent (sharing a side), they are supplementary.

Looking at the description: "two lines intersect", and angles labeled 135° and (45 + 3w)°.

If they are adjacent, then sum to 180°.

If they are vertical, then equal.

In the absence of a diagram, but based on standard labeling, often the angles that are next to each other are adjacent.

But in this case, if (45 + 3w) is adjacent to 135, then as above, w=0.

But let me double-check with the calculation.

Set up equation assuming adjacent:

(45 + 3w) + 135 = 180

180 + 3w = 180

3w = 0

w = 0

If they are vertical angles, then:

45 + 3w = 135

3w = 90

w = 30

Which one is it?

In the original problem statement, it says "find the value of each variable", and for problem 6, it's likely that (45 + 3w) and 135 are adjacent, because if they were vertical, it would be straightforward, but w=0 is valid.

However, let's think: if w=0, then the angle is 45°, which is fine.

But perhaps in the diagram, the (45 + 3w) is not adjacent but vertical.

I recall that in some diagrams, the angles are labeled such that the expression is for the vertical angle.

To resolve this, let's consider the context.

In problem 4, we had vertical angles, and we set them equal.

In problem 6, if the two angles are vertical, then 45 + 3w = 135, w=30.

If adjacent, w=0.

Now, w=0 might be acceptable, but let's see the answer.

Perhaps I can look for consistency.

Another way: the sum of all angles around a point is 360°.

If two lines intersect, they form two pairs of vertical angles.

Suppose one pair is 135° each, the other pair is x each.

Then 2*135 + 2*x = 360

270 + 2x = 360

2x = 90

x = 45

So the other angles are 45°.

Now, if (45 + 3w) is one of those 45° angles, then 45 + 3w = 45, w=0.

If it's labeled as the 135° angle, but it's already given as 135, so probably not.

In the diagram, likely (45 + 3w) is the measure of the angle that is adjacent to 135°, so it should be 45°, hence w=0.

But let's confirm with the calculation.

Perhaps the problem intends for (45 + 3w) to be the vertical angle, but that would be redundant since vertical angles are equal, and 135 is given, so why express it as 45+3w unless it's different.

In most textbook problems, when they give an expression and a number for intersecting lines, if they are adjacent, they sum to 180; if vertical, they are equal.

Here, since 45 + 3w is likely not 135 (unless w=30), but if w=30, 45+90=135, so it could be vertical.

But in the diagram, how are they positioned?

Upon re‐examining the user's description: "6. w = ____" and the diagram has two lines intersecting, with one angle 135°, and another angle labeled (45 + 3w)°.

In standard position, if the 135° is in one corner, the adjacent angle is 45°, and the vertical is 135°.

So if (45 + 3w) is written in the adjacent position, it should be 45°, so w=0.

If it's in the vertical position, it should be 135°, so w=30.

I think there's ambiguity, but let's look at the expression: 45 + 3w. If w=0, it's 45, which is nice. If w=30, it's 135, also nice.

But in the context of the other problems, w=0 might be intended.

Perhaps I can calculate both ways.

Another thought: in the diagram, the angle labeled (45 + 3w) might be the one that is vertically opposite to the 135°, but that would mean they are equal, so 45 + 3w = 135, w=30.

Or adjacent, w=0.

I recall that in some systems, the angle expressions are for the non-given angles.

Let's assume that the 135° is given, and (45 + 3w) is the adjacent angle, so they sum to 180.

So 45 + 3w + 135 = 180

As before, 180 + 3w = 180, w=0.

Perhaps the 135° is not adjacent but vertical, but then why label another angle.

I think the safest bet is to assume they are adjacent, as is common in such problems.

Moreover, in problem 4, we had vertical angles explicitly.

For problem 6, let's go with adjacent.

So w = 0.

But let's write it as 0.

Perhaps the problem has a typo, but based on math, w=0 is correct if adjacent.

Another way: the difference between 135 and 45 is 90, but not helpful.

I'll go with w=0.

But let's box it.

No, let's think again.

In the initial request, for problem 6, if the two angles are on a straight line, they sum to 180.

So (45 + 3w) + 135 = 180

3w = 180 - 180 = 0

w = 0

Yes.

So final answers:

1. k = 22

2. r = 32

3. t = 35

4. a = 28

5. d = 13

6. w = 0

Now, to present.

Final Answer:
1. k = 22
2. r = 32
3. t = 35
4. a = 28
5. d = 13
6. w = 0