Properties of Triangles - Angle bisectors Worksheets - Free Printable

We are given:

- $ m\angle 2 = 18x - 1 $
- $ m\angle SUT = 34x $
- We are to find $ x $

From the diagram (described in the image), we can infer the following:

- Point $ P $ lies on segment $ ST $
- Point $ T $ is on segment $ SU $
- There is a triangle $ \triangle SUT $
- Angle $ \angle 2 $ is at point $ T $, and it's part of angle $ \angle SUT $
- A line from $ P $ to $ U $ creates two angles at $ T $: $ \angle 1 $ and $ \angle 2 $
- $ \angle 1 $ and $ \angle 2 $ appear to be adjacent angles forming $ \angle SUT $

But more importantly, angle $ \angle 2 $ is likely the same as $ \angle SUT $, or perhaps they are related by some geometric property.

Wait — let’s analyze carefully.

Looking at the labels:

- $ m\angle 2 = 18x - 1 $
- $ m\angle SUT = 34x $

And since $ \angle SUT $ is the angle at vertex $ U $, and $ \angle 2 $ is labeled at point $ T $, that suggests they are not the same angle.

But wait — in the diagram:

- $ \angle 2 $ is at point $ T $, between points $ S $, $ T $, and $ U $? Or is it between $ P $, $ T $, $ U $?

From the notation: $ \angle 2 $ is marked at $ T $, between $ PT $ and $ TU $. So $ \angle 2 = \angle PTU $

And $ \angle SUT $ is the angle at $ U $, between $ S $, $ U $, $ T $

So unless there's a triangle or parallel lines involved, we need more.

But here's a key observation: The problem may be based on angle relationships in a triangle or straight line.

Let’s suppose that $ \angle 1 $ and $ \angle 2 $ are adjacent angles forming a straight line at point $ T $, but that doesn't make sense because both are inside the triangle.

Alternatively, maybe $ \angle SUT $ is an exterior angle to triangle $ \triangle PTU $, and $ \angle 2 $ is one of the remote interior angles.

Wait — actually, let's consider this possibility:

Suppose that $ \angle SUT $ is an exterior angle to triangle $ \triangle PTU $, and $ \angle 2 $ is one of the interior angles. But that seems unlikely.

Alternatively, perhaps $ \angle 2 $ and $ \angle SUT $ are supplementary or equal due to some symmetry.

But the most plausible interpretation — especially with multiple-choice answers — is that $ \angle 2 $ and $ \angle SUT $ are equal, or related via a triangle angle sum.

Wait — another idea: perhaps $ \angle 2 $ is an angle in triangle $ \triangle SUT $ at vertex $ T $, and $ \angle SUT $ is the angle at vertex $ U $. Then the third angle is at $ S $.

But unless we have more information, we can’t proceed.

But here's a better clue: The angle $ \angle 2 $ is labeled at $ T $, and $ \angle SUT $ is at $ U $.

Wait — could $ \angle 2 $ be equal to $ \angle SUT $? That would only happen if the triangle is isosceles or something.

But let's look at the answer choices:

A) 5
B) 3
C) 1
D) 6

Try plugging in values.

Let’s suppose that $ \angle 2 $ and $ \angle SUT $ are equal? Then:

$$
18x - 1 = 34x \\
-1 = 16x \\
x = -\frac{1}{16}
$$

Not possible — negative, not among choices.

So they are not equal.

Perhaps $ \angle 2 $ and $ \angle SUT $ are supplementary? Then:

$$
(18x - 1) + 34x = 180 \\
52x - 1 = 180 \\
52x = 181 \\
x \approx 3.48
$$

Not matching any choice exactly.

Try $ x = 3 $:
Then $ m\angle 2 = 18(3) - 1 = 54 - 1 = 53^\circ $
$ m\angle SUT = 34(3) = 102^\circ $

Now check if these could be parts of a triangle.

Wait — perhaps $ \angle 2 $ and $ \angle SUT $ are two angles in a triangle, and the third angle is known?

But no info.

Another idea: Perhaps $ \angle 2 $ is half of $ \angle SUT $? Not likely.

Wait — maybe $ \angle 2 $ and $ \angle SUT $ are vertical angles or corresponding angles?

But no indication of parallel lines.

Wait — look again: point $ P $ is on $ ST $, and a line from $ P $ to $ U $ makes angle $ \angle 1 $ and $ \angle 2 $ at $ T $. So $ \angle 1 $ and $ \angle 2 $ are adjacent angles forming $ \angle STU $.

So:
$$
\angle STU = \angle 1 + \angle 2
$$

But we don’t know $ \angle 1 $.

But perhaps $ \angle SUT $ is an exterior angle?

Wait — here's a key idea: In triangle $ \triangle SUT $, $ \angle SUT $ is one angle, and $ \angle 2 $ is another angle?

But $ \angle 2 $ is at $ T $, so $ \angle 2 = \angle STU $

Ah! This is crucial.

If $ \angle 2 $ is the angle at $ T $ in triangle $ \triangle SUT $, then:

$$
\angle 2 = \angle STU = 18x - 1
$$
$$
\angle SUT = 34x
$$

And let’s say $ \angle S $ is the third angle.

But unless we know more, we can’t use angle sum.

But perhaps there's a special relationship.

Wait — could $ \angle SUT $ be twice $ \angle 2 $? Try:

$$
34x = 2(18x - 1) = 36x - 2 \\
34x = 36x - 2 \\
-2x = -2 \\
x = 1
$$

That gives $ x = 1 $

Check:

- $ m\angle 2 = 18(1) - 1 = 17^\circ $
- $ m\angle SUT = 34(1) = 34^\circ $

So $ \angle SUT = 2 \times \angle 2 $ — interesting.

Is that possible?

Maybe in an isosceles triangle?

But we don’t know.

But now try other options.

Try $ x = 3 $:

- $ \angle 2 = 18(3) - 1 = 53^\circ $
- $ \angle SUT = 34(3) = 102^\circ $

No obvious relation.

Try $ x = 5 $:

- $ \angle 2 = 90 - 1 = 89^\circ $
- $ \angle SUT = 170^\circ $

Too big for a triangle.

Try $ x = 6 $:

- $ \angle 2 = 108 - 1 = 107^\circ $
- $ \angle SUT = 204^\circ $ → impossible

So only $ x = 1 $ and $ x = 3 $ give reasonable angles.

But earlier we found that if $ \angle SUT = 2 \times \angle 2 $, then $ x = 1 $

But is there a reason to believe that?

Wait — perhaps $ \angle SUT $ is an exterior angle to triangle $ \triangle PTU $, and $ \angle 2 $ is one of the remote interior angles.

But without knowing $ \angle 1 $, hard to say.

Alternatively, perhaps $ \angle 2 $ and $ \angle SUT $ are vertical angles — but they’re at different vertices.

Wait — another idea: maybe $ \angle 2 $ and $ \angle SUT $ are equal because of parallel lines?

But no indication.

Wait — perhaps $ \angle 2 $ is equal to $ \angle SUT $ in a certain configuration.

But we already saw that leads to negative x.

Unless I made a mistake.

Wait — let's think differently.

Perhaps $ \angle 2 $ and $ \angle SUT $ are supplementary?

Try $ x = 3 $:

- $ \angle 2 = 53^\circ $
- $ \angle SUT = 102^\circ $
- Sum = 155 — not 180

$ x = 5 $: $ 89 + 170 = 259 $ — too big

$ x = 1 $: $ 17 + 34 = 51 $ — too small

No.

Wait — what if $ \angle 2 $ is half of $ \angle SUT $? Then:

$$
18x - 1 = \frac{1}{2}(34x) = 17x \\
18x - 1 = 17x \\
x = 1
$$

Again, $ x = 1 $

Same result.

So $ \angle 2 = 17^\circ $, $ \angle SUT = 34^\circ $, so $ \angle 2 = \frac{1}{2} \angle SUT $

This suggests that $ \angle SUT $ is twice $ \angle 2 $, which often happens in isosceles triangles or when dealing with angle bisectors.

But is there a geometric reason?

Wait — perhaps $ \angle 2 $ is an angle in a triangle, and $ \angle SUT $ is an exterior angle.

For example, if $ \angle SUT $ is an exterior angle to triangle $ \triangle PTU $, then:

$$
\angle SUT = \angle 1 + \angle 2
$$

But we don’t know $ \angle 1 $

But if $ \angle 1 = \angle 2 $, then $ \angle SUT = 2\angle 2 $

Which is exactly what we have!

So assume $ \angle 1 = \angle 2 $, then:

$$
\angle SUT = \angle 1 + \angle 2 = 2\angle 2
$$

So:

$$
34x = 2(18x - 1) = 36x - 2 \\
34x = 36x - 2 \\
-2x = -2 \\
x = 1
$$

Yes! So $ x = 1 $

Now check if this makes sense:

- $ \angle 2 = 18(1) - 1 = 17^\circ $
- $ \angle SUT = 34(1) = 34^\circ $
- If $ \angle 1 = \angle 2 = 17^\circ $, then $ \angle STU = \angle 1 + \angle 2 = 34^\circ $
- And $ \angle SUT = 34^\circ $, so in triangle $ \triangle SUT $, two angles are $ 34^\circ $ each — so it's isosceles with $ ST = SU $

Perfectly valid.

So the assumption that $ \angle 1 = \angle 2 $ (perhaps due to symmetry or isosceles triangle) leads us to $ x = 1 $

And $ x = 1 $ is option C.

✔ Answer: C) 1

Explanation:

We are told:
- $ m\angle 2 = 18x - 1 $
- $ m\angle SUT = 34x $

From the diagram, $ \angle 2 $ is part of $ \angle STU $, and $ \angle SUT $ is the angle at $ U $. If $ \angle 1 = \angle 2 $, then $ \angle STU = 2\angle 2 $. In triangle $ \triangle SUT $, if $ \angle STU = \angle SUT $, then the triangle is isosceles, and this relationship holds.

Assuming $ \angle SUT = 2\angle 2 $, we solve:

$$
34x = 2(18x - 1) \\
34x = 36x - 2 \\
-2x = -2 \\
x = 1
$$

Thus, the correct answer is:

> C) 1 ✔