Using angle relationships to find angle measures 1 - Free Printable

Let's go through the problems shown in your image and solve them step by step, explaining each one clearly.

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Problem 2:

Given: A right angle (90°) split into two angles: one is 68°, and the other is labeled $ x $.

We are to find $ x $.

#### Solution:
Since the total angle is 90° (right angle), and one part is 68°:
$$
x = 90^\circ - 68^\circ = 22^\circ
$$

✔ So, $ x = 22^\circ $

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Problem 3:

Given: A straight line with an angle of 124° and another angle labeled $ x $. The total is 180° because it’s a straight line.

We are to find $ x $.

#### Solution:
Angles on a straight line add up to 180°:
$$
x + 124^\circ = 180^\circ
\Rightarrow x = 180^\circ - 124^\circ = 56^\circ
$$

✔ So, $ x = 56^\circ $

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Problem 5:

Given: A circle with angles around a point. There are three angles:
- One is 90°,
- Another is 72°,
- The third is labeled $ y $,
- And there's a large angle labeled 108°, which appears to be opposite or adjacent?

Wait — actually, looking at the diagram:

It seems like there’s a full circle (360°) with several angles. But the figure shows:
- A 90° angle,
- A 72° angle,
- An unknown $ y $,
- And a 108° angle.

But notice that the sum of angles around a point is 360°, so we can write:

$$
90^\circ + 72^\circ + y + 108^\circ = 360^\circ
$$

Add known values:
$$
90 + 72 = 162,\quad 162 + 108 = 270
\Rightarrow y = 360^\circ - 270^\circ = 90^\circ
$$

But wait — the student wrote $ x = 108^\circ $, $ y = 72^\circ $, $ z = 157^\circ $? That doesn’t match.

Wait — perhaps this is not a full circle? Let's re-express.

Looking again: Maybe this is a triangle or a polygon?

Actually, it looks like a triangle with an exterior angle.

Wait — the diagram shows:
- A triangle with one interior angle marked 72°,
- Another angle labeled $ y $,
- And an exterior angle of 108°,
- Also a 90° angle?

Wait — perhaps it's a triangle with angles inside.

Alternatively, maybe the 108° is an exterior angle, and the adjacent interior angle is $ y $, and the triangle has a 90° and 72°?

Wait — let's try to interpret correctly.

Actually, from the student’s work:
They wrote:
- $ x = 108^\circ $
- $ y = 72^\circ $
- $ z = 157^\circ $

But we don't see $ z $ in the diagram.

Wait — perhaps it's a triangle with one angle being 90°, and another angle is 72°, so the third angle would be:

$$
180^\circ - 90^\circ - 72^\circ = 18^\circ
$$

But that doesn't match.

Alternatively, if 108° is an exterior angle, then the interior angle adjacent to it is:

$$
180^\circ - 108^\circ = 72^\circ
$$

So if that interior angle is 72°, and another angle in the triangle is 90°, then the third angle is:

$$
180^\circ - 90^\circ - 72^\circ = 18^\circ
$$

Still not matching the student’s answers.

Wait — maybe the figure is showing angles around a point.

Let’s assume the angles around a point are:
- 90°,
- 72°,
- 108°,
- and $ y $

Then:
$$
90 + 72 + 108 + y = 360 \Rightarrow 270 + y = 360 \Rightarrow y = 90^\circ
$$

So $ y = 90^\circ $

But the student says $ x = 108^\circ $, $ y = 72^\circ $, $ z = 157^\circ $. This seems inconsistent.

Possibly the diagram is different.

Wait — perhaps the 108° is not one of the angles around the point, but rather an angle formed by lines intersecting?

Alternatively, maybe it's a quadrilateral or polygon.

But without seeing the exact image, I’ll go by what’s written.

The student wrote:
> 5. [Diagram]
> x = 108°
> y = 72°
> z = 157°

And they have:
> 108 = ?

Wait — perhaps the problem is about vertical angles or linear pairs.

Another possibility: the 108° and $ y $ are vertical angles → so $ y = 108^\circ $? But they say $ y = 72^\circ $

Alternatively, maybe the 108° is supplementary to $ y $, so:

If $ y + 108^\circ = 180^\circ $, then $ y = 72^\circ $

Ah! That makes sense.

So suppose:
- Two lines intersect.
- One angle is 108°,
- Then its supplementary angle (adjacent) is $ 180^\circ - 108^\circ = 72^\circ $
- So $ y = 72^\circ $
- Then $ x = 108^\circ $ (vertical angle to 108°)
- And $ z $ might be another angle?

Wait — if $ z = 157^\circ $, how?

Unless it's a different configuration.

Alternatively, maybe $ z $ is an angle in a triangle.

But since the image is not fully clear, let’s skip to the ones we can definitively solve.

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Problem 7:

> ∠A and ∠B are complementary angles. If the measure of ∠A is 42°, find the measure of ∠B.

#### Definition:
Complementary angles add up to 90°.

So:
$$
∠A + ∠B = 90^\circ
\Rightarrow 42^\circ + ∠B = 90^\circ
\Rightarrow ∠B = 90^\circ - 42^\circ = 48^\circ
$$

✔ So, $ \boxed{48^\circ} $

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Problem 9:

> ∠1 and ∠2 form a linear pair. If the measure of ∠1 is 113°, find the measure of ∠2.

#### Definition:
A linear pair means two adjacent angles that form a straight line → their sum is 180°.

So:
$$
∠1 + ∠2 = 180^\circ
\Rightarrow 113^\circ + ∠2 = 180^\circ
\Rightarrow ∠2 = 180^\circ - 113^\circ = 67^\circ
$$

✔ So, $ \boxed{67^\circ} $

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Summary of Solutions:

| Problem | Answer |
|--------|--------|
| 2 | $ x = 22^\circ $ |
| 3 | $ x = 56^\circ $ |
| 7 | $ \angle B = 48^\circ $ |
| 9 | $ \angle 2 = 67^\circ $ |

For Problem 5, the student’s answer seems inconsistent unless more context is given. Based on common configurations:
- If 108° and $ y $ are supplementary, $ y = 72^\circ $
- $ x = 108^\circ $ (vertical angle)
- But $ z = 157^\circ $ doesn't follow unless it's in a triangle or another setup.

But since the student wrote $ x = 108^\circ $, $ y = 72^\circ $, $ z = 157^\circ $, perhaps $ z $ is in a different part.

But without the full diagram, we can’t verify.

However, Problems 2, 3, 7, and 9 are clear.

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✔ Final Answers:

- Problem 2: $ x = 22^\circ $
- Problem 3: $ x = 56^\circ $
- Problem 7: $ \angle B = 48^\circ $
- Problem 9: $ \angle 2 = 67^\circ $

Let me know if you want help interpreting Problem 5 with a description of the diagram.