Math worksheet focused on finding angle measures and solving for variables in geometric diagrams.

Let’s solve each problem step by step.

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Problem 1: Find m∠ABC

We see two angles next to each other that together make ∠ABC. One is labeled 50°, the other is 40°.

So, add them:
50° + 40° = 90°

→ m∠ABC = 90°

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Problem 2: Find m∠ABC

Again, two adjacent angles form ∠ABC: one is 30°, the other is 60°.

Add them:
30° + 60° = 90°

→ m∠ABC = 90°

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Problem 3: Find m∠ABC

Two angles: 70° and 20°.

Add:
70° + 20° = 90°

→ m∠ABC = 90°

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Problem 4: Find m∠ABC

Three angles here: 25°, 35°, and 30°.

Add all three:
25 + 35 = 60; 60 + 30 = 90°

→ m∠ABC = 90°

*(Note: All four angle problems so far are right angles — 90° — which makes sense with the title “Angles, Angles, Everywhere!”)*

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Now let’s do the rectangle and circle problems.

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Problem 5: Find x in a square with diagonal

In a square, the diagonal splits the 90° corner into two equal parts.

So, each part is half of 90° → 90 ÷ 2 = 45°

But wait — look at the diagram. The angle marked “x” is NOT the small triangle angle — it’s the big angle from the bottom-left corner going up along the diagonal to the top-right. That’s actually the full corner plus the diagonal? Wait — no.

Actually, looking again: In a square, when you draw a diagonal, it creates two 45-45-90 triangles.

The angle labeled “x” is the angle between the bottom side and the diagonal — that’s one of the 45° angles.

Wait — but in the diagram, it looks like they’re labeling the angle inside the triangle at the bottom-left corner as “x”. Since it’s a square, that angle is split by the diagonal into two 45° angles? No — actually, the diagonal goes from corner to corner, so the angle at the bottom-left corner is still 90°, but the diagonal divides it into two 45° angles.

Looking at the diagram description (since we can’t see image), typically in such problems, if a diagonal is drawn in a square, and an angle is labeled at the corner between the side and the diagonal, then x = 45°.

BUT — wait! Problem 5 says “Find the value of x” and shows a square with a diagonal, and x is likely the angle formed at the corner between the base and the diagonal — which is 45°.

However, sometimes these diagrams label the angle opposite or something else. But based on standard problems:

→ In a square, diagonal makes 45° with sides.

So x = 45

Wait — let me double-check with problem 6.

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Problem 6: Rectangle with two lines from top-left corner

It says: one angle is 28°, another is 32°, and x is the remaining angle at that corner.

A rectangle has 90° corners.

So total angle at top-left corner = 90°

Given two parts: 28° and 32°

Add them: 28 + 32 = 60°

Then x = 90 - 60 = 30°

→ x = 30

That makes sense.

Back to problem 5: If it’s a square, and diagonal is drawn, then the angle between side and diagonal is 45°. So x should be 45.

But let’s confirm logic: Square → all angles 90°. Diagonal bisects the corner angle → 90/2 = 45°. Yes.

→ x = 45

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Problem 7: Circle with central angle 135°, find x

This is a circle with a sector labeled 135°, and x is the rest of the circle.

Full circle = 360°

So x = 360 - 135 = 225°

→ x = 225

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Problem 8: Circle with two sectors: 120° and 90°, find x

Total circle = 360°

Add known parts: 120 + 90 = 210°

x = 360 - 210 = 150°

→ x = 150

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Let me verify all answers once more:

1. 50 + 40 = 90 ✔️
2. 30 + 60 = 90 ✔️
3. 70 + 20 = 90 ✔️
4. 25 + 35 + 30 = 90 ✔️
5. Square diagonal → 45° ✔️
6. Rectangle corner: 90 - (28+32) = 30 ✔️
7. Circle: 360 - 135 = 225 ✔️
8. Circle: 360 - (120+90) = 150 ✔️

All correct.

Final Answer:
1. 90°
2. 90°
3. 90°
4. 90°
5. 45
6. 30
7. 225
8. 150