Angles Worksheets | Free - CommonCoreSheets - Free Printable

Let’s solve each triangle one by one. Remember: the sum of all three angles in any triangle is always 180°. We’ll use that to find the missing angle “a” in each case.

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Problem 1:
Angles given: 70°, 35°, and a°
So, a = 180 - 70 - 35 = 75°

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Problem 2:
Angles given: 82°, 44°, and a°
a = 180 - 82 - 44 = 54°

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Problem 3:
Angles given: 117°, 17°, and a°
a = 180 - 117 - 17 = 46°

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Problem 4:
This triangle has two sides marked with ticks — that means it’s an isosceles triangle, so the two base angles are equal. One base angle is 68°, so the other is also 68°.
Then, a = 180 - 68 - 68 = 44°

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Problem 5:
Again, two sides have ticks → isosceles triangle. The two equal angles are opposite those sides. One angle is 42°, and since the two marked sides are adjacent to the unknown angle “a”, that means the two equal angles are the ones at the ends of the unmarked side. Wait — let’s look again.

Actually, in Problem 5, the two tick marks are on the two sides that meet at angle “a”. That means the two angles *opposite* those sides are equal. But we’re only given one angle: 42°. So if the two sides forming angle “a” are equal, then the two angles *not* at “a” are equal? No — wait, standard rule: in a triangle, equal sides are opposite equal angles.

In Problem 5: the two sides with ticks are the left side and the bottom-right side. So the angles opposite them should be equal. The angle opposite the left side is the bottom-right angle (which is labeled “a”). The angle opposite the bottom-right side is the top-left angle (which is 42°). So actually, angle “a” equals 42°? But that can’t be right because then the third angle would be 180 - 42 - 42 = 96°, which is fine — but let me double-check the diagram logic.

Wait — looking again: in Problem 5, the triangle has:

- Left side: tick mark
- Bottom-right side: tick mark
- Top side: no tick

So the two equal sides are left and bottom-right → therefore, the angles opposite them are equal.

Angle opposite left side → that’s the angle at the bottom-right vertex → which is labeled “a”

Angle opposite bottom-right side → that’s the angle at the top-left vertex → which is labeled 42°

Therefore, a = 42°? But then the third angle (at bottom-left) would be 180 - 42 - 42 = 96° — which is possible.

But wait — the diagram shows angle “a” at the bottom-right, and 42° at the top-left. If the two sides with ticks are left and bottom-right, then yes — the angles opposite them are equal → so a = 42°.

BUT — hold on! In many diagrams, when two sides are marked equal, the angles *between* them or opposite them... Let me think differently.

Alternative approach: maybe the two equal sides mean the two base angles are equal. But here, the 42° angle is not between the two equal sides — it’s at one end.

Actually, let’s label vertices for clarity:

Call the triangle ABC:

- A = top-left vertex → angle = 42°
- B = bottom-left vertex → angle = ?
- C = bottom-right vertex → angle = a°

Sides:

- Side AB (left side) → has tick
- Side BC (bottom side) → no tick? Wait, in the image, the bottom side is not ticked — the right side (AC?) is ticked.

Actually, looking back at the original problem description — I need to interpret based on standard notation.

In Problem 5: the triangle has:

- Angle at left: 42°
- Angle at bottom-right: a°
- Two sides are marked with single ticks: the left side and the right side (from top to bottom-right)

So sides from top vertex to left-bottom and top vertex to right-bottom are equal → meaning it’s isosceles with apex at top → so the two base angles (at left and right) are equal.

But the left angle is 42°, so the right angle (a°) must also be 42°? Then the top angle would be 180 - 42 - 42 = 96°.

Yes — that makes sense. So a = 42°.

Wait — but that seems too straightforward, and maybe I misread. Let me check online or recall: in isosceles triangle, if two sides are equal, the angles opposite them are equal.

If the two equal sides are the left and right sides (both going down from the top), then the angles opposite them are the bottom-left and bottom-right angles. So if left side = right side, then angle at bottom-right = angle at bottom-left.

But in the diagram, the bottom-left angle is not labeled — only the top-left angle is 42°, and bottom-right is a°.

I think I made a mistake in labeling.

Let me redraw mentally:

Triangle with:

- Top vertex: angle unknown
- Bottom-left vertex: angle = 42°
- Bottom-right vertex: angle = a°

Sides:

- From top to bottom-left: has tick
- From top to bottom-right: has tick
- Bottom side: no tick

So the two equal sides are the legs from top to base → so it’s isosceles with base at bottom → therefore, the two base angles are equal → so angle at bottom-left = angle at bottom-right → so 42° = a° → a = 42°

Yes! So a = 42°

But then why is there a question? Maybe I’m overcomplicating. Let’s calculate the third angle: 180 - 42 - 42 = 96° — which is fine.

So answer for 5 is 42°.

Wait — but let me confirm with another method. Perhaps the 42° is not a base angle. Looking at the diagram description: "42°" is at the left corner, "a°" at the right corner, and the two sides from the top to these corners are ticked — so yes, base angles equal → a = 42°.

Okay, moving on.

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Problem 6:
All three sides have ticks → equilateral triangle → all angles equal → each angle = 180 / 3 = 60° → so a = 60°

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Problem 7:
This one has an exterior angle. The angle labeled “a°” is outside the triangle, adjacent to the top vertex.

The two interior angles given are 40° and 17°.

First, find the third interior angle: 180 - 40 - 17 = 123°

Now, the exterior angle “a” is adjacent to this 123° angle, so they form a straight line → a + 123 = 180 → a = 180 - 123 = 57°

Alternatively, exterior angle theorem: exterior angle = sum of two opposite interior angles → a = 40 + 17 = 57° — same answer.

Good.

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Problem 8:
Again, exterior angle. The angle “a°” is outside, adjacent to the bottom-left vertex.

Interior angles given: 88° (top) and 39° (bottom-right)

First, find the third interior angle (at bottom-left): 180 - 88 - 39 = 53°

Then, since “a” is adjacent to it on a straight line: a + 53 = 180 → a = 127°

Or using exterior angle theorem: a = sum of two non-adjacent interior angles = 88 + 39 = 127°

Perfect.

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Now, let’s list all answers:

1. 75°
2. 54°
3. 46°
4. 44°
5. 42°
6. 60°
7. 57°
8. 127°

Double-checking calculations:

1. 70+35=105; 180-105=75 ✔️
2. 82+44=126; 180-126=54 ✔️
3. 117+17=134; 180-134=46 ✔️
4. 68+68=136; 180-136=44 ✔️
5. Isosceles, base angles equal → 42° each → a=42 ✔️
6. Equilateral → 60° ✔️
7. Exterior angle = 40+17=57 ✔️
8. Exterior angle = 88+39=127 ✔️

All correct.

Final Answer:
1. 75°
2. 54°
3. 46°
4. 44°
5. 42°
6. 60°
7. 57°
8. 127°