Measuring angles in triangles worksheet with eight problems requiring students to find the missing angle 'a°' using the triangle angle sum property.

Let's solve each problem step by step using the angle sum property of triangles, which states:

> The sum of the interior angles in any triangle is 180°.

We'll also use properties of isosceles triangles (equal sides → equal angles) and exterior angles (an exterior angle equals the sum of the two non-adjacent interior angles).

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Problem 1

Given:
- One angle = 70°
- Another angle = 35°
- Find angle $ a^\circ $

$$
a = 180^\circ - 70^\circ - 35^\circ = 75^\circ
$$

✔ Answer: 75°

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Problem 2

Given:
- One angle = 82°
- Another angle = 44°
- Find angle $ a^\circ $

$$
a = 180^\circ - 82^\circ - 44^\circ = 54^\circ
$$

✔ Answer: 54°

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Problem 3

Given:
- One angle = 117°
- Another angle = 17°
- Find angle $ a^\circ $

$$
a = 180^\circ - 117^\circ - 17^\circ = 46^\circ
$$

✔ Answer: 46°

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Problem 4

This is an isosceles triangle (two sides marked with "x" — meaning equal sides).
So, the base angles are equal.
One angle given: 68°, and it’s at the base.

Since two sides are equal, the angles opposite them are equal. So the two base angles are both 68°.

Wait — but only one base angle is labeled as 68°. Let’s check:
If the two equal sides are the ones forming the apex, then the two base angles are equal. But here, the top angle is labeled $ a^\circ $, and the bottom right angle is 68°, and the two sides from the top to the bottom vertices are marked equal (the two legs), so the two base angles should be equal.

But only one base angle is given as 68°, so the other base angle is also 68°.

Then:
$$
a = 180^\circ - 68^\circ - 68^\circ = 44^\circ
$$

✔ Answer: 44°

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Problem 5

Triangle has:
- One angle = 42°
- Two sides marked equal (with tick marks), so it's isosceles.
- The side opposite $ a^\circ $ is marked equal to the side adjacent to the 42° angle.

Wait — let's interpret the markings.

The side between 42° and $ a^\circ $ is marked with a single tick, and the other side from $ a^\circ $ is also marked with a tick — so the two sides enclosing $ a^\circ $ are equal? Wait, actually:

Looking carefully:
- The side opposite the 42° angle is marked with a single tick.
- The side opposite $ a^\circ $ is also marked with a single tick.

Wait — actually, the two sides that are marked equal are the ones adjacent to the 42° angle? Let's re-analyze.

Actually, the two sides with tick marks are:
- One from the vertex with 42° to the bottom-left vertex
- The other from the same vertex to the bottom-right vertex?

No — the tick marks are on the two sides adjacent to $ a^\circ $ — i.e., the two sides forming angle $ a^\circ $ are equal.

So the triangle has two equal sides meeting at $ a^\circ $ → so the triangle is isosceles with vertex angle $ a^\circ $.

Therefore, the two base angles are equal.

But we are given one base angle = 42°, so the other base angle is also 42°.

Then:
$$
a = 180^\circ - 42^\circ - 42^\circ = 96^\circ
$$

✔ Answer: 96°

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Problem 6

This is an equilateral triangle?
All three sides have tick marks: two on each side? Wait — no.

Look:
- All three sides have tick marks:
- Left side: one tick
- Right side: one tick
- Base: one tick
But they’re all marked with one tick, so all sides are equal.

Yes! All three sides are equal → equilateral triangle.

In an equilateral triangle, all angles are 60°.

So $ a^\circ = 60^\circ $

✔ Answer: 60°

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Problem 7

We have a triangle with:
- Angles: 40° and 17° inside
- Angle $ a^\circ $ is outside, at the top vertex, forming a straight line with the interior angle.

So first, find the interior angle at the top vertex.

Sum of interior angles:
$$
\text{Top interior angle} = 180^\circ - 40^\circ - 17^\circ = 123^\circ
$$

Now, $ a^\circ $ is the exterior angle at that vertex, so it forms a straight line with the interior angle:
$$
a = 180^\circ - 123^\circ = 57^\circ
$$

Alternatively, exterior angle = sum of two remote interior angles:
$$
a = 40^\circ + 17^\circ = 57^\circ
$$

✔ Answer: 57°

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Problem 8

We have a triangle with:
- Interior angles: 88° and 39°
- $ a^\circ $ is an exterior angle at the third vertex (on the base)

So again, the exterior angle $ a^\circ $ is equal to the sum of the two non-adjacent interior angles:
$$
a = 88^\circ + 39^\circ = 127^\circ
$$

✔ Answer: 127°

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 75° |
| 2 | 54° |
| 3 | 46° |
| 4 | 44° |
| 5 | 96° |
| 6 | 60° |
| 7 | 57° |
| 8 | 127° |

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