Properties of Triangles Worksheet | Fun and Engaging Year 7 ... - Free Printable

Let’s solve this step by step.

We have two sections: Section A (filling a table with triangle properties) and Section B (labeling triangles based on their features).

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SECTION A: Fill the Table

We are given 15 statements about triangles, and we need to place each one under the correct triangle type:

Triangle types:
- Right Angled Triangle
- Scalene Triangle
- Equilateral Triangle
- Isosceles Triangle
- Right Angled Isosceles Triangle

Let’s go statement by statement and assign them correctly.

#### Statement List:

1. No equal sides → Scalene
2. Two equal sides → Isosceles, also Right Angled Isosceles (since it has two equal sides)
3. Three equal sides → Equilateral
4. Three equal angles → Equilateral (all 60°)
5. No equal angles → Scalene
6. One right angle → Right Angled, also Right Angled Isosceles
7. Two equal angles → Isosceles, also Right Angled Isosceles (has two 45° angles), also Equilateral? Wait — equilateral has three equal angles, so “two equal” is technically true but not specific. But since the instruction says “some may appear more than once”, we can include it in Isosceles and Right Angled Isosceles. Actually, better to be precise: “Two equal angles” is defining for Isosceles. Equilateral has *three* equal angles, so maybe not here. Let’s stick to standard definitions.
- Standard: Isosceles = at least two equal angles → yes.
- Right Angled Isosceles = two 45° angles → yes.
- Equilateral = all three equal → doesn’t fit “two equal” as primary descriptor → skip.
8. Two angles of 45° → Right Angled Isosceles only (because 45+45+90=180)
9. All angles are 60° → Equilateral
10. Can contain one obtuse angle → Scalene (can be obtuse scalene), Isosceles (can be obtuse isosceles). Not Right Angled (has 90°, can’t have >90° too), not Equilateral (all 60°), not Right Angled Isosceles (45-45-90). So: Scalene and Isosceles.
11. Cannot contain an obtuse angle → Right Angled (max 90°), Equilateral (all 60°), Right Angled Isosceles (45-45-90). Also, acute scalene or acute isosceles could, but the statement is general — so which triangle types *never* have obtuse? Right angled triangles can’t have another angle >90°, so they cannot contain an obtuse angle. Same for equilateral and right isosceles. Scalene and isosceles *can* have obtuse, so they don’t belong here.
- So: Right Angled, Equilateral, Right Angled Isosceles.
12. Has rotational symmetry → Only Equilateral (order 3). Others? Isosceles has line symmetry but not rotational (unless equilateral). Right angled isosceles has line symmetry but not rotational. So only Equilateral.
13. No lines of symmetry → Scalene (generally no symmetry). Others have at least one.
14. One line of symmetry → Isosceles, Right Angled Isosceles (if legs equal, then one line from right angle to hypotenuse midpoint). Also, some right angled triangles? No — only if isosceles. So: Isosceles and Right Angled Isosceles.
15. More than one line of symmetry → Equilateral (3 lines). That’s the only one.

Now let’s organize into the table columns.

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#### Right Angled Triangle:
- One right angle
- Cannot contain an obtuse angle (since one angle is 90°, others must sum to 90°, so both <90°)

→ So:
• One right angle
• Cannot contain an obtuse angle

*(Note: It might have no equal sides or two equal sides — but those are covered under scalene/isosceles subtypes. We’re listing properties that define or always apply.)*

Wait — actually, “Right Angled Triangle” as a category includes both scalene right triangles and isosceles right triangles. So properties that are always true for ANY right angled triangle:

- One right angle ✔
- Cannot contain an obtuse angle ✔ (since total is 180°, and one is 90°, other two sum to 90°, so each <90°)

Other properties like “no equal sides” or “two equal sides” depend on subtype — so we shouldn’t put them here unless specified. The table seems to want properties that are characteristic or always true.

Looking back at instructions: “Write each statement in the correct place in the table, some may appear in the table more than once.”

So we should assign each statement to every triangle type it applies to.

Let me make a grid mentally.

I’ll list each triangle type and collect applicable statements.

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#### Right Angled Triangle:
Applies to any triangle with a 90° angle.

Statements that apply:
- One right angle ✔
- Cannot contain an obtuse angle ✔ (as explained)
- Can it have no equal sides? Yes — e.g., 3-4-5 triangle → so “No equal sides” applies to some right angled triangles, but not all. The statement “No equal sides” is a property of scalene, which can be right angled. But the column is “Right Angled Triangle” — does it mean all right angled triangles have this? No. So probably we only put properties that are always true OR commonly associated.

This is tricky. Looking at typical curriculum, for “Right Angled Triangle”, they expect:
- One right angle
- Possibly “Cannot contain an obtuse angle”

But let’s check the other columns.

Perhaps better to think: which statements are DEFINING or ALWAYS TRUE for that type.

Standard approach:

- Right Angled Triangle: defined by having one 90° angle. Other properties vary.
→ So only: "One right angle"
→ And implicitly, since one angle is 90°, the other two are acute, so "Cannot contain an obtuse angle" is also always true.

Similarly, for Scalene: defined by no equal sides → so "No equal sides", "No equal angles" (in most cases, though technically a scalene could have two equal angles? No — if two angles equal, then two sides equal → isosceles. So scalene has no equal sides AND no equal angles.)

Actually, in Euclidean geometry:
- If two angles are equal, the opposite sides are equal → so isosceles.
- Therefore, scalene triangle has all sides different AND all angles different.

So:

#### Scalene Triangle:
- No equal sides
- No equal angles
- Can contain one obtuse angle (e.g., 20°, 30°, 130°)
- No lines of symmetry

Also, it can be right angled (like 3-4-5), but that’s a subtype. For pure "Scalene", we list properties that are always true for scalene triangles.

So:
- No equal sides
- No equal angles
- Can contain one obtuse angle (yes, because it can be obtuse scalene)
- No lines of symmetry

Does "Cannot contain an obtuse angle" apply? No — scalene can be obtuse.

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#### Equilateral Triangle:
- Three equal sides
- Three equal angles
- All angles are 60°
- Has rotational symmetry (order 3)
- More than one line of symmetry (3 lines)
- Cannot contain an obtuse angle (all 60°)
- Two equal angles? Well, it has three, so "two equal" is true but redundant. Probably not listed here since "three equal" is more accurate. But the instruction says "some may appear more than once", so maybe include? I think for clarity, we'll put only the most specific.

Typically, for equilateral:
- Three equal sides
- Three equal angles
- All angles are 60°
- Has rotational symmetry
- More than one line of symmetry
- Cannot contain an obtuse angle

"Not have no equal sides" etc. — obvious.

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#### Isosceles Triangle:
Defined as having at least two equal sides → which implies at least two equal angles.

Properties:
- Two equal sides
- Two equal angles
- One line of symmetry
- Can contain one obtuse angle (e.g., 20°, 20°, 140°)
- Cannot contain an obtuse angle? No — it can, so not this.
- Has rotational symmetry? No — only if equilateral.
- Three equal sides? No.

Also, it can be right angled (if the two equal angles are 45°), but that's a separate category.

So for "Isosceles Triangle" column:
- Two equal sides
- Two equal angles
- One line of symmetry
- Can contain one obtuse angle

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#### Right Angled Isosceles Triangle:
Specific type: right angled AND isosceles → so angles 45°, 45°, 90°.

Properties:
- Two equal sides (the legs)
- Two equal angles (45° each)
- One right angle
- Two angles of 45°
- One line of symmetry (from right angle to hypotenuse)
- Cannot contain an obtuse angle (angles are 45,45,90)
- Has rotational symmetry? No — only reflection symmetry.
- More than one line? No.

So:
- Two equal sides
- Two equal angles
- One right angle
- Two angles of 45°
- One line of symmetry
- Cannot contain an obtuse angle

Now, let’s compile the table.

I’ll write for each column the statements that belong.

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Right Angled Triangle:
- One right angle
- Cannot contain an obtuse angle

*(Note: Some might argue "Can contain one obtuse angle" is false for right angled, which is why "Cannot" is true.)*

Scalene Triangle:
- No equal sides
- No equal angles
- Can contain one obtuse angle
- No lines of symmetry

Equilateral Triangle:
- Three equal sides
- Three equal angles
- All angles are 60°
- Has rotational symmetry
- More than one line of symmetry
- Cannot contain an obtuse angle

Isosceles Triangle:
- Two equal sides
- Two equal angles
- One line of symmetry
- Can contain one obtuse angle

Right Angled Isosceles Triangle:
- Two equal sides
- Two equal angles
- One right angle
- Two angles of 45°
- One line of symmetry
- Cannot contain an obtuse angle

Now, check if all 15 statements are used:

List again:

1. No equal sides → Scalene
2. Two equal sides → Isosceles, Right Angled Isosceles
3. Three equal sides → Equilateral
4. Three equal angles → Equilateral
5. No equal angles → Scalene
6. One right angle → Right Angled, Right Angled Isosceles
7. Two equal angles → Isosceles, Right Angled Isosceles
8. Two angles of 45° → Right Angled Isosceles
9. All angles are 60° → Equilateral
10. Can contain one obtuse angle → Scalene, Isosceles
11. Cannot contain an obtuse angle → Right Angled, Equilateral, Right Angled Isosceles
12. Has rotational symmetry → Equilateral
13. No lines of symmetry → Scalene
14. One line of symmetry → Isosceles, Right Angled Isosceles
15. More than one line of symmetry → Equilateral

All accounted for. Good.

Now, for the table, we write under each column the statements that apply.

Since the table has blank cells, we fill them with the relevant statements.

In practice, for the answer, we can list per column.

But since the user will copy to worksheet, I’ll present it clearly.

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SECTION B: Label the Triangles

We have 9 triangles to label using names from Section A: Right Angled, Scalene, Equilateral, Isosceles, Right Angled Isosceles.

Look at each:

First row:

1. Triangle with two sides marked equal (ticks) and base angles marked equal → Isosceles Triangle
2. Triangle with all three sides marked equal (ticks) → Equilateral Triangle
3. Triangle with right angle symbol → Right Angled Triangle (and no other marks, so likely scalene right triangle, but we just say Right Angled unless specified)
- But looking: it has a right angle, and no equal sides indicated → so Right Angled Triangle (scalene subtype, but label as Right Angled)
4. Triangle with no marks, looks scalene → Scalene Triangle
5. Triangle with right angle and two sides marked equal → Right Angled Isosceles Triangle

Second row:

6. Triangle with two sides labeled 8 cm, and included angle 60° → since two sides equal, and angle between them 60°, then it must be equilateral? Let’s see: if two sides equal and included angle 60°, then the other two angles are equal, and sum to 120°, so each 60° → yes, equilateral. But labels show two sides 8cm, angle 60° — so actually, it is equilateral. But wait, the third side isn't labeled, but by SAS, if two sides and included angle, and angle is 60°, and sides equal, then yes, equilateral. However, in the diagram, it might be intended as isosceles with vertex angle 60°, which forces it to be equilateral. So label: Equilateral Triangle

But let's confirm: in a triangle, if two sides are equal and the angle between them is 60°, then the triangle is equilateral. Yes.

7. Triangle with angles 60° and 30°, and right angle symbol → so angles 30°, 60°, 90° → Right Angled Triangle (and scalene, since all angles different)

8. Triangle with two sides labeled 17 mm, base 9 mm → so two equal sides → Isosceles Triangle

9. Triangle with right angle and two 45° angles → Right Angled Isosceles Triangle

Now, list:

First row:
1. Isosceles Triangle
2. Equilateral Triangle
3. Right Angled Triangle
4. Scalene Triangle
5. Right Angled Isosceles Triangle

Second row:
6. Equilateral Triangle (since two sides 8cm and included angle 60° implies equilateral)
- But wait, in the diagram, it shows two sides 8cm and the angle between them is 60°, yes → equilateral.
- Alternatively, if it were isosceles with base angles 60°, same thing.
7. Right Angled Triangle (30-60-90)
8. Isosceles Triangle (two sides 17mm)
9. Right Angled Isosceles Triangle (45-45-90)

For #6: even though it has two sides labeled, the angle is 60°, so it must be equilateral. But the label might expect "Isosceles" if they don't realize, but mathematically it's equilateral. However, in many curricula, if two sides are equal and angle is 60°, they still call it isosceles, but strictly it's equilateral. To be precise, since all angles would be 60°, it's equilateral.

But let's see the diagram description: "8 cm, 8 cm, and angle 60° between them" — yes, equilateral.

However, in the image, it might be drawn as isosceles, but with calculation, it's equilateral. I think we should go with equilateral.

But to match common expectations, perhaps the problem intends it as isosceles? No, because with two sides equal and included angle 60°, the third side must also be equal.

Proof: by law of cosines, c² = a² + b² - 2ab cosC = 8² + 8² - 2*8*8*cos(60°) = 64 + 64 - 128*(0.5) = 128 - 64 = 64, so c=8. Yes, equilateral.

So label as Equilateral Triangle.

Similarly, #7: angles 30°, 60°, 90° — definitely Right Angled Triangle.

#8: two sides 17mm, base 9mm — Isosceles.

#9: 45°, 45°, 90° — Right Angled Isosceles.

Now, for #3: it has a right angle, no other marks — so Right Angled Triangle (which could be scalene, but we label as Right Angled).

Similarly, #4: no marks, appears scalene — Scalene Triangle.

So final labels:

Row 1:
1. Isosceles Triangle
2. Equilateral Triangle
3. Right Angled Triangle
4. Scalene Triangle
5. Right Angled Isosceles Triangle

Row 2:
6. Equilateral Triangle
7. Right Angled Triangle
8. Isosceles Triangle
9. Right Angled Isosceles Triangle

But for #6, is it possible they expect "Isosceles"? In some contexts, they might not realize it's equilateral. But mathematically, it is. Since the section says "using the names from the table", and equilateral is an option, we should use it.

To confirm, in the first row, #2 is clearly equilateral (all sides ticked), so #6 should also be equilateral.

Yes.

Now, compile final answers.

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Final Answer:

Section A Table:

- Right Angled Triangle:
One right angle
Cannot contain an obtuse angle

- Scalene Triangle:
No equal sides
No equal angles
Can contain one obtuse angle
No lines of symmetry

- Equilateral Triangle:
Three equal sides
Three equal angles
All angles are 60°
Has rotational symmetry
More than one line of symmetry
Cannot contain an obtuse angle

- Isosceles Triangle:
Two equal sides
Two equal angles
One line of symmetry
Can contain one obtuse angle

- Right Angled Isosceles Triangle:
Two equal sides
Two equal angles
One right angle
Two angles of 45°
One line of symmetry
Cannot contain an obtuse angle

Section B Labels:

First row (left to right):
1. Isosceles Triangle
2. Equilateral Triangle
3. Right Angled Triangle
4. Scalene Triangle
5. Right Angled Isosceles Triangle

Second row (left to right):
6. Equilateral Triangle
7. Right Angled Triangle
8. Isosceles Triangle
9. Right Angled Isosceles Triangle