Congruent Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve and explain the Congruent Triangle Notes worksheet step by step.

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Part 1: Fill in the Blanks

#### 1. Congruent: same ________, same ________
- Answer: *size, shape*
- Explanation: Two figures are congruent if they have the same size and shape. The symbol for congruence is ≅.

> ✔ So:
> Congruent: same size, same shape
> Symbol: ≅

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#### 2. Congruent Triangles: ________ that have 3 pairs of congruent corresponding ________ and 3 pairs of congruent corresponding ________
- Answer: *Triangles, sides, angles*
- Explanation: Congruent triangles are triangles where all three corresponding sides and all three corresponding angles are equal.

> ✔ So:
> Congruent Triangles: Triangles that have 3 pairs of congruent corresponding sides and 3 pairs of congruent corresponding angles

> Reminder: Corresponding means in the same position

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Triangle ABC and DEF (Given Diagram)

We are given two triangles:

- △ABC and △DEF
- Markings show:
- AB has one tick mark → corresponds to DE (one tick)
- BC has two tick marks → corresponds to EF (two ticks)
- CA has three tick marks → corresponds to FD (three ticks)
- Angles at A and D are marked with one arc → ∠A ≅ ∠D
- Angles at B and E are marked with two arcs → ∠B ≅ ∠E
- Angles at C and F are marked with three arcs → ∠C ≅ ∠F

So we can fill in:

```
AB ≅ DE
BC ≅ EF
CA ≅ FD

∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
```

Therefore, △ABC ≅ △DEF

> ✔ Final answer:
> Therefore, △ABC ≅ △DEF

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Examples

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#### Example 1: Triangles JKL and MON

We are given two triangles: △JKL and △MON

From markings:

- JK has two tick marks → matches MO (two tick marks)
- KL has one tick mark → matches ON (one tick mark)
- JL has one tick mark → matches MN? Wait — let’s look carefully.

Wait — actually, from diagram:

- JK has two tick marks → corresponds to MO (two tick marks) → so JK ≅ MO
- KL has one tick mark → corresponds to ON (one tick mark) → KL ≅ ON
- JL has no tick marks, but angle at J and M are both right angles (marked with square), and side JM is common?

Wait — better to match based on corresponding vertices.

But here's the key: The order of letters matters in congruence.

Look at the matching:

- ∠J and ∠M are both marked with a square → right angles → ∠J ≅ ∠M
- Side JK has two ticks → side MO has two ticks → JK ≅ MO
- Side JL has one tick → side MN has one tick → JL ≅ MN?
Wait — no. Let’s re-express.

Actually, triangle JKL and triangle MON:

- ∠J ≅ ∠M (both right angles)
- Side JK has two tick marks → MO has two tick marks → so JK ≅ MO
- Side KL has one tick mark → ON has one tick mark → KL ≅ ON
- Side JL has no tick marks → MN has no tick marks → JL ≅ MN?

But wait — the vertices must be matched properly.

Let’s analyze the correspondence:

- Right angle at J → right angle at M → so J ↔ M
- From J, side JK has two ticks → from M, side MO has two ticks → so K ↔ O
- Then L ↔ N

So correspondence: J ↔ M, K ↔ O, L ↔ N

Therefore:
△JKL ≅ △MON

Now fill in:

```
JK ≅ MO
ON ≅ KL (because O↔K, N↔L → ON ≅ KL)
MO ≅ JK (already above)
∠O ≅ ∠K (since O ↔ K)
∠K ≅ ∠O (same as above)
∠M ≅ ∠J (M ↔ J)
```

Wait — let's list what’s asked:

> JK ≅ ___
> ON ≅ ___
> MO ≅ ___
> ∠O ≅ ___
> ∠K ≅ ___
> ∠M ≅ ___

Using correspondence:
J ↔ M, K ↔ O, L ↔ N

So:

- JK ≅ MO (J→M, K→O)
- ON ≅ KL (O→K, N→L)
- MO ≅ JK (same as first)
- ∠O ≅ ∠K (since O ↔ K)
- ∠K ≅ ∠O (same)
- ∠M ≅ ∠J (M ↔ J)

✔ So answers:

- JK ≅ MO
- ON ≅ KL
- MO ≅ JK
- ∠O ≅ ∠K
- ∠K ≅ ∠O
- ∠M ≅ ∠J

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#### Example 2: △QRS ≅ △XYZ

Given: △QRS ≅ △XYZ

This means the vertices correspond in order:

- Q ↔ X
- R ↔ Y
- S ↔ Z

So:

- QR ≅ XY (Q→X, R→Y)
- RS ≅ YZ (R→Y, S→Z)
- ZX ≅ SQ? Wait — ZX is not a side of △QRS.

Wait — the question says:

> ZX ≅ ___

But ZX is part of △XYZ. Since S ↔ Z, and Q ↔ X, then QS ↔ ZX?

No — better: sides are between corresponding points.

We know:

- QR ≅ XY
- RS ≅ YZ
- QS ≅ XZ

But the blank says: ZX ≅ ___

ZX is the same as XZ → so ZX ≅ QS

Because:
- Z ↔ S
- X ↔ Q → so ZX ↔ SQ → ZX ≅ SQ

Similarly:

- ∠Q ≅ ∠X (Q ↔ X)
- ∠Y ≅ ∠R (Y ↔ R)
- ∠XZY ≅ ∠SQP? Wait — ∠XZY is at Z in △XYZ → corresponds to ∠S in △QRS

Wait — ∠XZY is the angle at Z → since Z ↔ S, then ∠XZY ≅ ∠QRS? No.

Wait: triangle XYZ → ∠XZY is angle at Z → between X and Y → so it's ∠Z

In △QRS, angle at S → ∠S

Since S ↔ Z → ∠S ≅ ∠Z

So ∠XZY ≅ ∠RSQ

But the blank says: ∠XZY ≅ ___

So ∠XZY ≅ ∠RSQ

But maybe they want the vertex name.

Alternatively, since △QRS ≅ △XYZ, then:

- ∠Q ≅ ∠X
- ∠R ≅ ∠Y
- ∠S ≅ ∠Z

So:

- QR ≅ XY
- RS ≅ YZ
- QS ≅ XZ → which is same as ZX → so ZX ≅ QS

And:

- ∠Q ≅ ∠X
- ∠Y ≅ ∠R
- ∠XZY ≅ ∠S (since Z ↔ S, and ∠XZY is ∠Z)

But ∠XZY = ∠Z → so ∠XZY ≅ ∠S

So:

> QR ≅ XY
> RS ≅ YZ
> ZX ≅ QS
> ∠Q ≅ ∠X
> ∠Y ≅ ∠R
> ∠XZY ≅ ∠S

✔ Final Answers:

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✔ Complete Worksheet Solution

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#### Fill-in-the-blank Section:

- Congruent: same size, same shape
Symbol: ≅

- Congruent Triangles: Triangles that have 3 pairs of congruent corresponding sides and 3 pairs of congruent corresponding angles
*Reminder: Corresponding means* in the same position

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#### Triangle ABC and DEF:

```
AB ≅ DE
BC ≅ EF
CA ≅ FD

∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
```

Therefore, △ABC ≅ △DEF

---

#### Examples:

##### 1. △JKL ≅ △MON (based on markings)

- JK ≅ MO
- ON ≅ KL
- MO ≅ JK
- ∠O ≅ ∠K
- ∠K ≅ ∠O
- ∠M ≅ ∠J

##### 2. △QRS ≅ △XYZ

- QR ≅ XY
- RS ≅ YZ
- ZX ≅ QS
- ∠Q ≅ ∠X
- ∠Y ≅ ∠R
- ∠XZY ≅ ∠S

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✔ Summary of Key Concepts:

- Congruent triangles have all sides and angles congruent.
- Corresponding parts are in the same relative position.
- When naming congruent triangles, order matters (e.g., △ABC ≅ △DEF means A→D, B→E, C→F).
- Use tick marks and angle arcs to identify congruent parts.

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