Congruent Triangles Notes and Worksheets - Lindsay Bowden - Free Printable

Here is the completed worksheet with explanations for each section.

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Main Notes Section

#### 1. Congruent: same ______, same ______
> Answer:
> - size, shape

Explanation:
Two figures are congruent if they have the exact same size and shape — meaning one can be moved (translated, rotated, or reflected) to perfectly match the other.

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#### 2. Symbol: ≅
> This is the standard symbol for congruence in geometry.

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#### 3. Congruent Triangles: ______ that have 3 pairs of congruent corresponding ______ and 3 pairs of congruent corresponding ______

> Answer:
> - Triangles
> - sides
> - angles

Explanation:
Congruent triangles are triangles that are identical in shape and size. That means all three sides of one triangle are equal in length to the corresponding sides of the other, and all three angles are equal in measure to the corresponding angles.

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#### 4. *Reminder: Corresponding means ______ ______*

> Answer:
> - matching parts (or “in the same position”)

Explanation:
When we say “corresponding,” we mean parts (sides or angles) that are in the same relative position in each triangle. For example, if angle A in triangle ABC corresponds to angle D in triangle DEF, then they are in the same corner position when the triangles are oriented similarly.

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#### 5. Matching Sides and Angles from Diagrams

You are given two triangles: △ABC and △DEF.

From the markings:
- AB has one tick → matches DE (also one tick)
- BC has two ticks → matches EF (also two ticks)
- AC has three ticks → matches DF (also three ticks)

Angles:
- ∠A has one arc → matches ∠D
- ∠B has two arcs → matches ∠E
- ∠C has three arcs → matches ∠F

> Therefore, △ABC ≅ △DEF

Fill in the blanks:

- AB ≅ DE
- BC ≅ EF
- CA ≅ FD (or DF — order matters based on correspondence; since A↔D, B↔E, C↔F, then CA ↔ FD)
- ∠A ≅ ∠D
- ∠B ≅ ∠E
- ∠C ≅ ∠F

✔ So: △ABC ≅ △DEF

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## Examples Section

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Example 1:

You’re given two triangles: △JKL and △MNO.

Look at the markings:

- Side JK has two ticks → matches side MN (two ticks) → so JK ≅ MN
- Side ON has one tick → matches side JL (one tick) → so ON ≅ JL
- Side MO has three ticks → matches side KL (three ticks) → so MO ≅ KL

Angles:

- ∠O has one arc → matches ∠J → so ∠O ≅ ∠J
- ∠K has two arcs → matches ∠N → so ∠K ≅ ∠N
- ∠M has three arcs → matches ∠L → so ∠M ≅ ∠L

> Note: The correspondence is J↔M, K↔N, L↔O (based on matching tick marks and arcs).

✔ So fill in:

- JK ≅ MN
- ON ≅ JL
- MO ≅ KL
- ∠O ≅ ∠J
- ∠K ≅ ∠N
- ∠M ≅ ∠L

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Example 2:

Given: △QRS ≅ △XYZ

This tells us the vertices correspond in order:
- Q ↔ X
- R ↔ Y
- S ↔ Z

So:

Sides:
- QR ↔ XY → QR ≅ XY
- RS ↔ YZ → RS ≅ YZ
- ZX ↔ SQ → ZX ≅ SQ (or SX? Wait — order matters!)

Actually, since △QRS ≅ △XYZ, the correspondence is:
- Q → X
- R → Y
- S → Z

So side QR corresponds to side XY
Side RS corresponds to side YZ
Side SQ corresponds to side ZX → so ZX ≅ SQ (but usually written as ZX ≅ QS to match order)

But in the blank it says ZX ≅ ___, so we write QS

Angles:
- ∠Q ↔ ∠X → ∠Q ≅ ∠X
- ∠Y ↔ ∠R → ∠Y ≅ ∠R
- ∠XZY ↔ ∠SQ R? Wait — let’s clarify.

Angle XZY is angle at vertex Z, between points X and Y → so it's ∠Z in triangle XYZ.

In triangle QRS, the corresponding angle is ∠S.

So: ∠XZY ≅ ∠QSR or simply ∠Z ≅ ∠S

But since the question writes ∠XZY, which is angle at Z, and since Z corresponds to S, then:

> ∠XZY ≅ ∠QSR — but more simply, since angles are named by their vertex, we can say ∠Z ≅ ∠S

However, to match the format, since it says “∠XZY”, we should write the corresponding angle as ∠QSR — but this is not standard. Better to simplify.

Actually, in standard notation, ∠XZY is the same as ∠Z.

So since Z ↔ S, then ∠XZY ≅ ∠QSR — but that’s messy.

Better answer:

Since △QRS ≅ △XYZ, then:

- ∠Q ≅ ∠X
- ∠R ≅ ∠Y
- ∠S ≅ ∠Z → so ∠XZY ≅ ∠QSR is technically correct, but simpler to say ∠S ≅ ∠Z

But the problem writes ∠XZY, so we must match it with the corresponding angle in △QRS, which is ∠QSR — but that’s not intuitive.

Alternatively, perhaps it’s a typo or meant to be ∠Z.

Let’s assume the problem intends:

> ∠XZY means the angle at Z → corresponds to angle at S → so ∠XZY ≅ ∠QSR

But since students usually learn vertex correspondence, we’ll go with:

✔ Fill in:

- QR ≅ XY
- RS ≅ YZ
- ZX ≅ SQ (or QS — both acceptable, but typically written as QS to match QRS order)
- ∠Q ≅ ∠X
- ∠Y ≅ ∠R
- ∠XZY ≅ ∠QSR (angle at Z corresponds to angle at S)

But to keep it simple and aligned with typical teaching, many teachers accept:

> ∠XZY ≅ ∠S — but since the problem wrote ∠XZY, we should write the full name: ∠QSR

However, ∠QSR is not standard — usually we say ∠S.

To avoid confusion, here’s the cleanest way:

Since △QRS ≅ △XYZ, then:

- ∠Q ≅ ∠X
- ∠R ≅ ∠Y
- ∠S ≅ ∠Z → so ∠XZY ≅ ∠S (if we interpret ∠XZY as ∠Z)

✔ Final Answer for Example 2:

- QR ≅ XY
- RS ≅ YZ
- ZX ≅ QS
- ∠Q ≅ ∠X
- ∠Y ≅ ∠R
- ∠XZY ≅ ∠S (assuming ∠XZY = ∠Z)

*(If your teacher expects full angle names, write ∠QSR — but ∠S is more common.)*

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## ✔ FINAL COMPLETED WORKSHEET ANSWERS:

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Main Notes:

- 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* matching parts

From diagrams:
- AB ≅ DE
- BC ≅ EF
- CA ≅ FD
- ∠A ≅ ∠D
- ∠B ≅ ∠E
- ∠C ≅ ∠F
- Therefore, △ABC ≅ △DEF

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Examples:

#### 1.
- JK ≅ MN
- ON ≅ JL
- MO ≅ KL
- ∠O ≅ ∠J
- ∠K ≅ ∠N
- ∠M ≅ ∠L

#### 2.
- QR ≅ XY
- RS ≅ YZ
- ZX ≅ QS
- ∠Q ≅ ∠X
- ∠Y ≅ ∠R
- ∠XZY ≅ ∠S *(or ∠QSR if required)*

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✔ You now have a fully completed and explained worksheet! Let me know if you’d like a printable version or further clarification.