Congruent Triangle Notes worksheet with definitions, examples, and diagrams.

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 Diagram

We’re given two triangles:
- Triangle $ \triangle ABC $ and $ \triangle DEF $

From the markings:
- Side $ AB $ has one tick mark, matches $ DE $
- Side $ BC $ has two tick marks, matches $ EF $
- Side $ CA $ has three tick marks, matches $ FD $
- Angles at $ A $, $ B $, $ C $ match angles at $ D $, $ E $, $ F $ respectively

So:

| Segment | Congruent to |
|--------|--------------|
| $ \overline{AB} $ | $ \overline{DE} $ |
| $ \overline{BC} $ | $ \overline{EF} $ |
| $ \overline{CA} $ | $ \overline{FD} $ |

| Angle | Congruent to |
|-------|-------------|
| $ \angle A $ | $ \angle D $ |
| $ \angle B $ | $ \angle E $ |
| $ \angle C $ | $ \angle F $ |

Therefore:
> $ \triangle ABC \cong \triangle DEF $

✔ Final answer:
Therefore, $ \triangle ABC \cong \triangle DEF $

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Examples

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#### Example 1: Triangles $ \triangle JKL $ and $ \triangle MON $

From the diagram:
- $ \overline{JK} $ has two tick marks, matches $ \overline{ON} $
- $ \overline{ON} $ has two tick marks, so it corresponds to $ \overline{JK} $
- $ \overline{MO} $ has one tick mark, matches $ \overline{JL} $
- $ \overline{MN} $ has three tick marks, matches $ \overline{KL} $
- Angles:
- $ \angle K $ (marked with a single arc) → matches $ \angle N $
- $ \angle M $ (double arc) → matches $ \angle L $
- $ \angle O $ (triple arc) → matches $ \angle J $

Now fill in:

| Expression | Answer |
|-----------|--------|
| $ \overline{JK} \cong $ | $ \overline{ON} $ |
| $ \overline{ON} \cong $ | $ \overline{JK} $ |
| $ \overline{MO} \cong $ | $ \overline{JL} $ |
| $ \angle O \cong $ | $ \angle J $ |
| $ \angle K \cong $ | $ \angle N $ |
| $ \angle M \cong $ | $ \angle L $ |

✔ Answers:
- $ \overline{JK} \cong \overline{ON} $
- $ \overline{ON} \cong \overline{JK} $
- $ \overline{MO} \cong \overline{JL} $
- $ \angle O \cong \angle J $
- $ \angle K \cong \angle N $
- $ \angle M \cong \angle L $

> Note: The order of naming matters based on correspondence. Since $ \triangle JKL \cong \triangle MON $, vertex $ J $ corresponds to $ M $, $ K $ to $ O $, $ L $ to $ N $. But from the tick marks, we can see:
> - $ JK \leftrightarrow ON $
> - $ JL \leftrightarrow MO $
> - $ KL \leftrightarrow MN $
>
> So actually, the correspondence is:
> - $ J \leftrightarrow O $
> - $ K \leftrightarrow N $
> - $ L \leftrightarrow M $
>
> Wait — let’s double-check the matching.

Let’s re-analyze the diagram carefully.

In triangle $ \triangle JKL $:
- $ \angle J $ has no special marking, but side $ JK $ has two ticks, $ JL $ has one tick
- $ \angle K $ has one arc, $ \angle L $ has two arcs

In triangle $ \triangle MON $:
- $ \angle M $ has two arcs, $ \angle O $ has one arc, $ \angle N $ has no arc
- Side $ MO $ has one tick, $ ON $ has two ticks, $ NM $ has three ticks

Matching:
- $ \angle K $ (one arc) ↔ $ \angle O $ (one arc)
- $ \angle L $ (two arcs) ↔ $ \angle M $ (two arcs)
- $ \angle J $ ↔ $ \angle N $

Sides:
- $ JK $ (two ticks) ↔ $ ON $ (two ticks)
- $ JL $ (one tick) ↔ $ MO $ (one tick)
- $ KL $ (three ticks) ↔ $ NM $ (three ticks)

So the correspondence is:
- $ J \leftrightarrow N $
- $ K \leftrightarrow O $
- $ L \leftrightarrow M $

Thus: $ \triangle JKL \cong \triangle NOM $

But the question doesn’t ask for that; it just asks for what each segment or angle is congruent to.

So now:

| Given | Matches |
|------|--------|
| $ \overline{JK} $ | $ \overline{ON} $ |
| $ \overline{ON} $ | $ \overline{JK} $ |
| $ \overline{MO} $ | $ \overline{JL} $ |
| $ \angle O $ | $ \angle K $ |
| $ \angle K $ | $ \angle O $ |
| $ \angle M $ | $ \angle L $ |

✔ Final answers for Example 1:
- $ \overline{JK} \cong \overline{ON} $
- $ \overline{ON} \cong \overline{JK} $
- $ \overline{MO} \cong \overline{JL} $
- $ \angle O \cong \angle K $
- $ \angle K \cong \angle O $
- $ \angle M \cong \angle L $

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#### Example 2: $ \triangle QRS \cong \triangle XYZ $

Given: $ \triangle QRS \cong \triangle XYZ $

This means:
- Vertex $ Q $ corresponds to $ X $
- Vertex $ R $ corresponds to $ Y $
- Vertex $ S $ corresponds to $ Z $

So:

| Segment | Congruent to |
|--------|--------------|
| $ \overline{QR} $ | $ \overline{XY} $ |
| $ \overline{RS} $ | $ \overline{YZ} $ |
| $ \overline{ZX} $ | $ \overline{SQ} $ ? Wait — $ ZX $ is not a side of $ \triangle QRS $

Wait: $ \overline{ZX} $ is not a side of $ \triangle QRS $. Let’s check.

In $ \triangle XYZ $, sides are:
- $ XY $, $ YZ $, $ ZX $

So $ \overline{ZX} $ is a side of $ \triangle XYZ $, which corresponds to $ \overline{SR} $ in $ \triangle QRS $, since:
- $ Z \leftrightarrow S $
- $ X \leftrightarrow Q $
- So $ ZX \leftrightarrow SQ $, i.e., $ \overline{ZX} \cong \overline{SQ} $

But $ \overline{SQ} = \overline{QS} $, so $ \overline{ZX} \cong \overline{QS} $

Similarly:
- $ \angle Q \cong \angle X $
- $ \angle Y \cong \angle R $
- $ \angle XZY \cong \angle QSR $

Because:
- $ \angle XZY $ is the angle at $ Z $ between $ X $ and $ Y $, so it's $ \angle Z $
- $ \angle Z \cong \angle S $
- So $ \angle XZY \cong \angle QSR $

Now fill in:

| Expression | Answer |
|-----------|--------|
| $ \overline{QR} \cong $ | $ \overline{XY} $ |
| $ \overline{RS} \cong $ | $ \overline{YZ} $ |
| $ \overline{ZX} \cong $ | $ \overline{SQ} $ or $ \overline{QS} $ |
| $ \angle Q \cong $ | $ \angle X $ |
| $ \angle Y \cong $ | $ \angle R $ |
| $ \angle XZY \cong $ | $ \angle QSR $ |

✔ Final answers for Example 2:
- $ \overline{QR} \cong \overline{XY} $
- $ \overline{RS} \cong \overline{YZ} $
- $ \overline{ZX} \cong \overline{QS} $
- $ \angle Q \cong \angle X $
- $ \angle Y \cong \angle R $
- $ \angle XZY \cong \angle QSR $

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✔ Final Completed Worksheet Answers:

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#### Definitions:

- 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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#### Diagram: $ \triangle ABC $ and $ \triangle DEF $

- $ \overline{AB} \cong \overline{DE} $
- $ \overline{BC} \cong \overline{EF} $
- $ \overline{CA} \cong \overline{FD} $
- $ \angle A \cong \angle D $
- $ \angle B \cong \angle E $
- $ \angle C \cong \angle F $

> Therefore, $ \triangle ABC \cong \triangle DEF $

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#### Example 1:

- $ \overline{JK} \cong \overline{ON} $
- $ \overline{ON} \cong \overline{JK} $
- $ \overline{MO} \cong \overline{JL} $
- $ \angle O \cong \angle K $
- $ \angle K \cong \angle O $
- $ \angle M \cong \angle L $

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#### Example 2: $ \triangle QRS \cong \triangle XYZ $

- $ \overline{QR} \cong \overline{XY} $
- $ \overline{RS} \cong \overline{YZ} $
- $ \overline{ZX} \cong \overline{QS} $
- $ \angle Q \cong \angle X $
- $ \angle Y \cong \angle R $
- $ \angle XZY \cong \angle QSR $

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✔ All filled in correctly based on congruence and correspondence.

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