Congruent and Similar Triangles Notes worksheet - Free Printable

Problem Analysis:

The image provides a worksheet on Congruent and Similar Triangles. The task involves filling in blanks related to congruence and similarity properties of triangles, as well as solving problems based on these concepts.

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Section 1: Congruent Triangles

#### Key Concepts:
- Congruent Triangles: Triangles that have the same shape and size. Corresponding angles are equal, and corresponding sides are equal.
- Notation: \( \triangle ABC \cong \triangle DEF \) means triangle \( ABC \) is congruent to triangle \( DEF \).

#### Task 1: Fill in the blanks for congruent triangles.

Given:
- \( \triangle ABC \cong \triangle DEF \)

From the notation, we know:
- Corresponding angles are equal: \( \angle A \cong \angle D \), \( \angle B \cong \angle E \), \( \angle C \cong \angle F \).
- Corresponding sides are equal: \( AB \cong DE \), \( BC \cong EF \), \( CA \cong FD \).

So, the completed table is:
| Angles | Sides |
|------------------|---------------|
| \( \angle A \cong \angle D \) | \( AB \cong DE \) |
| \( \angle B \cong \angle E \) | \( BC \cong EF \) |
| \( \angle C \cong \angle F \) | \( CA \cong FD \) |

#### Task 2: Write the congruence statement for the given triangles.

From the diagram:
- The triangles are labeled \( \triangle JIK \) and \( \triangle CAB \).
- The markings indicate that they are congruent.

Thus, the congruence statement is:
\[ \triangle JIK \cong \triangle CAB \]

#### Task 3: Complete each congruent statement for the given triangles.

From the diagram:
- The triangles are labeled \( \triangle IKJ \) and \( \triangle VWX \).
- The markings indicate that they are congruent.

The corresponding sides are:
- \( IK \cong VX \)
- \( KJ \cong WX \)
- \( JI \cong XV \)

So, the completed statements are:
\[ IK = VX, \quad KJ = WX, \quad JI = XV \]

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Section 2: Similar Triangles

#### Key Concepts:
- Similar Triangles: Triangles that have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are proportional.
- Notation: \( \triangle ABC \sim \triangle DEF \) means triangle \( ABC \) is similar to triangle \( DEF \).
- Scale Factor (k): The ratio of the lengths of corresponding sides.

#### Task 4: Fill in the blanks for similar triangles.

Given:
- \( \triangle ABC \sim \triangle DEF \)

From the notation, we know:
- Corresponding angles are equal: \( \angle A \cong \angle D \), \( \angle B \cong \angle E \), \( \angle C \cong \angle F \).
- Corresponding sides are proportional: \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \).

So, the completed table is:
| Angles | Sides |
|------------------|---------------|
| \( \angle A \cong \angle D \) | \( \frac{AB}{DE} \) |
| \( \angle B \cong \angle E \) | \( \frac{BC}{EF} \) |
| \( \angle C \cong \angle F \) | \( \frac{CA}{FD} \) |

#### Task 5: Write the similarity statement for the given triangles.

From the diagram:
- The triangles are labeled \( \triangle KLM \) and \( \triangle FED \).
- The markings indicate that they are similar.

Thus, the similarity statement is:
\[ \triangle KLM \sim \triangle FED \]

#### Task 6: Find the missing side in the given similar triangles.

Given:
- \( \triangle BCD \sim \triangle LMN \)
- Side lengths: \( BC = 169 \), \( LM = 28 \), \( MN = 26 \)
- We need to find the missing side \( CD \).

Since the triangles are similar, the ratios of corresponding sides are equal. Let the missing side \( CD \) be \( x \). The corresponding sides are:
- \( BC \) corresponds to \( LM \)
- \( CD \) corresponds to \( MN \)

The proportion is:
\[
\frac{BC}{LM} = \frac{CD}{MN}
\]

Substitute the known values:
\[
\frac{169}{28} = \frac{x}{26}
\]

Solve for \( x \):
\[
x = \frac{169 \times 26}{28}
\]

Calculate:
\[
x = \frac{4394}{28} = 157
\]

So, the missing side \( CD \) is:
\[ CD = 157 \]

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Final Answers:

1. Congruent Triangles:
- Angles: \( \angle A \cong \angle D \), \( \angle B \cong \angle E \), \( \angle C \cong \angle F \)
- Sides: \( AB \cong DE \), \( BC \cong EF \), \( CA \cong FD \)
- Congruence statement: \( \triangle JIK \cong \triangle CAB \)
- Congruent sides: \( IK = VX \), \( KJ = WX \), \( JI = XV \)

2. Similar Triangles:
- Angles: \( \angle A \cong \angle D \), \( \angle B \cong \angle E \), \( \angle C \cong \angle F \)
- Sides: \( \frac{AB}{DE} \), \( \frac{BC}{EF} \), \( \frac{CA}{FD} \)
- Similarity statement: \( \triangle KLM \sim \triangle FED \)
- Missing side: \( CD = 157 \)

\[
\boxed{157}
\]