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Recognizing congruent shapes

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visibility 102update 7 months agobookmarkshare

🎯 In this topic you will

Identify congruent shapes

🧠 Key Words

congruent
corresponding sides
corresponding angles
different orientation

Show Definitions

congruent: Geometric figures that have identical shape and size, with all corresponding angles and sides equal.
corresponding sides: Sides of two or more polygons that occupy the same relative position in each figure.
corresponding angles: Angles in two or more polygons that occupy the same relative position at each vertex.
different orientation: A spatial arrangement where congruent shapes are rotated or flipped relative to each other but maintain identical dimensions.

There are two congruent right triangles, $\triangle LMN$ and $\triangle XYZ$, shown with green fill and marked right angles. The triangles are congruent, meaning they have identical shape and size.

In these congruent triangles:

$\overline{LM}$ and $\overline{XY}$ are corresponding sides
$\angle MLN$ and $\angle YXZ$ are corresponding angles

In congruent figures, all corresponding sides and angles are equal. Therefore, triangles $LMN$ and $XYZ$ have:

Three pairs of equal sides: $\overline{LM} = \overline{XY}$, $\overline{MN} = \overline{YZ}$, $\overline{NL} = \overline{ZX}$
Three pairs of equal angles: $\angle L = \angle X$, $\angle M = \angle Y$, $\angle N = \angle Z$

💡 Angle Notation

Angles are named using three letters:

The middle letter represents the vertex
Example: $\angle MLN$ has vertex at $L$
First and last letters are points on the rays

📘 Worked example

a. Which of the arrow shapes A to H are congruent to shape A?

b. Two congruent triangles are shown with sides and angles labeled.

Answer:

a. Shapes C, E, and G are congruent to shape A. They have identical shape and size but different orientation.

b. i. $BC = 10.9$ cm, $JL = 5.1$ cm

b. ii. $\angle BAC = 108^\circ$, $\angle KLJ = 45^\circ$
Corresponding sides and angles are equal in congruent triangles.

Congruent shapes have identical size and shape, regardless of orientation. Rotated or reflected shapes are still congruent if all corresponding sides and angles are equal.

In congruent triangles, corresponding sides have equal lengths and corresponding angles have equal measures.

For triangle congruence, we match vertices in order: $A$ corresponds to $K$, $B$ to $L$, and $C$ to $J$.

🧠 PROBLEM-SOLVING Strategy

Determining Congruent Shapes

Use the following steps to identify and prove geometric shapes are congruent.

Identify the type of shapes (triangles, rectangles, etc.) and their given properties.
For triangles, check for congruence using:
- SSS (Side-Side-Side): All three corresponding sides are equal
- SAS (Side-Angle-Side): Two sides and the included angle are equal
- ASA (Angle-Side-Angle): Two angles and the included side are equal
- RHS (Right angle-Hypotenuse-Side): For right-angled triangles
Measure and compare all corresponding sides of the shapes, ensuring $AB = DE$, $BC = EF$, etc.
Compare corresponding angles where given, ensuring $\angle ABC = \angle DEF$, etc.
Use tracing paper to verify congruence by:
- Tracing one shape
- Attempting to match it exactly with another shape
- Rotating or flipping the tracing if needed
Remember that congruent shapes have:
- Identical corresponding angles
- Identical corresponding side lengths
- Equal perimeters
- Equal areas
For classification problems, group shapes by:
- Number of sides
- Side lengths and proportions
- Angle measures
- Symmetry properties

Congruence Rule	Requirements	Example
SSS	Three equal sides	$AB=DE$, $BC=EF$, $AC=DF$
SAS	Two sides and included angle	$AB=DE$, $\angle ABC=\angle DEF$, $BC=EF$
ASA	Two angles and included side	$\angle ABC=\angle DEF$, $BC=EF$, $\angle ACB=\angle DFE$

❓ EXERCISES

🧠 Reasoning Tip

Use tracing paper to check for congruence.

1. Which triangles are congruent to triangle A?

👀 Show answer

Triangles B and D are congruent to triangle A. They have identical size and shape but may be rotated or reflected.

2. Which shapes are congruent to shape A?

👀 Show answer

Shapes C and F are congruent to shape A. They match exactly when superimposed, regardless of orientation.

3. Rectangle ABCD is congruent to rectangle EFGH.

a. Find the length of side AB.

b. Calculate the area of rectangle ABCD.

👀 Show answer

a. $AB = 5$ cm

b. Area = $AB \times BC = 5 \times 3 = 15$ cm²

❓ EXERCISES

4. Triangles $\triangle ABC$ and $\triangle DEF$ are congruent.

a. Write the lengths of the sides of $\triangle ABC$.

b. Write the sizes of the angles of $\triangle ABC$.

c. Write the lengths of the sides of $\triangle DEF$.

d. Write the sizes of the angles of $\triangle DEF$.

👀 Show answer

a. $AB = 13$ cm, $BC = 12$ cm, $AC = 5$ cm

b. $\angle A = 35^\circ$, $\angle B = 55^\circ$, $\angle C = 90^\circ$

c. $DE = 13$ cm, $EF = 12$ cm, $DF = 5$ cm

d. $\angle D = 35^\circ$, $\angle E = 55^\circ$, $\angle F = 90^\circ$

5. Triangles $\triangle UVW$ and $\triangle XYZ$ are congruent.

a. Write the lengths of the sides of $\triangle UVW$.

b. Write the sizes of the angles of $\triangle XYZ$.

👀 Show answer

a. $UV = 7$ cm, $VW = 24$ cm, $UW = 25$ cm

b. $\angle X = 90^\circ$, $\angle Y = 73.7^\circ$, $\angle Z = 16.3^\circ$

6. Sofia drew two triangles. Are they congruent?

👀 Show answer

The triangles are not congruent. Although they have identical angles ($57^\circ$, $42^\circ$, $81^\circ$), there is no information about corresponding sides being equal. For congruence, we need at least one pair of corresponding sides to be equal in addition to the angles.

🧠 Think like a Mathematician

Question: In an equilateral triangle all the angles are 60°. This means that all equilateral triangles must be congruent, as all the angles are the same size. Is it correct? Discuss.

Equipment: Paper, pencil, ruler, protractor

Method:

Draw two equilateral triangles of different sizes (e.g., one with 3cm sides, another with 6cm sides).
Measure all angles in both triangles using a protractor.
Compare the angle measurements between the two triangles.
Compare the side lengths between the two triangles.
Apply the definition of congruence to determine if the triangles are congruent.
Explain your reasoning about whether having equal angles guarantees congruence.

Follow-up Questions:

1. Are all equilateral triangles congruent? Why or why not?

2. What additional information would be needed to prove two equilateral triangles are congruent?

3. How does the concept of similarity differ from congruence in this context?

Show Answers

1: No, not all equilateral triangles are congruent. While they all have identical angles (60° each), they can have different side lengths. Congruence requires both identical shape and size.
2: To prove two equilateral triangles are congruent, you would need to know that at least one pair of corresponding sides are equal in length. Since all sides in an equilateral triangle are equal, knowing one side length determines the entire triangle's size.
3: Similarity means shapes have identical angles but proportional sides. All equilateral triangles are similar because they all have 60° angles. Congruence requires both identical angles AND identical side lengths. So while all equilateral triangles are similar, only those with matching side lengths are congruent.

❓ EXERCISES

🧠 Reasoning Tip

For congruence, check if all corresponding sides and angles are equal. Remember: SSS (Side-Side-Side) is a valid congruence criterion.

8. Two triangles are shown with side lengths:

Triangle 1: $5.3$ cm, $7.1$ cm, $9.5$ cm

Triangle 2: $7.1$ cm, $5.3$ cm, $9.5$ cm

a. Sofia says the triangles are congruent because corresponding sides are equal. Zara disagrees, saying we need to know the angles. Who is correct? Explain your reasoning.

b. Arun mentions that congruent shapes have the same perimeter. Calculate the perimeter of each triangle and explain how this relates to congruence.

👀 Show answer

a. Sofia is correct. By the SSS (Side-Side-Side) congruence criterion, if all three sides of one triangle are equal to the corresponding sides of another triangle, then the triangles are congruent. The angles are not needed to establish congruence in this case.

b. Perimeter of Triangle 1 = $5.3 + 7.1 + 9.5 = 21.9$ cm
Perimeter of Triangle 2 = $7.1 + 5.3 + 9.5 = 21.9$ cm
Since the triangles are congruent, they have identical perimeters. Congruent shapes always have the same perimeter because all corresponding sides are equal.

9. Explain why knowing only the side lengths is sufficient to determine congruence in this case, but might not be sufficient in other cases.

👀 Show answer

Knowing all three side lengths is sufficient because of the SSS congruence criterion. However, if we only knew two sides and the non-included angle (SSA), this would not guarantee congruence because two different triangles could be formed with those measurements. Similarly, knowing only two sides and the included angle (SAS) would be sufficient, but knowing only three angles (AAA) would not guarantee congruence as the triangles could be different sizes.

❓ EXERCISES

10. Classify these shapes into groups. Describe the properties that characterise each of your groups.

👀 Show answer

Group 1: Circles (A, I, L)
Properties: Round shape with no straight sides or angles. All points are equidistant from the center. No vertices.

Group 2: Triangles (C, D, F, G, K, M)
Properties: Three straight sides and three angles. Sum of interior angles is $180^\circ$. Can be classified by side lengths (equilateral, isosceles, scalene) or angles (acute, right, obtuse).

Group 3: Squares and Rectangles (B, N, P)
Properties: Four sides and four right angles ($90^\circ$). Opposite sides are equal and parallel. Squares have all sides equal; rectangles have opposite sides equal. Sum of interior angles is $360^\circ$.

Group 4: Hexagons (E, J)
Properties: Six straight sides and six angles. Sum of interior angles is $720^\circ$. Can be regular (all sides and angles equal) or irregular.

Group 5: Trapezoids (H, Q)
Properties: Four sides with exactly one pair of parallel sides. The parallel sides are called bases. Sum of interior angles is $360^\circ$.

📘 What we've learned

Congruent shapes are identical in size and shape, with all corresponding sides and angles equal.
For triangles, congruence can be proven using four key rules:
- SSS (Side-Side-Side): All three corresponding sides equal
- SAS (Side-Angle-Side): Two sides and included angle equal
- ASA (Angle-Side-Angle): Two angles and included side equal
- RHS (Right angle-Hypotenuse-Side): For right-angled triangles
Corresponding parts of congruent shapes are always equal: $\angle A = \angle D$, $\overline{AB} = \overline{DE}$, etc.
Tracing paper is a practical tool to verify congruence by matching shapes through rotation or flipping.
Congruent shapes have identical perimeters and areas despite possible orientation differences.
Shapes can be classified by properties like side lengths, angle measures, and symmetry.

Recognizing congruent shapes

🎯 In this topic you will

🧠 Key Words

💡 Angle Notation

🧠 PROBLEM-SOLVING Strategy

Determining Congruent Shapes

❓ EXERCISES

🧠 Reasoning Tip

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

🧠 Reasoning Tip

❓ EXERCISES

📘 What we've learned

Related Past Papers

Related Tutorials

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