Congruence And Similarity

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Notes

Congruence

Two shapes are **congruent** if they are identical in **shape and size**.
One may be a **reflection**, **rotation**, or **translation** of the other.
If a shape is an enlargement, it is **not congruent** (different size).
To prove congruence, show **corresponding sides** are equal and **corresponding angles** are equal.
Tracing paper can help check congruence if shapes are drawn to scale.

Similarity

Two shapes are **similar** if they have the same shape and corresponding sides are **in proportion**.
One shape is an **enlargement** of the other; similarity does **not** imply congruence.
For triangles, prove similarity by showing **all three corresponding angles are equal**.
For non-triangular shapes, show that the **scale factor** is the same for all corresponding sides.
Look for equal angles using properties like **vertically opposite**, **alternate**, and **corresponding angles** on parallel lines.

Similar Lengths – Scale Factor

The **scale factor** links corresponding lengths on similar shapes.
If the second shape is larger, scale factor > 1; if smaller, $0 <$ scale factor $< 1$ .
To find scale factor: divide a length on the second shape by the corresponding length on the first.
Method 1: Find scale factor from first to second (may $be <1)$ , then multiply/divide accordingly.
Method 2: Find scale factor from smaller to larger $(always >1)$ , then multiply/divide accordingly.

Similar Lengths – Finding Missing Lengths

To find a missing length on the **second** shape: multiply the corresponding length on the first by the scale factor.
To find a missing length on the **first** shape: divide the corresponding length on the second by the scale factor.
If using the smaller-to-larger scale factor: to find a length on the larger, multiply the smaller by the scale factor; to find a length on the smaller, divide the larger by the scale factor.
Redraw overlapping similar shapes separately to avoid confusion.

Congruent Shapes (Reflection and Rotation)

Similar Triangles (Angle Proof)

Similar Rectangles (Scale Factor)

Finding Missing Length (Method 1)

Practice questions

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1.Which word describes two polygons that are the same shape and size?
Easy
- ACongruent
- BSimilar
- CRegular
- DIsosceles
2.Two shapes are similar. Which of the following is always true?
Easy
- AThey are the same size
- BTheir corresponding sides are in proportion
- CThey are congruent
- DThey have the same area
3.Which of the following transformations always produces a shape congruent to the original?
Easy
- AEnlargement
- BRotation
- CShear
- DStretch
4.Two triangles are similar. Their corresponding angles are:
Easy
- AEqual
- BSupplementary
- CDifferent
- DSum to 180°
5.What is the scale factor from a shape of side 4 cm to a similar shape of side 10 cm?
Easy
- A2.5
- B0.4
- C6
- D14
6.Two similar rectangles have widths 3 cm and 5 cm. The smaller rectangle has length 7 cm. What is the length of the larger rectangle?
Medium
- A11.67 cm
- B4.2 cm
- C8 cm
- D10 cm
7.In the diagram, triangles ABC and DEF are similar. $AB = 6$ cm, $DE = 9$ cm, $BC = 8$ cm. Find EF.
Medium
- A12 cm
- B5.33 cm
- C6 cm
- D15 cm
8.A shape with side 12 cm is enlarged to a similar shape with side 4 cm. What is the scale factor?
Medium
- A $\frac{1}{3}$
- B3
- C8
- D48

Summary

## Congruence & Similarity ### Congruence - **Definition**: Two shapes are congruent if they are identical in shape and size. One may be a reflection, rotation, or translation of the other. - **Not congruent**: If one shape is an enlargement of the other, they are not congruent. - **Proof**: Show corresponding sides are equal in length and corresponding angles are equal in size. ### Similarity - **Definition**: Two shapes are similar if they have the same shape and corresponding sides are in proportion. One is an enlargement of the other. - **Similarity does not imply congruence**. - **Triangles**: To prove two triangles are similar, show that their corresponding angles are equal. If angles are equal, sides are automatically in proportion. - **Other shapes**: Show that corresponding sides are in proportion by finding the scale factor. If the scale factor is the same for all corresponding sides, the shapes are similar. ### Similar Lengths - **Scale factor**: - If second shape is bigger: scale factor > 1. - If second shape is smaller: 0 < scale factor < 1. - **Finding scale factor**: Divide a length on the second shape by the corresponding length on the first shape. - **Method 1**: 1. Find scale factor (second/first). 2. Multiply first shape length by scale factor to get missing length on second; divide first shape length by scale factor to get missing length on first. - **Method 2**: 1. Find scale factor from smaller to bigger (bigger/smaller). 2. Multiply smaller by scale factor to get bigger; divide bigger by scale factor to get smaller. ### Key Tips - Tracing paper can help check congruence. - For similar triangles in an hourglass formation, look for vertically opposite angles and alternate angles on parallel lines. - Redraw overlapping similar shapes separately. ### Formulas - Scale factor = (length on second shape) / (corresponding length on first shape) - Missing length = corresponding length × scale factor (or ÷ scale factor for opposite direction)

Key terms

Congruent: Two shapes are congruent if they are identical in shape and size. One may be a reflection, rotation, or translation of the other.
Similar: Two shapes are similar if they have the same shape and their corresponding sides are in proportion. One shape is an enlargement of the other.
Scale factor: The ratio of corresponding lengths on two similar shapes. If the second shape is bigger, scale factor > 1; if smaller, 0 < scale factor < 1.
Corresponding sides: Sides that are in the same relative position in two similar or congruent shapes.
Corresponding angles: Angles that are in the same relative position in two similar or congruent shapes.

Slides

Congruence

Two shapes are congruent if they are identical in shape and size.
One may be a reflection, rotation, or translation of the other.
If one shape is an enlargement of the other, they are not congruent.

Which word describes two polygons that are the same shape and size?

hexagon
regular
perpendicular
congruent
isosceles

Proving Congruence

Show that corresponding sides are the same length.
Show that corresponding angles are the same size.
You do not need to show they face the same direction.

Which of these are true about congruent shapes?

They are identical in shape and size.
They can be enlargements of each other.
A translation of a shape is congruent to the original.

Similarity

Two shapes are similar if they have the same shape and corresponding sides are in proportion.
One shape is an enlargement of the other.
Similarity does not imply congruence.

Which condition is necessary for two triangles to be similar?

Their sides are equal.
Their angles are equal.
They are the same size.
They are reflections.

Proving Triangles Similar

Show that corresponding angles are equal.
Use angle properties: vertically opposite angles, alternate angles on parallel lines, etc.
If all three angles match, the triangles are similar.

Triangle ABC and triangle PQR are similar because angle ABC equals angle PQR (given), angle BAC equals angle QPR (given), and the third angles are equal because angles in a triangle sum $to ____$ degrees.

Scale Factor for Lengths

Find the scale factor by dividing a length on the second shape by the corresponding length on the first.
If the second shape is larger, scale factor > 1; if smaller, $0 <$ scale factor $< 1$ .
Multiply or divide by the scale factor to find missing lengths.

Two similar rectangles have widths 2 cm and 5 cm. What is the scale factor from the smaller to the larger?

Finding Missing Lengths (Method 1)

STEP 1: Find the scale factor from first to second shape.
STEP 2: Multiply the corresponding length on the first by the scale factor to find a missing length on the second.
To find a missing length on the first, divide the corresponding length on the second by the scale factor.

Shape A has side 6 cm. Shape B is similar with scale factor 0.5 from A to B. The corresponding side in B $is ____ cm$ .

Finding Missing Lengths (Method 2)

STEP 1: Find the scale factor from smaller to larger $(always > 1)$ .
STEP 2: Multiply a length on the smaller by the scale factor to get the corresponding length on the larger.
To find a length on the smaller, divide the corresponding length on the larger by the scale factor.

Two similar triangles have sides 3 cm and 9 cm. What is the scale factor from the smaller to the larger?

3
$\frac{1}{3}$
6
0.33

Worked Example: Similar Rectangles

Two rectangles have lengths 15 cm and 6 cm, widths 5 cm and 2 cm.
Scale factor for lengths: $\frac{15}{6} =$ 2.5; for widths: $\frac{5}{2} = 2.5$ .
Since the scale factor is the same for both pairs, the rectangles are similar.

Which statements about similar shapes are true?

Corresponding sides are in proportion.
They are always the same size.
If two triangles have equal angles, they are similar.

Exam Tips

Tracing paper can help check congruence if shapes are drawn to scale.
For similar triangles in an hourglass formation, look for vertically opposite angles.
Redraw similar shapes separately if they overlap.

To prove two triangles are similar, you must show that all three corresponding angles $are ____, or$ that two pairs are equal and the third follows from the angle sum.

Flashcards

What does it mean for two shapes to be congruent?

They are identical in shape and size; one may be a reflection, rotation, or translation of the other.

What does it mean for two shapes to be similar?

They have the same shape and corresponding sides are in proportion; one is an enlargement of the other.

How do you prove two triangles are similar?

Show that all three corresponding angles are equal.

How do you prove two non-triangular shapes are similar?

Show that all corresponding sides are in the same proportion (same scale factor).

What is the scale factor from shape A to shape B if A has side 4 cm and B has corresponding side 10 cm?

Scale factor $= \frac{10}{4} = 2.5$ .

If two similar shapes have a scale factor of 3 from the smaller to the larger, how many times bigger is the area of the larger?

Area scale factor $= (linear$ scale $factor)^{2} = 3^{2} = 9$ times bigger.

In similar triangles ABC and $DEF, AB = 6$ cm, $DE = 9$ cm, and $BC = 8$ cm. Find EF.

Scale factor $= \frac{9}{6} = 1.5$ , so $EF = 8 \times 1.5 = 12$ cm.

Triangle ABC is similar to triangle PQR. Angle $A = 50^{\circ}$ , angle $B = 60^{\circ}$ . Find angle R.

Angle $C = 180^{\circ} - 50^{\circ} - 60^{\circ} = 70^{\circ}$ , so angle $R =$ angle $C = 70^{\circ}$ .

What is the condition for two polygons to be congruent?

Corresponding sides are equal in length and corresponding angles are equal in size.

If a shape is reflected, rotated, or translated, is its image congruent to the original?

Yes, the image is congruent because only the position/orientation changes, not size or shape.

In the diagram, AB is parallel to CD. Triangles ABX and CDX share vertex X. Why are they similar?

Vertically opposite angles at X are equal; alternate angles on parallel lines give two other equal pairs; all three angles equal.

Two rectangles have sides 15 cm by 5 cm and 6 cm by 2 cm. Are they similar?

Yes, scale factor for length $= \frac{15}{6} = 2.5$ , for width $= \frac{5}{2} = 2.5$ , so corresponding sides are in proportion.

How do you find a missing length on a similar shape using the scale factor?

Multiply the corresponding length on the first shape by the scale factor (if second is larger) or divide (if second is smaller).

What is the scale factor if the second shape is smaller than the first?

The scale factor is between 0 and 1.

In similar shapes, if a side of the first is 12 cm and the corresponding side of the second is 4 cm, what is the scale factor from first to second?

Scale factor $= \frac{4}{12} = \frac{1}{3}$ .

What does 'mathematically similar' mean?

It means the shapes are similar: same shape, corresponding sides in proportion.

How can you check if two triangles are congruent?

Check that all corresponding sides are equal and all corresponding angles are equal.

If two shapes are similar and the scale factor is 2, what is the ratio of their areas?

Area ratio $= (scale factor)^{2} = 4:1$ .

In similar triangles, if one side of the first is 5 cm and the corresponding side of the second is 7.5 cm, find the scale factor from first to second.

Scale factor $= \frac{7.5}{5} = 1.5$ .

Why are two triangles similar if they have two pairs of equal angles?

Because the third angles will also be equal (angles in a triangle sum to 180°).

What is the difference between congruent and similar?

Congruent: same shape and size. Similar: same shape but different size (enlargement).

If shape A is enlarged by scale factor 0.5 to get shape B, are A and B similar?

Yes, they are similar because one is an enlargement of the other (scale factor 0.5).

How do you find the scale factor from the larger shape to the smaller shape?

Divide a length on the smaller shape by the corresponding length on the larger shape.

Two similar plant pots have heights 21.6 cm and h, and diameters 27 cm and 33 cm. Find h.

Scale factor $= \frac{33}{27} = \frac{11}{9}$ , so $h = 21.6 \times (\frac{11}{9}) = 26.4$ cm.

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