Congruence And Similarity

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Lesson 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.

Congruent Shapes

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 Triangles in an Hourglass

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 Rectangles

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.

Finding a Missing Length on a Similar Shape

Slides

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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?
Easy
- A1/3
- B3
- C8
- D48

Revision notes

## 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)

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.

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 = 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 = 3² = 9 times bigger.

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

Scale factor = 9/6 = 1.5, so EF = 8 × 1.5 = 12 cm.

Triangle ABC is similar to triangle PQR. Angle A = 50°, angle B = 60°. Find angle R.

Angle C = 180° - 50° - 60° = 70°, so angle R = angle C = 70°.

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 = 15/6 = 2.5, for width = 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 = 4/12 = 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 = 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 = 33/27 = 11/9, so h = 21.6 × (11/9) = 26.4 cm.

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