Congruence And Similarity

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Notes de leçon

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

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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

Plus de questions

9.Which condition is sufficient to prove two triangles are similar?
Medium
- ATwo pairs of corresponding sides are in proportion
- BTwo angles are equal
- CAll three sides are proportional
- DTwo sides are equal
10.Two similar triangles have areas 16 cm² and 36 cm². What is the scale factor of their corresponding sides?
Medium
- A1.5
- B2.25
- C0.75
- D2
11.AB and CD are parallel lines. A straight line crosses both, making an angle at A (on line AB) and an angle at D (on line CD) that are on opposite sides of the transversal, between the two parallel lines. What type of angle pair is angle A and angle D?
Easy
- ACorresponding angles
- BAlternate angles
- CVertically opposite angles
- DCo-interior angles
12.Two similar plant pots have heights 21.6 cm and 33 cm. The smaller pot has diameter 27 cm. What is the diameter of the larger pot?
Medium
- A41.25 cm
- B17.67 cm
- C35 cm
- D39.6 cm
13.Triangle ABC is similar to triangle PQR. AB = 5 cm, BC = 7 cm, PQ = 12 cm. Find QR.
Medium
- A16.8 cm
- B2.92 cm
- C8.4 cm
- D19 cm
14.Shape ABCD is similar to shape PQRS. AB = 8 cm, BC = 6 cm, PQ = 12 cm. Find QR.
Medium
- A9 cm
- B4 cm
- C16 cm
- D10 cm
15.Two similar triangles have perimeters 30 cm and 45 cm. The smaller triangle has a side of length 8 cm. What is the corresponding side in the larger triangle?
Medium
- A12 cm
- B5.33 cm
- C13.5 cm
- D16 cm
16.In the diagram, triangle TUV is similar to triangle XYZ. TU = 9 cm, UV = 12 cm, XY = 15 cm. Find YZ.
Medium
- A20 cm
- B7.2 cm
- C18 cm
- D11.25 cm
17.Two triangles have angles 50°, 60°, 70° and 50°, 60°, 70° respectively. Which statement is true?
Medium
- AThey are congruent but not similar.
- BThey are similar but not congruent.
- CThey are both congruent and similar.
- DThey are neither congruent nor similar.
18.Which of the following is NOT a condition for two triangles to be congruent?
Medium
- ATwo sides and the included angle are equal (SAS).
- BThree sides are equal (SSS).
- CTwo angles and a non-included side are equal (AAS).
- DTwo sides and a non-included angle are equal (SSA).
19.The diagram shows two similar triangles. Triangle ABC has sides AB = 4 cm, BC = 6 cm, AC = 8 cm. Triangle PQR has side PQ = 6 cm. What is the length of PR?
Medium
- A9 cm
- B12 cm
- C10 cm
- D8 cm
20.Two rectangles are similar. The smaller rectangle has width 5 cm and length 8 cm. The larger rectangle has width 7.5 cm. What is the length of the larger rectangle?
Medium
- A12 cm
- B10 cm
- C11 cm
- D13.5 cm
21.Triangles ABC and DEF are similar. AB = 9 cm, BC = 12 cm, DE = 6 cm. Find EF.
Medium
- A8 cm
- B18 cm
- C4.5 cm
- D13.5 cm
22.In the diagram, triangles ABC and PQR are similar. AB = 5 cm, BC = 7 cm, PR = 10.5 cm. What is the length of PQ?
Medium
- A7.5 cm
- B8 cm
- C9 cm
- D6 cm
23.Shape ABCD is similar to shape PQRS. AB = 10 cm, BC = 8 cm, PQ = 15 cm. Find QR.
Medium
- A12 cm
- B10 cm
- C13 cm
- D11 cm
24.Two similar triangles have perimeters 40 cm and 60 cm. The smaller triangle has a side of length 10 cm. What is the corresponding side in the larger triangle?
Medium
- A15 cm
- B20 cm
- C12 cm
- D18 cm
25.In the diagram, triangle TUV is similar to triangle XYZ. TU = 8 cm, UV = 10 cm, XY = 12 cm. Find YZ.
Medium
- A15 cm
- B9.6 cm
- C14 cm
- D16 cm
26.Two similar triangles have areas 25 cm² and 100 cm². What is the scale factor of their corresponding sides?
Easy
- A2
- B4
- C1.5
- D3
27.Triangle ABC is similar to triangle PQR. AB = 6 cm, BC = 9 cm, QR = 12 cm. Find PQ.
Medium
- A8 cm
- B10 cm
- C7.5 cm
- D9 cm
28.Shape ABCD is similar to shape PQRS. AB = 12 cm, BC = 9 cm, PQ = 8 cm. Find QR.
Medium
- A6 cm
- B7 cm
- C5 cm
- D4.5 cm
29.Two similar plant pots have heights 24 cm and 36 cm. The smaller pot has diameter 30 cm. What is the diameter of the larger pot?
Medium
- A45 cm
- B40 cm
- C50 cm
- D48 cm
30.Triangle ABC is similar to triangle PQR. AB = 8 cm, BC = 10 cm, PQ = 12 cm, AC = 6 cm. Find PR.
Medium
- A9 cm
- B7.5 cm
- C10 cm
- D8 cm
31.Two similar triangles have perimeters 24 cm and 36 cm. The larger triangle has a side of length 15 cm. What is the corresponding side in the smaller triangle?
Medium
- A10 cm
- B12 cm
- C8 cm
- D9 cm
32.In the diagram, triangle ABC is similar to triangle DEF. AB = 5 cm, AC = 7 cm, DE = 8 cm. Find DF.
Medium
- A11.2 cm
- B10 cm
- C9.8 cm
- D12 cm
33.Two similar triangles have areas 50 cm² and 72 cm². What is the scale factor of their corresponding sides?
Medium
- A1.2
- B1.44
- C0.833
- D1.5
34.A model car is built to a scale of 1:24. The real car has a length of 4.8 m. What is the length of the model in cm?
Easy
- A20 cm
- B2 cm
- C200 cm
- D0.2 cm
35.In the diagram, triangles ABC and DEF are similar. AB = 9 cm, DE = 6 cm, BC = 12 cm. Find EF.
Medium
- A8 cm
- B18 cm
- C4.5 cm
- D6 cm
36.Two similar cylinders have heights 10 cm and 14 cm. The smaller cylinder has volume 500 cm³. What is the volume of the larger cylinder?
Medium
- A980 cm³
- B700 cm³
- C1372 cm³
- D500 cm³
37.Triangle ABC is similar to triangle PQR. AB = 8 cm, BC = 10 cm, PQ = 12 cm. Find QR.
Medium
- A15 cm
- B6.67 cm
- C9.6 cm
- D20 cm
38.Two polygons are congruent. Which of the following is true?
Easy
- AThey have the same shape but different size.
- BThey have the same shape and size.
- CThey have the same size but different shape.
- DThey are both regular polygons.
39.A rectangle has length 12 cm and width 8 cm. A similar rectangle has length 18 cm. What is its width?
Medium
- A12 cm
- B14 cm
- C10 cm
- D16 cm
40.Two similar cones have radii 5 cm and 8 cm. The smaller cone has surface area 200 cm². What is the surface area of the larger cone?
Medium
- A512 cm²
- B320 cm²
- C125 cm²
- D800 cm²
41.Which of the following transformations always produces a shape that is congruent to the original?
Easy
- AEnlargement
- BRotation
- CReflection
- DBoth rotation and reflection
42.Two shapes are similar. Which of the following is always true about their corresponding sides?
Easy
- AThey are equal in length
- BThey are in the same ratio
- CThey are parallel
- DThey are perpendicular
43.Which of the following are true about congruent shapes? (Select all that apply)
Medium
- AThey have the same shape and size
- BThey have the same shape but not necessarily the same size
- COne can be a reflection of the other
- DOne can be an enlargement of the other
44.Two triangles with all three pairs of corresponding sides in the same ratio are always similar.
Easy
True or false?
45.Two shapes are congruent. Which of the following is always true?
Easy
- AThey have the same shape but different size.
- BThey have the same size but different shape.
- CThey are identical in shape and size.
- DThey are enlargements of each other.
46.If a shape is reflected, its image is congruent to the original shape.
Easy
True or false?
47.Match each condition to whether it proves congruence or similarity of two triangles.
Medium
- All three pairs of corresponding sides are equal
- All three pairs of corresponding angles are equal
- Two sides and the included angle are equal
- Congruence
- Similarity
- Congruence
48.Arrange the steps in order to find a missing length on a similar shape using the scale factor method.
Medium
- Divide the length on the second shape by the corresponding length on the first shape to find the scale factor.
- Use the scale factor to multiply or divide the known length.
- Identify two known corresponding sides.
- State the missing length.
49.Which of the following statements are true about similar shapes? (Select all that apply)
Medium
- ACorresponding angles are equal.
- BCorresponding sides are in proportion.
- CThey are always congruent.
- DOne shape is an enlargement of the other.
- EThe scale factor is always greater than 1.
50.Two rectangles are similar. The smaller rectangle has width 4 cm and length 6 cm. The larger rectangle has width 10 cm. What is the length of the larger rectangle?
Medium
- A12 cm
- B15 cm
- C20 cm
- D24 cm
51.A rectangle has length 12 cm and width 8 cm. A similar rectangle has length 18 cm. Find its width.
Easy
52.Two similar rectangles have widths 3 cm and 5 cm. The smaller rectangle has length 7 cm. Find the length of the larger rectangle.
Medium
53.Complete the sentence: Two shapes are ______ if they are identical in shape and size.
Easy
Two shapes are ____ if they are identical in shape and size.
54.State the condition for two triangles to be similar (in terms of angles).
Easy
55.Two similar cylinders have heights 10 cm and 14 cm. The smaller cylinder has volume 500 cm³. Find the volume of the larger cylinder.
Medium
56.Two similar triangles have corresponding sides of lengths 8 cm and 12 cm. What is the scale factor from the smaller triangle to the larger triangle?
Easy
57.Complete the sentence.
Easy
Two shapes are ____ if they have the same shape and their corresponding sides are in proportion.
58.In the diagram, AB is parallel to DC. Triangles ABX and CDX share vertex X. Name the pair of equal angles formed by the parallel lines.
Medium
59.Two similar triangles have corresponding sides of lengths 8 cm and 12 cm. What is the scale factor from the smaller triangle to the larger triangle?
Easy
- A3.5
- B0.5
- C2.5
- D1.5

Notes de révision

## 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.

Cartes mémo

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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