Congruence.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Congruence

Similarity and congruence

Lesson 1 of 5

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Similarity and congruence

Lesson 1 of 5

Answer each one, then check.

1

What do the angles in a triangle add up to?

2

What does isosceles mean?

3

What is a right-angled triangle's longest side called?

4

Do reflections keep the size of a shape?

5

What do the angles on a straight line add up to?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Similarity and congruence

Lesson 1 of 5

1

What do the angles in a triangle add up to?

180°

2

What does isosceles mean?

Two equal sides

3

What is a right-angled triangle's longest side called?

The hypotenuse

4

Do reflections keep the size of a shape?

Yes

5

What do the angles on a straight line add up to?

180°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Similarity and congruence

Lesson 1 of 5

1

Explain what congruent means.

2

Use SSS, SAS, ASA and RHS to show triangles are congruent.

3

Explain why AAA and SSA do not prove congruence.

4

Use congruence to find missing sides and angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity and congruence

Lesson 1 of 5

CONGRUENT

Two shapes are congruent if they are exactly the same shape and size.

One can be moved onto the other by a rotation, a reflection, a translation, or a combination.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Four Congruence Conditions

Similarity and congruence

Lesson 1 of 5

If two triangles satisfy any one of these, they are congruent.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Four Conditions

Similarity and congruence

Lesson 1 of 5

Learn the letters and what they mean.

Condition

Means

Example

SSS

All three sides equal

5, 6, 7 and 5, 6, 7

SAS

Two sides and the included angle equal

5, 40°, 7 and 5, 40°, 7

ASA

Two angles and a corresponding side equal

50°, 6, 70°

RHS

Right angle, hypotenuse and one other side equal

Right angle, 10, 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Proving Two Triangles Congruent

Similarity and congruence

Lesson 1 of 5

Triangle ABC has AB = 5 cm, BC = 7 cm and angle ABC = 50°. Triangle PQR has PQ = 5 cm, QR = 7 cm and angle PQR = 50°. Are the triangles congruent?

1

Two pairs of equal sides

AB = PQ and BC = QR

2

The angle between them is equal too

Angle B = angle Q = 50°

3

Two sides and the included angle

SAS

ANSWER

Yes, the triangles are congruent by SAS.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Conditions That Do NOT Work

Similarity and congruence

Lesson 1 of 5

AAA

Three equal angles give similar triangles, but they can be different sizes.

SSA

Two sides and a non-included angle can give two different triangles.

Only sides

Two sides alone are not enough.

Angles not between the sides

If the angle is not between the two sides, it is not SAS.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Using Congruence to Find a Length

Similarity and congruence

Lesson 1 of 5

Triangles ABC and DEF are congruent by SSS, with A matching D, B matching E and C matching F. AB = 6 cm, BC = 8 cm, DF = 10 cm. Find AC and DE.

1

Corresponding sides are equal

AB = DE, BC = EF, AC = DF

2

Find AC

AC = DF = 10 cm

3

Find DE

DE = AB = 6 cm

ANSWER

AC = 10 cm and DE = 6 cm.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Writing the Reason

Similarity and congruence

Lesson 1 of 5

A congruence answer needs three things.

The equal parts

Write which sides or angles are equal, with the reason for each.

The condition

State SSS, SAS, ASA or RHS.

The conclusion

"So triangle ABC is congruent to triangle DEF."

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Similarity and congruence

Lesson 1 of 5

Congruent

Exactly the same shape and size.

Corresponding

In matching positions in two shapes.

Included angle

The angle between two given sides.

Hypotenuse

The longest side of a right-angled triangle.

SSS, SAS, ASA, RHS

The four conditions for congruent triangles.

Similar

The same shape but a different size.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity and congruence

Lesson 1 of 5

YOUR TASK

Congruent or Not?

12 minutes

Decide which pairs of triangles are congruent and name the condition. (a) 4, 5, 6 and 6, 4, 5 (b) sides 5 and 8 with an angle of 30° between them, and sides 5 and 8 with an angle of 40° between them (c) angles 40°, 60°, 80° in both triangles, but one has a side of 3 cm and the other a side of 6 cm.

1

Match the equal parts.

2

Name the condition.

WHAT A GOOD ANSWER SHOWS

(a) Congruent by SSS. (b) Not congruent: the included angles differ. (c) Not congruent: AAA does not prove congruence; the sides are different sizes, so the triangles are similar.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Similarity and congruence

Lesson 1 of 5

Say what congruent means.

Recall SSS, SAS, ASA and RHS.

Decide whether two triangles are congruent.

Explain why AAA is not enough.

Explain why SSA is not enough.

Find corresponding sides and angles.

Use congruence to find a length.

Write a clear reason.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Congruent means identical in shape and size.

Conditions: SSS, SAS, ASA (or AAS), RHS.

AAA and SSA do not prove congruence.

Always give the condition as your reason.

EXAM FOCUS

Triangle ABC has AB = 6 cm, AC = 8 cm and angle A = 50°. Triangle PQR has PQ = 6 cm, PR = 8 cm and angle P = 50°. Show that the two triangles are congruent. (3 marks)

Name the pairs of equal parts and check the angle is between the two sides. Then state SAS.

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