Geometric proof and congruence - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Geometric proof and congruence

Similarity and congruence

Lesson 2 of 5

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Similarity and congruence

Lesson 2 of 5

Answer each one, then check.

1

Name the four conditions for congruent triangles.

2

What are alternate angles?

3

What do the diagonals of a parallelogram do?

4

What is a proof?

5

Are vertically opposite angles equal?

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Similarity and congruence

Lesson 2 of 5

1

Name the four conditions for congruent triangles.

SSS, SAS, ASA, RHS

2

What are alternate angles?

Equal angles between parallel lines, in a Z shape

3

What do the diagonals of a parallelogram do?

Bisect each other

4

What is a proof?

A logical argument that shows a statement is always true

5

Are vertically opposite angles equal?

Yes

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Similarity and congruence

Lesson 2 of 5

1

Set out a geometric proof with statements and reasons.

2

Prove triangles congruent.

3

Use congruent triangles to prove other facts.

4

Use standard reasons for angles in parallel lines and triangles.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Writing a Proof

Similarity and congruence

Lesson 2 of 5

A proof is a chain of statements, each with a reason.

1

State what you are given

Write down the facts from the question

2

Find equal sides and angles

Give a reason for each: "opposite sides of a parallelogram are equal"

3

Name the congruence condition

SSS, SAS, ASA or RHS

4

State the conclusion

"Therefore triangle ABC is congruent to triangle DEF"

5

Use it

Corresponding parts of congruent triangles are equal

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Diagonals of a Parallelogram

Similarity and congruence

Lesson 2 of 5

Triangles AMB and CMD are congruent, which shows the diagonals bisect each other.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Proving Diagonals Bisect

Similarity and congruence

Lesson 2 of 5

ABCD is a parallelogram with diagonals AC and BD meeting at M. Prove that AM = MC and BM = MD.

1

AB = DC

Opposite sides of a parallelogram are equal

2

Angle BAM = angle DCM

Alternate angles (AB parallel to DC)

3

Angle ABM = angle CDM

Alternate angles (AB parallel to DC)

4

Triangles AMB and CMD are congruent

ASA (two angles and the corresponding side)

5

So AM = CM and BM = DM

Corresponding sides of congruent triangles

ANSWER

The diagonals of a parallelogram bisect each other, because triangles AMB and CMD are congruent (ASA).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Base Angles of an Isosceles Triangle

Similarity and congruence

Lesson 2 of 5

In triangle ABC, AB = AC. D is the midpoint of BC. Prove that angle ABC = angle ACB.

1

AB = AC

Given

2

BD = DC

D is the midpoint of BC

3

AD = AD

Common side

4

Triangles ABD and ACD are congruent

SSS

5

So angle ABD = angle ACD

Corresponding angles of congruent triangles

ANSWER

The base angles are equal because triangles ABD and ACD are congruent (SSS).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Good Proof, Weak Proof

Similarity and congruence

Lesson 2 of 5

A GOOD PROOF

• Every statement has a reason.

• Uses correct mathematical language.

• Names the congruence condition.

• Ends with a clear conclusion.

A WEAK PROOF

• Says "it is obvious" or "it looks equal".

• Uses measurements from a drawing.

• Misses out a step.

• Does not say why triangles are congruent.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reasons You Can Use

Similarity and congruence

Lesson 2 of 5

Choose the exact words the mark scheme expects.

Parallel lines

Alternate angles are equal; corresponding angles are equal; co-interior angles add to 180°.

Triangles

Angles in a triangle add to 180°; base angles of an isosceles triangle are equal.

Parallelograms

Opposite sides are equal and parallel; diagonals bisect each other.

Common side

Write "AD is common" when two triangles share a side.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Similarity and congruence

Lesson 2 of 5

Proof

A logical argument showing a statement is always true.

Statement

A fact written in a proof.

Reason

The rule that justifies a statement.

Corresponding

In matching positions in congruent triangles.

Common side

A side shared by two triangles.

Bisect

Cut exactly in half.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity and congruence

Lesson 2 of 5

YOUR TASK

Complete the Proof

15 minutes

PQRS is a kite with PQ = PS and RQ = RS. Prove that triangles PQR and PSR are congruent, and hence that angle PQR = angle PSR.

1

List equal sides.

2

Name SSS.

3

Conclude.

WHAT A GOOD ANSWER SHOWS

PQ = PS (given), RQ = RS (given), PR = PR (common side). The triangles are congruent by SSS. Corresponding angles are equal, so angle PQR = angle PSR.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Similarity and congruence

Lesson 2 of 5

Set out statements and reasons.

Use alternate angles in a proof.

Name a common side.

Choose SSS, SAS, ASA or RHS.

Use corresponding parts of congruent triangles.

Prove diagonals bisect.

Prove base angles are equal.

Finish with a conclusion.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Every step needs a reason.

Give the congruence condition in words or letters.

Corresponding parts of congruent triangles are equal.

End with a conclusion that answers the question.

EXAM FOCUS

ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent. (4 marks)

Write each fact with its reason: opposite sides equal, alternate angles equal. Then name the condition (ASA here).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Geometric Proof and Congruence

Similarity and congruence

Lesson 2 of 5

Answer all questions. Give reasons for every step. · 30 minutes

Question 1 · 4 marks · Prove

ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent.

Question 2 · 2 marks · Prove

Using the congruent triangles from the last question, prove that AM = MC.

Question 3 · 4 marks · Prove

Triangle ABC is isosceles with AB = AC. D is the midpoint of BC. Prove that angle ABC = angle ACB.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Geometric Proof and Congruence (cont.)

Similarity and congruence

Lesson 2 of 5

Question 4 · 3 marks · Prove

PQRS is a kite with PQ = PS and RQ = RS. Prove that angle PQR = angle PSR.

Question 5 · 2 marks · Explain

Explain why "the triangles look the same" is not a good reason in a proof.

Question 6 · 3 marks · Prove

ABCD is a rectangle. The diagonals AC and BD are drawn. Prove that AC = BD.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 4 marks · Prove

Similarity and congruence

Lesson 2 of 5

ABCD is a parallelogram. The diagonals AC and BD meet at M. Prove that triangles AMB and CMD are congruent. (4 marks)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Similarity and congruence

Lesson 2 of 5

4 marks available. Award a mark for each point made.

AB = DC with a reason

B1

One pair of equal angles with a reason

M1

The second pair of equal angles with a reason

M1

Congruent by ASA

A1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Prove

Similarity and congruence

Lesson 2 of 5

“

Using the congruent triangles from the last question, prove that AM = MC.

HOW TO ANSWER IT

Command word: Prove. Worth 2 marks, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Similarity and congruence

Lesson 2 of 5

2 marks available. Award a mark for each point made.

Corresponding sides of congruent triangles

M1

AM = MC

A1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 4 marks · Prove

Similarity and congruence

Lesson 2 of 5

Triangle ABC is isosceles with AB = AC. D is the midpoint of BC. Prove that angle ABC = angle ACB. (4 marks)

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Similarity and congruence

Lesson 2 of 5

4 marks available. Award a mark for each point made.

AB = AC and BD = DC with reasons

M1

AD common

M1

SSS

A1

Conclusion about angles

C1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Prove

Similarity and congruence

Lesson 2 of 5

“

PQRS is a kite with PQ = PS and RQ = RS. Prove that angle PQR = angle PSR.

HOW TO ANSWER IT

Command word: Prove. Worth 3 marks, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Similarity and congruence

Lesson 2 of 5

3 marks available. Award a mark for each point made.

Equal sides and common side

M1

SSS

M1

Conclusion

A1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Explain

Similarity and congruence

Lesson 2 of 5

“

Explain why "the triangles look the same" is not a good reason in a proof.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Similarity and congruence

Lesson 2 of 5

2 marks available. Award a mark for each point made.

Not appearances or measurements

M1

Each step needs a mathematical reason

C1

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 3 marks · Prove

Similarity and congruence

Lesson 2 of 5

“

ABCD is a rectangle. The diagonals AC and BD are drawn. Prove that AC = BD.

HOW TO ANSWER IT

Command word: Prove. Worth 3 marks, so plan before writing.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Similarity and congruence

Lesson 2 of 5

3 marks available. Award a mark for each point made.

Equal sides and right angles

M1

SAS

M1

AC = BD

A1

openrevise.com