PAPER · REASONING · PRACTICE

Congruence: correspondence, SSS and SAS — Teacher reference

Read the vertex order, check included angles and build short justified proofs.

40 minutes · Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9

Textbook reference: Year 8, upper · Chapter 12: congruent triangles · lessons 7–9

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Student classroom page Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

Rotate one cut-out triangle before matching it to another. Which vertex order and angle must remain paired?

Organise the key idea

SSS uses three side pairs. SAS uses two side pairs and their included angle; rotating a diagram changes none of these conditions.

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

AB=DE, AC=DF and ∠BAC=∠EDF. Prove △ABC≅△DEF, explaining why SAS applies.

Geometry diagram; not drawn to scale except for construction tasks.

List the given equal sides and angles; preserve vertex correspondence.
Check the exact conditions of SAS; do not assume an unmarked fact.
Write a criterion with reasons, followed by a corresponding-side or angle conclusion.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Check your work

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Guided attempt2

The marked side pairs are equal. Complete an SSS proof in the order A↔D, B↔E, C↔F.

AB=___, AC=___, BC=___.
Therefore △ABC≅△___ by ___.

Geometry diagram; not drawn to scale except for construction tasks.

List the given equal sides and angles; preserve vertex correspondence.
Check the exact conditions of SSS; do not assume an unmarked fact.
Write a criterion with reasons, followed by a corresponding-side or angle conclusion.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Check your work

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Prerequisite3

Given △ABC≅△DEF, which side corresponds to BC?

Geometry diagram; not drawn to scale except for construction tasks.

The order gives B↔E and C↔F.
Thus side BC corresponds to EF.
Compare with: EF

The order gives B↔E and C↔F.
Thus side BC corresponds to EF.

Check your work

Choose the statement justified by the stated conditions and explain why.

Prerequisite4

Which angle is between AB and AC?

∠ABC
∠ACB
Every angle is included.
∠BAC

Geometry diagram; not drawn to scale except for construction tasks.

The shared endpoint of AB and AC is A.
The included angle therefore has A in the middle: ∠BAC.
Compare with: ∠BAC

The shared endpoint of AB and AC is A.
The included angle therefore has A in the middle: ∠BAC.

Check your work

Choose the statement justified by the stated conditions and explain why.

Independent practice5

AB=DE and AC=DF. Which additional equality completes an SSS test?

BC=EF
∠A=∠D
AB=AB
BC=DE

Geometry diagram; not drawn to scale except for construction tasks.

SSS needs all three corresponding side pairs.
The remaining pair is BC and EF.
Compare with: BC=EF

SSS needs all three corresponding side pairs.
The remaining pair is BC and EF.

Check your work

Choose the statement justified by the stated conditions and explain why.

Independent practice6

AB=DE, AC=DF and ∠BAC=∠EDF. Which criterion uses exactly these facts directly?

Geometry diagram; not drawn to scale except for construction tasks.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Compare with: SAS

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Check your work

Choose the statement justified by the stated conditions and explain why.

Independent practice7

Given △ABC≅△DEF, which angle corresponds to ∠ACB?

∠EDF
∠ABC
∠DFE
∠DEF

Geometry diagram; not drawn to scale except for construction tasks.

A↔D, C↔F and B↔E.
The vertex C becomes F, so ∠ACB corresponds to ∠DFE.
Compare with: ∠DFE

A↔D, C↔F and B↔E.
The vertex C becomes F, so ∠ACB corresponds to ∠DFE.

Check your work

Choose the statement justified by the stated conditions and explain why.

Independent practice8

Use only the three marked equal side pairs to prove the rotated triangles congruent. Write the vertex order.

Geometry diagram; not drawn to scale except for construction tasks.

List the given equal sides and angles; preserve vertex correspondence.
Check the exact conditions of SSS; do not assume an unmarked fact.
Write a criterion with reasons, followed by a corresponding-side or angle conclusion.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Check your work

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Exit question9

AB=DE, AC=DF and ∠ABC=∠DEF. Is this automatically SAS?

No: the given angle is not between the two given sides.
Yes: any equal angle works.
Yes: two sides always determine a triangle.
It is SSS.

Geometry diagram; not drawn to scale except for construction tasks.

AB and AC meet at A, not B.
The equal angle at B is a non-included angle; the supplied pair provides an SSA counterexample.
Compare with: No: the given angle is not between the two given sides.

AB and AC meet at A, not B.
The equal angle at B is a non-included angle; the supplied pair provides an SSA counterexample.

Check your work

Choose the statement justified by the stated conditions and explain why.

Exit question10

The three corresponding angles are equal. Must the triangles have equal side lengths?

Yes, if they are rotated.
No: one triangle may be an enlargement of the other.
Yes, by AAA congruence.
Only the drawing colour matters.

Geometry diagram; not drawn to scale except for construction tasks.

Equal angles preserve shape but do not fix scale.
Here the second triangle has twice every side length, so it is not congruent.
Compare with: No: one triangle may be an enlargement of the other.

Equal angles preserve shape but do not fix scale.
Here the second triangle has twice every side length, so it is not congruent.

Check your work

Choose the statement justified by the stated conditions and explain why.

Further practice11

The marked correspondence is A↔D, B↔E, C↔F. Which congruence statement respects it?

△ABC≅△EDF
△ABC≅△FED
△ABC≅△DEF
△ABC≅△DFE

Geometry diagram; not drawn to scale except for construction tasks.

Replace A,B,C in sequence by D,E,F.
Write △ABC≅△DEF; the order carries mathematical information.
Compare with: △ABC≅△DEF

Replace A,B,C in sequence by D,E,F.
Write △ABC≅△DEF; the order carries mathematical information.

Check your work

Choose the statement justified by the stated conditions and explain why.

Further practice12

Fill in why the marked angle is included, then give the SAS conclusion.

Geometry diagram; not drawn to scale except for construction tasks.

List the given equal sides and angles; preserve vertex correspondence.
Check the exact conditions of SAS; do not assume an unmarked fact.
Write a criterion with reasons, followed by a corresponding-side or angle conclusion.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Check your work

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice13

Write three side equalities and one congruence conclusion for the diagram.

Geometry diagram; not drawn to scale except for construction tasks.

List the given equal sides and angles; preserve vertex correspondence.
Check the exact conditions of SSS; do not assume an unmarked fact.
Write a criterion with reasons, followed by a corresponding-side or angle conclusion.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Check your work

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice14

After proving △ABC≅△DEF, which equality follows from corresponding sides?

BC=DF
AC=DF
AC=DE
AB=EF

Geometry diagram; not drawn to scale except for construction tasks.

Use A↔D and C↔F.
The corresponding side to AC is DF.
Compare with: AC=DF

Use A↔D and C↔F.
The corresponding side to AC is DF.

Check your work

Choose the statement justified by the stated conditions and explain why.

Extension15

The two triangles meet the marked SSA facts but have different third sides. What does this show about SSA?

SAS is therefore invalid.
The vertex order can be ignored.
SSA alone does not guarantee congruence.
All SSA triangles fail to exist.

Geometry diagram; not drawn to scale except for construction tasks.

The two supplied coordinate constructions have AB=DE and AC=DF and equal angles at B and E.
Their BC and EF lengths differ, so one counterexample disproves the proposed general guarantee.
Compare with: SSA alone does not guarantee congruence.

The two supplied coordinate constructions have AB=DE and AC=DF and equal angles at B and E.
Their BC and EF lengths differ, so one counterexample disproves the proposed general guarantee.

Check your work

Choose the statement justified by the stated conditions and explain why.

Extension16

Explain why replacing ∠ABC=∠DEF by ∠BAC=∠EDF repairs this two-side proof.

It adds a third side.
It changes SAS into AAA.
It only makes the picture prettier.
It supplies the angle between the two given side pairs, completing SAS.

Geometry diagram; not drawn to scale except for construction tasks.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Compare with: It supplies the angle between the two given side pairs, completing SAS.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Check your work

Choose the statement justified by the stated conditions and explain why.

Preparation and assessment notes

Prerequisites

Triangle names, corresponding sides and angles, and nondegenerate triangles.

Look for in students’ work

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

All reference answers

Worked example1

AB=DE, AC=DF and ∠BAC=∠EDF. Prove △ABC≅△DEF, explaining why SAS applies.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SAS, with A↔D, B↔E and C↔F.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Guided attempt2

The marked side pairs are equal. Complete an SSS proof in the order A↔D, B↔E, C↔F.

AB=___, AC=___, BC=___.
Therefore △ABC≅△___ by ___.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Prerequisite3

Given △ABC≅△DEF, which side corresponds to BC?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

The order gives B↔E and C↔F.
Thus side BC corresponds to EF.

Choose the statement justified by the stated conditions and explain why.

Prerequisite4

Which angle is between AB and AC?

∠ABC
∠ACB
Every angle is included.
∠BAC

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

∠BAC

The shared endpoint of AB and AC is A.
The included angle therefore has A in the middle: ∠BAC.

Choose the statement justified by the stated conditions and explain why.

Independent practice5

AB=DE and AC=DF. Which additional equality completes an SSS test?

BC=EF
∠A=∠D
AB=AB
BC=DE

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

BC=EF

SSS needs all three corresponding side pairs.
The remaining pair is BC and EF.

Choose the statement justified by the stated conditions and explain why.

Independent practice6

AB=DE, AC=DF and ∠BAC=∠EDF. Which criterion uses exactly these facts directly?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

SAS

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Choose the statement justified by the stated conditions and explain why.

Independent practice7

Given △ABC≅△DEF, which angle corresponds to ∠ACB?

∠EDF
∠ABC
∠DFE
∠DEF

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

∠DFE

A↔D, C↔F and B↔E.
The vertex C becomes F, so ∠ACB corresponds to ∠DFE.

Choose the statement justified by the stated conditions and explain why.

Independent practice8

Use only the three marked equal side pairs to prove the rotated triangles congruent. Write the vertex order.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Exit question9

AB=DE, AC=DF and ∠ABC=∠DEF. Is this automatically SAS?

No: the given angle is not between the two given sides.
Yes: any equal angle works.
Yes: two sides always determine a triangle.
It is SSS.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No: the given angle is not between the two given sides.

AB and AC meet at A, not B.
The equal angle at B is a non-included angle; the supplied pair provides an SSA counterexample.

Choose the statement justified by the stated conditions and explain why.

Exit question10

The three corresponding angles are equal. Must the triangles have equal side lengths?

Yes, if they are rotated.
No: one triangle may be an enlargement of the other.
Yes, by AAA congruence.
Only the drawing colour matters.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No: one triangle may be an enlargement of the other.

Equal angles preserve shape but do not fix scale.
Here the second triangle has twice every side length, so it is not congruent.

Choose the statement justified by the stated conditions and explain why.

Further practice11

The marked correspondence is A↔D, B↔E, C↔F. Which congruence statement respects it?

△ABC≅△EDF
△ABC≅△FED
△ABC≅△DEF
△ABC≅△DFE

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF

Replace A,B,C in sequence by D,E,F.
Write △ABC≅△DEF; the order carries mathematical information.

Choose the statement justified by the stated conditions and explain why.

Further practice12

Fill in why the marked angle is included, then give the SAS conclusion.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SAS, with A↔D, B↔E and C↔F.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice13

Write three side equalities and one congruence conclusion for the diagram.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice14

After proving △ABC≅△DEF, which equality follows from corresponding sides?

BC=DF
AC=DF
AC=DE
AB=EF

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

AC=DF

Use A↔D and C↔F.
The corresponding side to AC is DF.

Choose the statement justified by the stated conditions and explain why.

Extension15

The two triangles meet the marked SSA facts but have different third sides. What does this show about SSA?

SAS is therefore invalid.
The vertex order can be ignored.
SSA alone does not guarantee congruence.
All SSA triangles fail to exist.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

SSA alone does not guarantee congruence.

The two supplied coordinate constructions have AB=DE and AC=DF and equal angles at B and E.
Their BC and EF lengths differ, so one counterexample disproves the proposed general guarantee.

Choose the statement justified by the stated conditions and explain why.

Extension16

Explain why replacing ∠ABC=∠DEF by ∠BAC=∠EDF repairs this two-side proof.

It adds a third side.
It changes SAS into AAA.
It only makes the picture prettier.
It supplies the angle between the two given side pairs, completing SAS.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

It supplies the angle between the two given side pairs, completing SAS.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Choose the statement justified by the stated conditions and explain why.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Chapter-core practice after the mapped textbook lessons; these tasks do not replace every first-teaching period.

大湾区初中数学三维课堂（八年级上册）正文.pdf

Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Congruence: correspondence, SSS and SAS

Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

AB=DE, AC=DF and ∠BAC=∠EDF. Prove △ABC≅△DEF, explaining why SAS applies.

Geometry diagram; not drawn to scale except for construction tasks.

Guided attempt2

The marked side pairs are equal. Complete an SSS proof in the order A↔D, B↔E, C↔F.

AB=___, AC=___, BC=___.
Therefore △ABC≅△___ by ___.

Geometry diagram; not drawn to scale except for construction tasks.

Prerequisite3

Given △ABC≅△DEF, which side corresponds to BC?

Geometry diagram; not drawn to scale except for construction tasks.

Prerequisite4

Which angle is between AB and AC?

∠ABC
∠ACB
Every angle is included.
∠BAC

Geometry diagram; not drawn to scale except for construction tasks.

Congruence: correspondence, SSS and SAS

Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

AB=DE and AC=DF. Which additional equality completes an SSS test?

BC=EF
∠A=∠D
AB=AB
BC=DE

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice6

AB=DE, AC=DF and ∠BAC=∠EDF. Which criterion uses exactly these facts directly?

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice7

Given △ABC≅△DEF, which angle corresponds to ∠ACB?

∠EDF
∠ABC
∠DFE
∠DEF

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice8

Use only the three marked equal side pairs to prove the rotated triangles congruent. Write the vertex order.

Geometry diagram; not drawn to scale except for construction tasks.

Exit question9

AB=DE, AC=DF and ∠ABC=∠DEF. Is this automatically SAS?

No: the given angle is not between the two given sides.
Yes: any equal angle works.
Yes: two sides always determine a triangle.
It is SSS.

Geometry diagram; not drawn to scale except for construction tasks.

Exit question10

The three corresponding angles are equal. Must the triangles have equal side lengths?

Yes, if they are rotated.
No: one triangle may be an enlargement of the other.
Yes, by AAA congruence.
Only the drawing colour matters.

Geometry diagram; not drawn to scale except for construction tasks.

Congruence: correspondence, SSS and SAS — Further practice

Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9

Further practice11

The marked correspondence is A↔D, B↔E, C↔F. Which congruence statement respects it?

△ABC≅△EDF
△ABC≅△FED
△ABC≅△DEF
△ABC≅△DFE

Geometry diagram; not drawn to scale except for construction tasks.

Further practice12

Fill in why the marked angle is included, then give the SAS conclusion.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice13

Write three side equalities and one congruence conclusion for the diagram.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice14

After proving △ABC≅△DEF, which equality follows from corresponding sides?

BC=DF
AC=DF
AC=DE
AB=EF

Geometry diagram; not drawn to scale except for construction tasks.

Extension15

The two triangles meet the marked SSA facts but have different third sides. What does this show about SSA?

SAS is therefore invalid.
The vertex order can be ignored.
SSA alone does not guarantee congruence.
All SSA triangles fail to exist.

Geometry diagram; not drawn to scale except for construction tasks.

Extension16

Explain why replacing ∠ABC=∠DEF by ∠BAC=∠EDF repairs this two-side proof.

It adds a third side.
It changes SAS into AAA.
It only makes the picture prettier.
It supplies the angle between the two given side pairs, completing SAS.

Geometry diagram; not drawn to scale except for construction tasks.

Congruence: correspondence, SSS and SAS — Teacher answers

Year 8, upper volume · text pp. 13–18 · foundation lessons 7–9

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Worked example1

AB=DE, AC=DF and ∠BAC=∠EDF. Prove △ABC≅△DEF, explaining why SAS applies.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SAS, with A↔D, B↔E and C↔F.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Guided attempt2

The marked side pairs are equal. Complete an SSS proof in the order A↔D, B↔E, C↔F.

AB=___, AC=___, BC=___.
Therefore △ABC≅△___ by ___.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Prerequisite3

Given △ABC≅△DEF, which side corresponds to BC?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

The order gives B↔E and C↔F.
Thus side BC corresponds to EF.

Choose the statement justified by the stated conditions and explain why.

Prerequisite4

Which angle is between AB and AC?

∠ABC
∠ACB
Every angle is included.
∠BAC

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

∠BAC

The shared endpoint of AB and AC is A.
The included angle therefore has A in the middle: ∠BAC.

Choose the statement justified by the stated conditions and explain why.

Independent practice5

AB=DE and AC=DF. Which additional equality completes an SSS test?

BC=EF
∠A=∠D
AB=AB
BC=DE

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

BC=EF

SSS needs all three corresponding side pairs.
The remaining pair is BC and EF.

Choose the statement justified by the stated conditions and explain why.

Independent practice6

AB=DE, AC=DF and ∠BAC=∠EDF. Which criterion uses exactly these facts directly?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

SAS

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Choose the statement justified by the stated conditions and explain why.

Independent practice7

Given △ABC≅△DEF, which angle corresponds to ∠ACB?

∠EDF
∠ABC
∠DFE
∠DEF

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

∠DFE

A↔D, C↔F and B↔E.
The vertex C becomes F, so ∠ACB corresponds to ∠DFE.

Choose the statement justified by the stated conditions and explain why.

Independent practice8

Use only the three marked equal side pairs to prove the rotated triangles congruent. Write the vertex order.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Exit question9

AB=DE, AC=DF and ∠ABC=∠DEF. Is this automatically SAS?

No: the given angle is not between the two given sides.
Yes: any equal angle works.
Yes: two sides always determine a triangle.
It is SSS.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No: the given angle is not between the two given sides.

AB and AC meet at A, not B.
The equal angle at B is a non-included angle; the supplied pair provides an SSA counterexample.

Choose the statement justified by the stated conditions and explain why.

Exit question10

The three corresponding angles are equal. Must the triangles have equal side lengths?

Yes, if they are rotated.
No: one triangle may be an enlargement of the other.
Yes, by AAA congruence.
Only the drawing colour matters.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No: one triangle may be an enlargement of the other.

Equal angles preserve shape but do not fix scale.
Here the second triangle has twice every side length, so it is not congruent.

Choose the statement justified by the stated conditions and explain why.

Further practice11

The marked correspondence is A↔D, B↔E, C↔F. Which congruence statement respects it?

△ABC≅△EDF
△ABC≅△FED
△ABC≅△DEF
△ABC≅△DFE

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF

Replace A,B,C in sequence by D,E,F.
Write △ABC≅△DEF; the order carries mathematical information.

Choose the statement justified by the stated conditions and explain why.

Further practice12

Fill in why the marked angle is included, then give the SAS conclusion.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SAS, with A↔D, B↔E and C↔F.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice13

Write three side equalities and one congruence conclusion for the diagram.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

△ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

AB=DE, AC=DF and BC=EF are the three given pairs of corresponding sides.
Therefore △ABC≅△DEF by SSS, with A↔D, B↔E and C↔F.

Use only given or independently proved facts.
Keep the vertex order consistent. Other complete justified proofs are accepted.

Further practice14

After proving △ABC≅△DEF, which equality follows from corresponding sides?

BC=DF
AC=DF
AC=DE
AB=EF

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

AC=DF

Use A↔D and C↔F.
The corresponding side to AC is DF.

Choose the statement justified by the stated conditions and explain why.

Extension15

The two triangles meet the marked SSA facts but have different third sides. What does this show about SSA?

SAS is therefore invalid.
The vertex order can be ignored.
SSA alone does not guarantee congruence.
All SSA triangles fail to exist.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

SSA alone does not guarantee congruence.

The two supplied coordinate constructions have AB=DE and AC=DF and equal angles at B and E.
Their BC and EF lengths differ, so one counterexample disproves the proposed general guarantee.

Choose the statement justified by the stated conditions and explain why.

Extension16

Explain why replacing ∠ABC=∠DEF by ∠BAC=∠EDF repairs this two-side proof.

It adds a third side.
It changes SAS into AAA.
It only makes the picture prettier.
It supplies the angle between the two given side pairs, completing SAS.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

It supplies the angle between the two given side pairs, completing SAS.

AB=DE and AC=DF are given. Their included angles are ∠BAC and ∠EDF.
The given ∠BAC=∠EDF supplies the included-angle condition.
Thus △ABC≅△DEF by SAS, with the stated vertex order.

Choose the statement justified by the stated conditions and explain why.