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Congruence: ASA, AAS and right-triangle HL

Select a criterion from the actual givens and complete a short proof without circular assumptions.

40 minutes · Year 8, upper volume · text pp. 19–24 · foundation lessons 10–12

Textbook reference: Year 8, upper · Chapter 12: congruent triangles · lessons 10–12

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.

Teacher referenceArrange in My Classroom

  1. 4′Check prerequisites
  2. 5′Observe and predict
  3. 8′Worked example and guided attempt
  4. 15′Independent practice
  5. 5′Discuss and correct
  6. 3′Exit questions

Observe · explain

A question before practising

Cover one marked fact at a time. Which criterion remains available, and when is the information insufficient?

Organise the key idea

ASA uses an included side; AAS uses a non-included corresponding side. HL additionally requires both triangles to be right triangles.

Open the original chapter

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

Worked example1

Use the two marked angle pairs and AB=DE to prove congruence by AAS.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Guided attempt2

Both triangles have the marked right angles. Complete an HL proof using the marked side pairs.

  1. The hypotenuses are ___ and ___.
  2. Equal hypotenuses ___=___ and legs ___=___ give ___.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Prerequisite3

Which side lies between ∠BAC and ∠ABC?

  1. BC
  2. There is no included side.
  3. AB
  4. AC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Prerequisite4

∠BAC=90°. Which side is the hypotenuse of △ABC?

  1. AB
  2. AC
  3. Any side may be the hypotenuse.
  4. BC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Independent practice5

The equal angles are at A,D and B,E, with AB=DE. Which criterion uses exactly these facts directly?

  1. ASA
  2. SSS
  3. HL
  4. AAA
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Independent practice6

The equal angles are at A,D and C,F, with AB=DE. Which direct criterion applies?

  1. SSA
  2. AAS
  3. ASA using AB as the included side
  4. SSS
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Independent practice7

In the marked right triangles, BC=EF and AB=DE. Which conclusion follows?

  1. All triangle angles are 90°.
  2. The information is AAA.
  3. AC=DF by HL congruence.
  4. AC=DE because the drawing looks equal.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Independent practice8

Use the given two angle pairs and their included side to prove congruence. Name your criterion and reasons.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Exit question9

Why does an HL proof need both right-angle conditions before identifying the hypotenuses?

  1. HL is a criterion for two right triangles; a longest-looking side is insufficient.
  2. Any two triangles have hypotenuses.
  3. The right angles may be assumed from the picture.
  4. HL only needs one arbitrary equal side.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Exit question10

Can three equal angle pairs replace the equal-side requirement in ASA or AAS?

  1. Yes if the angles are acute.
  2. No: scale is not fixed without a corresponding side.
  3. Yes: AAA is stronger than ASA.
  4. Yes if the triangles are labelled.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Further practice11

Write the two equal-angle pairs, their included side and the ASA conclusion.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice12

Write an AAS proof and explain why AB is a non-included side here.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice13

List the right angles, hypotenuses and equal legs before the HL conclusion.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice14

After the AAS proof, which additional angle equality follows?

  1. ∠ABC=90°
  2. ∠ABC=∠DEF
  3. ∠ABC=∠DFE
  4. ∠ABC=∠EDF
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Extension15

The marked two side pairs and their included angle already give SAS. Which extra side equality would ALSO allow a direct SSS proof?

  1. The triangles have equal areas only.
  2. The shapes look the same.
  3. BC=EF
  4. ∠ABC=∠DEF only
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Choose the statement justified by the stated conditions and explain why.
Extension16

Give an alternative proof of the marked AAS result: first obtain the third angle pair, then use ASA.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Preparation and assessment notes

Prerequisites

SSS/SAS, included sides, corresponding vertices and hypotenuses.

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.

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.

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. 19–24 · foundation lessons 10–12

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

Congruence: ASA, AAS and right-triangle HL

Year 8, upper volume · text pp. 19–24 · foundation lessons 10–12 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Use the two marked angle pairs and AB=DE to prove congruence by AAS.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2

Both triangles have the marked right angles. Complete an HL proof using the marked side pairs.

  1. The hypotenuses are ___ and ___.
  2. Equal hypotenuses ___=___ and legs ___=___ give ___.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3

Which side lies between ∠BAC and ∠ABC?

  1. BC
  2. There is no included side.
  3. AB
  4. AC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite4

∠BAC=90°. Which side is the hypotenuse of △ABC?

  1. AB
  2. AC
  3. Any side may be the hypotenuse.
  4. BC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
g8-congruence-angle-hl · 2026-10-14-upper-core-v1 · Diagrams are not to scale. Conditions and reasons matter; measurement is not proof.

Congruence: ASA, AAS and right-triangle HL

Year 8, upper volume · text pp. 19–24 · foundation lessons 10–12 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

The equal angles are at A,D and B,E, with AB=DE. Which criterion uses exactly these facts directly?

  1. ASA
  2. SSS
  3. HL
  4. AAA
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6

The equal angles are at A,D and C,F, with AB=DE. Which direct criterion applies?

  1. SSA
  2. AAS
  3. ASA using AB as the included side
  4. SSS
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7

In the marked right triangles, BC=EF and AB=DE. Which conclusion follows?

  1. All triangle angles are 90°.
  2. The information is AAA.
  3. AC=DF by HL congruence.
  4. AC=DE because the drawing looks equal.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8

Use the given two angle pairs and their included side to prove congruence. Name your criterion and reasons.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9

Why does an HL proof need both right-angle conditions before identifying the hypotenuses?

  1. HL is a criterion for two right triangles; a longest-looking side is insufficient.
  2. Any two triangles have hypotenuses.
  3. The right angles may be assumed from the picture.
  4. HL only needs one arbitrary equal side.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10

Can three equal angle pairs replace the equal-side requirement in ASA or AAS?

  1. Yes if the angles are acute.
  2. No: scale is not fixed without a corresponding side.
  3. Yes: AAA is stronger than ASA.
  4. Yes if the triangles are labelled.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
g8-congruence-angle-hl · 2026-10-14-upper-core-v1 · Diagrams are not to scale. Conditions and reasons matter; measurement is not proof.

Congruence: ASA, AAS and right-triangle HL — Further practice

Year 8, upper volume · text pp. 19–24 · foundation lessons 10–12

Further practice11

Write the two equal-angle pairs, their included side and the ASA conclusion.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12

Write an AAS proof and explain why AB is a non-included side here.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13

List the right angles, hypotenuses and equal legs before the HL conclusion.

Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14

After the AAS proof, which additional angle equality follows?

  1. ∠ABC=90°
  2. ∠ABC=∠DEF
  3. ∠ABC=∠DFE
  4. ∠ABC=∠EDF
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Extension15

The marked two side pairs and their included angle already give SAS. Which extra side equality would ALSO allow a direct SSS proof?

  1. The triangles have equal areas only.
  2. The shapes look the same.
  3. BC=EF
  4. ∠ABC=∠DEF only
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Extension16

Give an alternative proof of the marked AAS result: first obtain the third angle pair, then use ASA.

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

Congruence: ASA, AAS and right-triangle HL — Teacher answers

Year 8, upper volume · text pp. 19–24 · foundation lessons 10–12

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

Use the two marked angle pairs and AB=DE to prove congruence by AAS.

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

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

  1. ∠BAC=∠EDF and ∠ACB=∠DFE are given.
  2. AB=DE is not the side between those angles.
  3. The two angle pairs and this corresponding side give △ABC≅△DEF by AAS.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Guided attempt2

Both triangles have the marked right angles. Complete an HL proof using the marked side pairs.

  1. The hypotenuses are ___ and ___.
  2. Equal hypotenuses ___=___ and legs ___=___ give ___.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

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

  1. ∠BAC=∠EDF=90°, so both triangles are right triangles.
  2. BC=EF are the hypotenuses, and AB=DE are a pair of corresponding legs.
  3. Thus △ABC≅△DEF by HL.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Prerequisite3

Which side lies between ∠BAC and ∠ABC?

  1. BC
  2. There is no included side.
  3. AB
  4. AC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

AB

  1. The angle vertices are A and B.
  2. The side joining them is AB.
  • Choose the statement justified by the stated conditions and explain why.
Prerequisite4

∠BAC=90°. Which side is the hypotenuse of △ABC?

  1. AB
  2. AC
  3. Any side may be the hypotenuse.
  4. BC
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

BC

  1. The hypotenuse is opposite the right angle.
  2. The side opposite A is BC.
  • Choose the statement justified by the stated conditions and explain why.
Independent practice5

The equal angles are at A,D and B,E, with AB=DE. Which criterion uses exactly these facts directly?

  1. ASA
  2. SSS
  3. HL
  4. AAA
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

ASA

  1. ∠BAC=∠EDF and ∠ABC=∠DEF are given.
  2. AB=DE is the side between those two angles.
  3. Thus △ABC≅△DEF by ASA.
  • Choose the statement justified by the stated conditions and explain why.
Independent practice6

The equal angles are at A,D and C,F, with AB=DE. Which direct criterion applies?

  1. SSA
  2. AAS
  3. ASA using AB as the included side
  4. SSS
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

AAS

  1. ∠BAC=∠EDF and ∠ACB=∠DFE are given.
  2. AB=DE is not the side between those angles.
  3. The two angle pairs and this corresponding side give △ABC≅△DEF by AAS.
  • Choose the statement justified by the stated conditions and explain why.
Independent practice7

In the marked right triangles, BC=EF and AB=DE. Which conclusion follows?

  1. All triangle angles are 90°.
  2. The information is AAA.
  3. AC=DF by HL congruence.
  4. AC=DE because the drawing looks equal.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

AC=DF by HL congruence.

  1. ∠BAC=∠EDF=90°, so both triangles are right triangles.
  2. BC=EF are the hypotenuses, and AB=DE are a pair of corresponding legs.
  3. Thus △ABC≅△DEF by HL.
  4. Corresponding remaining legs give AC=DF.
  • Choose the statement justified by the stated conditions and explain why.
Independent practice8

Use the given two angle pairs and their included side to prove congruence. Name your criterion and reasons.

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

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

  1. ∠BAC=∠EDF and ∠ABC=∠DEF are given.
  2. AB=DE is the side between those two angles.
  3. Thus △ABC≅△DEF by ASA.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Exit question9

Why does an HL proof need both right-angle conditions before identifying the hypotenuses?

  1. HL is a criterion for two right triangles; a longest-looking side is insufficient.
  2. Any two triangles have hypotenuses.
  3. The right angles may be assumed from the picture.
  4. HL only needs one arbitrary equal side.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

HL is a criterion for two right triangles; a longest-looking side is insufficient.

  1. ∠BAC=∠EDF=90°, so both triangles are right triangles.
  2. BC=EF are the hypotenuses, and AB=DE are a pair of corresponding legs.
  3. Thus △ABC≅△DEF by HL.
  • Choose the statement justified by the stated conditions and explain why.
Exit question10

Can three equal angle pairs replace the equal-side requirement in ASA or AAS?

  1. Yes if the angles are acute.
  2. No: scale is not fixed without a corresponding side.
  3. Yes: AAA is stronger than ASA.
  4. Yes if the triangles are labelled.
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

No: scale is not fixed without a corresponding side.

  1. All corresponding angles can stay equal under enlargement.
  2. The displayed counterexample has doubled sides and is not congruent.
  • Choose the statement justified by the stated conditions and explain why.
Further practice11

Write the two equal-angle pairs, their included side and the ASA conclusion.

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

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

  1. ∠BAC=∠EDF and ∠ABC=∠DEF are given.
  2. AB=DE is the side between those two angles.
  3. Thus △ABC≅△DEF by ASA.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice12

Write an AAS proof and explain why AB is a non-included side here.

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

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

  1. ∠BAC=∠EDF and ∠ACB=∠DFE are given.
  2. AB=DE is not the side between those angles.
  3. The two angle pairs and this corresponding side give △ABC≅△DEF by AAS.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice13

List the right angles, hypotenuses and equal legs before the HL conclusion.

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

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

  1. ∠BAC=∠EDF=90°, so both triangles are right triangles.
  2. BC=EF are the hypotenuses, and AB=DE are a pair of corresponding legs.
  3. Thus △ABC≅△DEF by HL.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.
Further practice14

After the AAS proof, which additional angle equality follows?

  1. ∠ABC=90°
  2. ∠ABC=∠DEF
  3. ∠ABC=∠DFE
  4. ∠ABC=∠EDF
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

∠ABC=∠DEF

  1. ∠BAC=∠EDF and ∠ACB=∠DFE are given.
  2. AB=DE is not the side between those angles.
  3. The two angle pairs and this corresponding side give △ABC≅△DEF by AAS.
  4. B corresponds to E, so the remaining angle equality is ∠ABC=∠DEF.
  • Choose the statement justified by the stated conditions and explain why.
Extension15

The marked two side pairs and their included angle already give SAS. Which extra side equality would ALSO allow a direct SSS proof?

  1. The triangles have equal areas only.
  2. The shapes look the same.
  3. BC=EF
  4. ∠ABC=∠DEF only
Geometry diagram; not drawn to scale except for construction tasks.ABCDEF
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer

BC=EF

  1. The existing two side pairs and included angle already justify SAS.
  2. A direct SSS proof additionally needs the third corresponding side pair BC=EF.
  3. With that extra equality, AB=DE, AC=DF and BC=EF also give △ABC≅△DEF by SSS.
  • Choose the statement justified by the stated conditions and explain why.
Extension16

Give an alternative proof of the marked AAS result: first obtain the third angle pair, then use ASA.

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

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

  1. The two given angle pairs are equal. Subtract each pair from 180° to obtain ∠ABC=∠DEF.
  2. Now ∠BAC=∠EDF, ∠ABC=∠DEF and the included side AB=DE give ASA congruence.
  • Use only given or independently proved facts.
  • Keep the vertex order consistent. Other complete justified proofs are accepted.