Triangle Congruence (SSS, SAS, ASA, AAS, HL)

Triangle Congruence (SSS, SAS, ASA, AAS, HL)

by Angela Preston

​The Idea of Congruence

​Two geometric figures with exactly the same size and shape.​

​Corresponding Parts

​In a previous lesson, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent

Match

Match the congruent parts

• Angle A

• $$\overline{EF}$$

• $$\overline{FE}$$

• $$\angle B$$

• $$\angle F$$

Angle D

• Angle D

• $$\overline{BC}$$

• $$\overline{CB}$$

• $$\angle E$$

• $$\angle C$$

​Corresponding Parts

Congruence statements are written such that the corresponding, congruent parts are listed in the same order.

​In this figure, the following congruence statements are also true:
Δ BCA ≅ Δ EFD

Δ ACB ≅ Δ DFE

Multiple Choice

If ∆LMK ≅ ∆LMT, then ∠K ≅ ____

• MT

• ∠T

• ∠M

• ∠L

Multiple Choice

Choose the correct congruence statement for the figures shown.

• ∆TUV ≅ ∆EUV

• ∆VUT ≅ ∆UVE

• ∆TUV ≅ ∆UEV

• ∆UTV ≅ ∆EUV

​Do you need to show that all six parts are congruent???​

​Triangle Congruence Shortcuts

​Side-Side-Side (SSS)​

​If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.​

​Side-Side-Side (SSS)​

​If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.​

​Side-Side-Side (SSS)​

​If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.​

​Label the sides of the first triangle on your paper to illustrate SSS!

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• SAS
• SSS
• ASA
• HL

What is an ​Included Angle?​

Name the Included Angle...

​***For this activity, complete problem #2 on your paper!

​Name the Included Angle...

​Name the Included Angle...

​Name the Included Angle...

​Side-Angle-Side (SAS)​

​If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.​

​​Side-Angle-Side (SAS)

​​Side-Angle-Side (SAS)

​Label the sides of the second triangle on your paper to illustrate SAS!

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• SSS
• ASA
• AAS
• SAS

​What is an Included Side?

​The side between two angles

​

​

​

​

​

​

​​This combo is called angle-side-angle, or just ASA.​

​Name the Included Side...

​***For this activity, complete problem #3 on your paper!

​Name the Included Side...

​Name the Included Side...

​Name the Included Side...

​Angle-Side-Angle (ASA)​

​If two angles and the included side of one triangle are congruent to the two angles and the Included side of another triangle, then the triangles are congruent.​

​Angle-Side-Angle (ASA)​

​Angle-Side-Angle (ASA)​

​Label the sides of the third triangle on your paper to illustrate ASA!

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• AAA

• SSS

• ASA

• N/A

​Angle-Angle-Side (AAS)​

​If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.

​​Angle-Angle-Side (AAS)

​​Angle-Angle-Side (AAS)

​Label the sides of the fourth triangle on your paper to illustrate AAS!

What other congruence statements can I write?

What other congruence statements can I write?

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• AAS

• AAA

• SAS

• SSS

​Warning: No SSA Postulate​

​Warning: No SSA Postulate​

​BUT: SSA DOES work in one​ situation!​

​WE CALL THIS...

​Hypotenuse-Leg (HL)​

​If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

​​Hypotenuse-Leg (HL)

​​Hypotenuse-Leg (HL)

​Label the sides of the fifth triangle on your paper to illustrate HL!

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• SSS

• ASA

• SSA

• HL

Multiple Choice

Which is NOT a criteria for triangles to be congruent by the HL Theorem?

• Right triangles
• congruent hypotenuses
• Two pairs of congruent legs
• One pair of congruent legs

​Warning: No AAA Postulate​

​Congruence Postulates ​and Theorems​

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS
• SAS
• ASA
• Not Possible

Multiple Choice

Which triangle congruence theorem can be used to prove the triangles are congruent?

• SSS

• ASA

• AAS

• NOT ENOUGH INFORMATION!

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SAS

• AAS

• ASA

• NOT ENOUGH INFORMATION!

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SAS
• ASA
• AAS
• Not Possible

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS

• ASA

• AAS

• NOT ENOUGH INFORMATION!

​Vertical Angles, ​Reflexive Sides and Angles​

​Let’s Practice​...

​​Indicate the additional information needed to enable us to apply the specified congruence postulate.​ This activity coincides with problem #5 on your paper!

​

​For ASA:

​

​For SAS:

​​

For AAS:​

​Let’s Practice​...

​Let’s Practice​...

​Let’s Practice​...

Multiple Choice

How are the triangles congruent?

• HL
• SSS
• SAS
• AAS

Multiple Choice

Which postulate can we use to prove the two triangles are congruent?

• SSS

• SAS

• ASA

• AAS

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS
• SAS
• ASA
• Not Possible

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS
• ASA
• AAS
• Not Possible

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS
• SAS
• AAS
• Not Possible

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SAS
• ASA
• AAS
• Not Possible

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS
• SAS
• ASA
• AAS

Multiple Choice

Name the postulate, if possible, that makes the triangles congruent.

• SSS

• SAS

• AAS

• Not Possible

​Class Assignment

Go to your online textbook (clever.com).

​

Complete the following assignment by the due date:

​

"Triangle Congruence"​