Triangle Congruence Theorems

​​Triangle Similarity and Congruence Review

Things to Remember!!

The sum of all the interior angles of a triangle will always be 180.​

Triangle Angle Sum Theorem

Triangle Congruence Theorems

Two triangles are congruent if

• all of their corresponding sides and angles are congruent;

• there exists a finite sequence of rigid motions that maps one triangle to the other; or

• one of the triangle congruence criteria described in the coming slides

SAS Triangle Congruence Theorem: If two triangles have two pairs of congruent sides, and the corresponding included angles are congruent, then the two triangles are congruent.

SAS Congruence

​AAS Triangle Congruence Theorem: If two triangles have two pairs of congruent angles and a pair of nonincluded corresponding congruent sides, then the two triangles are congruent.

AAS Congruence

ASA Congruence

ASA Triangle Congruence Theorem: If two triangles have two pairs of congruent angles, and the corresponding included sides of the angles are congruent, then the triangles are congruent

SSS Triangle Congruence Theorem: If two triangles have three pairs of congruent sides, then the two triangles are congruent.

SSS Congruence

HL Triangle Congruence Theorem: Given two right triangles, if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of another triangle, then the triangles are congruent.

HL Congruence

Things to Remember!!

Vertical Angles

1) ​Congruent

2) Formed by any pair of intersecting lines​

3) Share only one point, the vertex​

Multiple Choice

How are the triangles congruent?

• HL
• SSS
• SAS
• AAS

Multiple Choice

How are the triangles congruent?

• SAS
• SSS
• HL
• ASA

Multiple Choice

How are the triangles congruent?

• ASA
• SAS
• AAS
• SSS

Multiple Choice

How are the triangles congruent?

• SAS
• ASA
• HL
• SSS

Multiple Choice

Are these triangles congruent?

• Yes, by ASA
• Yes, by SAS
• No they are not congruent
• Not enough information to know for sure

Multiple Choice

• A
• B
• C
• D

Multiple Choice

For which situation could you prove

∆1≅ ∆2 using the H-L Theorem?

• I only

• II only

• I & II

• I & III

• III only

Multiple Choice

Are there triangle congruent? If so, by which Triangle Congruence Theorem?

• ASA

• AAS

• HL

• SAS

Multiple Choice

Are there triangle congruent? If so, by which Triangle Congruence Theorem?

• ASA

• AAA

• AAS

• SAS

Multiple Choice

Are there triangle congruent? If so, by which Triangle Congruence Theorem?

• SAS

• SSS

• ASA

• AAS

​Two shapes are similar if their corresponding angles are equal and their corresponding line segments are proportional.

​​Triangle Similarity

We know two fractions are proportional when they reduce to be the same.

Proportional

For Congruence, we had 5 shortcuts:

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

AAA (or AA) Similarity

If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

SAS Similarity

If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

SSS Similarity

​For similarity we have 3 Shortcuts

Things to Remember!!

Vertical Angles

1) ​Congruent

2) Formed by any pair of intersecting lines​

3) Share only one point, the vertex​

Things to Remember!!

The sum of all the interior angles of a triangle will always be 180.​

Triangle Angle Sum Theorem

Multiple Choice

Are these triangles similar? If so, by which similarity theorem.

• Yes by AA Similarity

• Yes, by SSS Similarity

• Yes, by SAS Similarity

• Not Similar

Multiple Choice

Are these triangles similar? If so, by which similarity theorem.

• Yes, by AA

• Yes, by SAS

• Yes, by SSS

• Not Similar

Multiple Choice

Are these triangles similar? If so, by which similarity theorem.

• Yes, by AA

• Yes, by SSS

• Yes, by SAS

• Not Similar

Multiple Choice

Are these triangles similar? If so, by which similarity theorem.

• Yes, by AA Similarity

• Yes, by SAS Similarity

• Yes, by SSS Similarity

• Not Similar

Multiple Choice

Are the triangles similar by the AA shortcut? If yes, write the similarity statement.

• Similar; $$\triangle LMN\ \sim\triangle RPQ$$  

• Similar;   $$\triangle LMN\sim\triangle PQR$$  

• Similar; $$\triangle LMN\sim\triangle QRP$$  

• Not Similar

Multiple Choice

Are the triangles similar by the AA shortcut? If yes, write the similarity statement.

• Not Similar

• Similar; $$\triangle DEF\sim\triangle CBA$$

• Similar; $$\triangle DEF\sim\triangle ACB$$

• Similar; $$\triangle DEF\sim\triangle CAB$$

Multiple Choice

Are the triangles similar by the AA shortcut? If yes, write the similarity statement.

• Similar; $$\triangle BCD\sim\triangle BUV$$  

• Not Similar

• Similar; $$\triangle BCD\sim\triangle BVU$$  

• Similar; $$\triangle BCD\sim\triangle VUB$$  

Multiple Choice

If two figures are similar,

then the angles are _________________.

• proportional

• congruent

• complementary

• supplementary

Multiple Choice

Sides in similar figures must be __________.  

• Proportional

• Congruent

• Opposite Reciprocals

• Parallel

Multiple Choice

State if the triangles are similar by SSS.

• Yes, SSS Similarity

• Similar but not SSS

• Not Similar

Multiple Choice

State if the triangles are similar by SSS.

• Yes, SSS Similarity

• Similar but not SSS

• Not Similar

Multiple Choice

State if the triangles are similar by SAS.

• Yes, SAS Similarity

• Similar but not SAS

• Not Similar

Multiple Choice

State if the triangles are similar by SAS.

• Yes, SAS Similarity

• Similar but not SAS

• Not Similar

Multiple Choice

State if the two triangles are similar and the rule that applies.

• Not Similar

• Similar by SAS

• Similar by AA

• Similar by SSS

Multiple Choice

State if the two triangles are similar and the rule that applies.

• Not Similar

• Similar by SAS

• Similar by AA

• Similar by SSS