Triangle Congruence

Math-Essentials

Lesson 6-2

Naming TrianglesTriangles are named using a small triangle symbol and the three vertices of the triangles.

The order of the vertices does not matter for NAMING a triangle

Examples

A C

BΔ𝐴𝐵𝐶Δ𝐴𝐶𝐵Δ𝐵𝐴𝐶Δ𝐵𝐶𝐴Δ𝐶𝐴𝐵

Δ𝐶𝐵𝐴

X

Z

Y

Δ𝑋𝑌𝑍

Your TurnGive all six names of the triangles

P R

Q

Δ𝑃𝑄𝑅Δ𝑃𝑅𝑄Δ𝑅𝑃𝑄Δ𝑅𝑄𝑃Δ𝑄𝑃𝑅

Δ𝑄𝑅𝑃

D

F

E

1) 2)Δ𝐷𝐸𝐹Δ𝐷𝐹𝐸Δ𝐸𝐷𝐹Δ𝐸𝐹𝐷Δ𝐹𝐷𝐸

Δ𝐹𝐸𝐷

Correspondence

In triangles, there are two types of correspondence that we consider:• Corresponding angles

• Corresponding sides

Corresponding Angles of Triangles: an angle in one triangle that has the same position (relative to its sides) as an angle in another triangle (relative to its sides).

Corresponding Sides of Triangles: a side in one triangle that has the same position (relative to its angles) as a side in another triangle (relative to its angles).

∠𝐴 corresponds to ∠𝐷 since they are opposite the longest side of their triangles

𝐵𝐶 corresponds to 𝐸𝐹 since they are opposite the largest angle of their triangles

Your Turn:

1) What angle does ∠𝐴 correspond to?

2) What angle does ∠𝑋 correspond to?

3) What side does 𝑋𝑌 correspond to?

4) What side does 𝐴𝐶 correspond to?

∠𝑍

∠𝐶

𝐶𝐵

𝑍𝑋

Congruence

• Angles• two angles are congruent if they have the same measure (degrees)• ∠𝐴 ≅ ∠𝐵 if 𝑚∠𝐴 = 𝑚∠𝐵

• Segments (sides)• two line segments are congruent if they have the same length• 𝐴𝐵 ≅ 𝐶𝐷 if 𝐴𝐵 = 𝐶𝐷

• Triangles• two triangles are congruent if each angle in one triangle is congruent to its

corresponding angle in the other triangle AND if each side in one triangle is congruent to its corresponding side in the other polygon.

• In short we say “corresponding parts of congruent triangles are congruent” or “CPCTC”

Congruence Statements

• When naming a triangle, the order of the vertices is not important

• But when stating congruence, the order is important• The vertices of must be put in order so that the corresponding parts in the

names of the triangles match the corresponding parts in the triangles themselves

• For Example, Δ𝐴𝐵𝐶 ≅ Δ𝑍𝑌𝑋 because… • ∠𝐴 corresponds to ∠𝑍

∠𝐵 corresponds to ∠𝑌∠𝐶 corresponds to ∠𝑋

• 𝐴𝐵 corresponds to 𝑍𝑌𝐵𝐶 corresponds to 𝑌𝑋𝐶𝐴 corresponds to 𝑋𝑍

Your Turn• Express the congruence of the following pairs of triangles

1) 2)

Δ𝐷𝐸𝐹 ≅ Δ𝑃𝑅𝑄 Δ𝑅𝑆𝑇 ≅ Δ𝐺𝐹𝐻

Congruence Conditions

Why are Δ𝑅𝑆𝑇 and Δ𝑍𝑌𝑋 congruent? (That is, how do we prove it?)

• The corresponding parts are congruent (CPCTC)

• This is just the definition of congruence

Are there other ways we can know triangles are congruent?...

Background Vocab

• Included side: If two angles in a triangle are given, the included side is the side that is between the two angles or side that both of the angles have in common.

• 𝑅𝑆 is the included side of ∠𝑅 and ∠𝑆

• Included angle: If two sides of a triangle are given, the included angle is the angle formed by those two sides.

• ∠𝑇 is the included angle of 𝑅𝑇 and 𝑇𝑆

D

F

E

Your Turn

1) ∠𝐷 is the included angle of which two sides?

2) What is the included angle of sides 𝐷𝐹 and 𝐸𝐹?

3) 𝐷𝐹 is the included side of which two angles?

4) What is the included side of ∠𝐷 and ∠𝐸

𝐷𝐹 and 𝐷𝐸

∠𝐹

∠𝐷 and ∠𝐹

𝐷𝐸

More Congruence Conditions

Side-Angle-Side (SAS) Congruence Axiom: if an angle in one triangle is congruent to an angle in another triangle and if the sides of the angle in the first triangle are congruent to the sides of the angle in the other triangle, then the two triangles are congruent.

• 𝑋𝑍 ≅ 𝑄𝑅

• ∠𝑍𝑋𝑌 ≅ ∠𝑅𝑄𝐹

• 𝑋𝑌 ≅ 𝑄𝐹

• Therefore, Δ𝑋𝑌𝑍 ≅ Δ𝑄𝐹𝑅by SAS

More Congruence Conditions

• Angle-Side-Angle (ASA) Congruence Theorem: if two angles and their included side are congruent, then the two triangles are congruent.

• ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐺

• 𝐵𝐶 ≅ 𝐸𝐺

• ∠𝐵𝐶𝐴 ≅ ∠𝐸𝐺𝐷

• Therefore, Δ𝐴𝐵𝐶 ≅ Δ𝐷𝐸𝐺by ASA

More Congruence Conditions

Side-Side-Side (SSS) Congruence Theorem: if all corresponding sides of a triangle are congruent, then the triangles are congruent

• 𝐴𝐵 ≅ 𝐷𝐸

• 𝐵𝐶 ≅ 𝐸𝐹

• 𝐶𝐴 ≅ 𝐹𝐷

• Therefore, Δ𝐴𝐵𝐶 ≅ Δ𝐷𝐸𝐹by SSS

More Congruence Conditions

• Angle-Angle-Side (AAS) Congruency Theorem: If two corresponding angles are congruent between two triangles and a pair of corresponding sides are congruent (which are NOT the included side), then the two triangles are congruent.

• ∠𝑍𝑋𝑌 ≅ ∠𝐸𝐹𝐷

• ∠𝑋𝑌𝑍 ≅ ∠𝐹𝐷𝐸

• 𝑋𝑍 ≅ 𝐹𝐸

• Therefore, Δ𝑋𝑌𝑍 ≅ Δ𝐹𝐷𝐸by AAS

Your Turn• Determine which congruence condition proves the congruence for each

the following pairs of triangles.

• Write a congruence statement for each of the following pairs of triangles.

1) 2)

3) 4)

Your Turn

You may have notice a pattern of needing 3 parts of a triangle.

What other 3-part groups are we missing?

• AAA

• ASS (usually referred to by the more appropriate SSA)

Angle-Angle-Angle (AAA) Condition

Let’s look at an example of AAA

• ∠𝐴 ≅ ∠𝐴

• ∠𝐴𝐵𝐶 ≅ ∠𝐴𝐷𝐸 (why?)

• ∠𝐴𝐶𝐵 ≅ ∠𝐴𝐸𝐷 (why?)

• Does this mean that Δ𝐴𝐵𝐶 ≅ Δ𝐴𝐷𝐸? How do you know?

Angle-Side-Side (ASS) Condition

Let’s look at an example of ASS

• ∠𝐴 ≅ ∠𝐷

• 𝐴𝐵 ≅ 𝐷𝐸

• 𝐵𝐶 ≅ 𝐸𝐹

• Does this mean that Δ𝐴𝐵𝐶 ≅ Δ𝐷𝐸𝐹? How do you know?

Congruence Conditions

• Conditions that work (axioms)• SSS

• ASA

• SAS

• AAS

• Conditions that do not work• AAA

• ASS