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Title: Congruent Triangles

Objective: Students will be able to identify congruent triangles when given few measurements.

Language Objective: Students will be able to describe the different types of triangle congruency.

Essential Question: “Why is knowing about triangle congruency important?”

Congruent Triangle Overview:CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Congruency Statement: ΔABC ≅ ΔXYZ (Note: Order of symbols and letters matters)

Congruent Corresponding Parts:Angles: ∠A ≅ ∠X ∠B ≅ ∠Y ∠C ≅ ∠ZSides: AB ≅ XY BC ≅ YZ AC ≅ XZ

(Note: Use the Congruency Statement)

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Congruent Triangle Overview:Problem solving:

ΔDEF ≅ ΔMNP. Calculate the values of x, y, and m∠D

For x:

5x + 10 = 60

-10 -10

5x = 50

5 5

x = 10

Congruent Triangle Overview:Problem solving:

ΔDEF ≅ ΔMNP. Calculate the values of x, y, and m∠D

Length of side NP:

NP = 4(10) – 12 = 28

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Congruent Triangle Overview:Problem solving:

ΔDEF ≅ ΔMNP. Calculate the values of x, y, and m∠D

For y:

4y + 4 = 28

-4 -4

4y = 24

4 4

y = 6

Congruent Triangle Overview:Problem solving:

ΔDEF ≅ ΔMNP. Calculate the values of x, y, and m∠D

For m∠D:

m∠D = 5(6) = 30°

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Triangle Congruency Shortcuts:SSS

Description:

If the sides of one triangle are congruent to the sides of another triangle, the triangles are congruent.

Diagram:

SASDescription:

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

Diagram:

ΔABC ≅ ΔEFDΔABC ≅ ΔEDF

Triangle Congruency Shortcuts:ASA

Description:

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

Diagram:

AAS

Description:

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

Diagram:

ΔABC ≅ ΔFEDΔABC ≅ ΔDFE

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Triangle Congruency Shortcuts:LL

Description:

In a right triangle, if the legs of one triangle are congruent to the corresponding legs of another triangle, the triangles are congruent.

Diagram:

HL

Description:

In a right triangle, if the hypotenuse and the leg of one triangle are congruent to the hypotenuse and corresponding leg of another triangle, the triangles are congruent.

Diagram:

ΔABC ≅ ΔDEFΔABC ≅ ΔDEF

Triangle Congruency Shortcuts:LA

Description:

In a right triangle, if one acute angle and leg of one triangle are congruent to the corresponding acute angle and corresponding leg of another triangle, the triangles are congruent.

Diagram:

HA

Description:

In a right triangle, if the hypotenuse and one acute angle of the triangle are congruent to the corresponding hypotenuse and acute angle of another triangle, the triangles are congruent.

Diagram:

ΔABC ≅ ΔDEFΔABC ≅ ΔFDE

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SAS AASVertical Angles … congruent

Shared side…congruent

HL Not ≅

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AASSSSShared side…congruent

Vertical Angles … congruent

Not ≅ ASA Shared side…congruent

Alternate Interior Angles…congruent

Alternate Interior Angles…congruent

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Not ≅ HL Shared side…or in this case it’s a hypotenuse…congruent

SAS AASShared side…congruent

Alternate Interior Angles…congruent

Shared side…congruentAlternate Interior

Angles…congruent

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SASSSSMidpoints cut things in half…congruentShared side…

congruent

Shared side…congruent

Not ≅ASA or AASAlternate Interior Angles…congruent

Alternate Interior Angles…congruent

Vertical Angles … congruent

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HLASA Bisecting…cutting angles in half…both sets…congruent

Shared side…congruent

Shared side…Leg in this case…congruent

Not ≅SASCorresponding angles…congruent

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Not ≅ AASShared side…congruent

AASSSSVertical Angles … congruent

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