Download File - Ms. K. Owens Mathematics

Lesson 5.1 and 5.2 Triangle
Congruence: SSS, SAS, ASA, AAS,
HL
EQ: What information about two triangles allows you to conclude the triangles are
congruent?
Did you know…
Triangle congruency is used in blueprints by construction engineers and managers?
Two triangles are congruent if they have…
 exactly the same three sides and
 exactly the same three angles
 all corresponding sides are congruent
 all corresponding angles ae congruent
We do not have to know that all three sides and all three angles are exactly the
same to conclude that the triangles are congruent.
Sometimes knowing that THREE out of six corresponding parts are congruent is
enough to conclude that two triangles are congruent.
There are five ways to prove or show that two triangles are congruent.
1) Side-Side-Side Congruence (SSS) – Two triangles with ALL THREE
CORRESPONDING SIDES congruent.
2) Side-Angle-Side (SAS) – Two triangles with TWO CORRESPONDING SIDES and the
INCLUDED ANGLE congruent.
3) Angle-Side-Angle (ASA) – Two triangles with TWO CORRESPONDING ANGLES and
the INCLUDED SIDE congruent.
4) Angle-Angle-Side (AAS) – Two triangles with TWO CORRESPONDING ANGLES and
the NONINCLUDED SIDE congruent.
5) Hypotenuse-Leg (HL)* – Two RIGHT triangles whose HYPOTENUSE and ONE LEG
congruent.
*Applies only to right triangles
Understanding the difference between an included angle, an included side, and a
non-included side.
An included angle is an angle that is made up of the intersection of two sides. It is
between the two sides.
Summary:
An included side is a side that is between two angles.
A non-included side is a side that is not between two angles.
Practice: Use the diagram to answer the questions that follow.
1. Given: ̅̅̅̅
𝐶𝐴 and ̅̅̅̅
𝐴𝑇, what angle is an included angle? ________
2. Given: ∠𝐶 and ∠𝑇, what side is an included side? ________
3. Given: ∠𝐴 and ∠𝑇, what side is a non-included side? _______
4. Given: ∠𝐶 and ∠𝑇, what side is the included side? _______
5. Given: ̅̅̅̅
𝑇𝐶 and ̅̅̅̅
𝐴𝑇, what angle is the included angle? _______
6. Given: ∠𝑇 and ∠𝐴, what side is the non-included side? _______
Side-Side-Side Congruence Postulate
Side-Angle-Side Congruence Postulate
Angle-Side-Angle Congruence Postulate
If two angles and the included side of
one triangle is congruent to two corresponding angles and the included
side of another triangle, then the
triangles are congruent.
Abbreviation: ASA
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
Angle-Angle-Side Congruence Postulate
Hypotenuse-Leg Congruence Postulate
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse of another right triangle, then
the triangles are congruent.
Abbreviation: HL
∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍
Guided Practice
Use SSS to explain why the triangles in each pair
are congruent.
1. JKM > LKM
2.ABC > CDA
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3.Use SAS to explain why WXY > WZY.
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Show that the triangles are congruent for the given value of the
variable.
4. BCD ≅ FGH, x  6
5.PQR ≅ VWX, n  3
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Determine whether you can use ASA to prove the
triangles congruent. Explain.
1. KLM and NPQ
2.EFG and XYZ
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3. KLM and PNM, given that M is the
midpoint of NL
4.STW and UTV
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