Download Lesson 3.2:Proving Triangles Congruent (The SSS Postulate)

Lesson 3.2: Proving Triangles Congruent
(The SSS Postulate)
If three sides of one triangle are congruent to three sides of another triangle (SSS), must the
two triangles be congruent? Let's find out.
We say that two triangles are congruent if their six corresponding parts are congruent.
However, we can have congruent triangles without having all the corresponding parts of the two
triangles congruent. The postulate below will indicate one condition for any two triangles to be
congruent.
Postulate 3.2.1. The SSS Postulate
If three sides of one triangle are congruent to the corresponding sides of the other triangle,
then the two angles are congruent.
Challenge:
Look at the figure, tell me why ∆ABC is congruent to ∆DEF.
Let's Practice:
Which triangles are congruent based on the markings in the given figure?
Example 3.2.1.
Example 3.2.2.
Example 3.2.3.
Lesson 3.3: The SAS and ASA Congruence
Postulate
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another, must the two triangles be congruent?
Postulate 3.3.1. The SAS Postulate
If two sides and the included angle of one triangle are congruent to the corresponding two
sides and the included of another triangle, then the two triangles are congruent.
Question:
As consumers of Geometry, what do we mean by "included angle"?
Definition 3.3.1. The Included Angle
The term "included angle" is the angle formed by the two given sides.
Task:
From the two figures below, identify the included angle.
If two angles and the included side of one triangle are congruent to two angles and the
included side of another triangle, must the two angles be congruent?
Postulate 3.3.2. The ASA Postulate
If two angles and the included side of one triangle are congruent to the corresponding two
angles and the included side of another triangle, then the two triangles are congruent.
Question:
As consumers of Geometry, what do we mean by "included side"?
Definition 3.3.2. The Included Side
The term "included side" is the side whose endpoints are the vertices of the two given
angles.
Let's Practice:
Determine which postulates can be used to prove that the triangles are congruent. If it is not
possible to prove that they are not congruent, write not possible.
1.
3.
4.
2.
Example 3.3.1.
Example 3.3.2.
Theorem 3.3.1. The Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a segment, then the point equidistant form
the endpoints of the segment.
Example 3.3.3.
Example 3.3.4.
Example 3.3.5.
Assignment:
Direction: Draw and label a related diagram. List the given and what is to be proved. Then write a
complete proof on a yellow paper.
1. In an isosceles triangle, if the angle formed by the congruent sides is bisected, then two
congruent triangles are formed.
2. In an isosceles triangle, if a median is drawn from the vertex angle, then two congruent
triangles are formed.
Lesson 3.4: Other Methods of Proving Congruent
Triangles
If two angles and a non-included side of one triangle are congruent to the corresponding
parts of another triangle, must the two triangles be congruent?
Theorem 3.4.1 The SAA or AAS Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding
parts of another triangle, then the triangles are congruent.
Theorem 3.4.2. The L-L Theorem (The Leg-Leg Theorem)
If two legs of one right triangle are congruent to the corresponding legs of another right
triangle, then the two triangles are congruent.
Theorem 3.4.3. The L-AA Theorem (The Leg-Acute Angle Theorem)
If a leg and an acute angle of one right triangle are congruent to the corresponding leg and
an acute angle of another, then the two triangles are congruent.
Theorem 3.4.4. The H-AA Theorem (The Hypotenuse-Acute Angle Theorem)
If the hypotenuse and an acute angle of one right triangle are congruent to the
corresponding hypotenuse and an acute angle of another, then the two triangles are congruent.
Theorem 3.4.5. The H-L Theorem (The Hypotenuse-Leg Theorem)
If the hypotenuse and leg of one right triangle are congruent to the corresponding
hypotenuse and leg of another, then the two triangles are congruent.
Assignment: Provide a proof for Theorem 3.4.5.
Example 3.4.1.
Example 3.4.2.
Theorem 3.4.6.
If a point lies on the bisector of an angle, then the point is equidistant from the sides of
angle.
Theorem 3.4.7.
If a point is equidistant from the sides of an angle, then the point lies on the bisector of the
angle.