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3.2 Three Ways to Prove
Triangles Congruent
Objectives:
1. identify included angles and included sides
2. apply the SSS postulate
3. apply the SAS postulate
4. apply the ASA postulate
Triangles
Use a ruler and protractor to construct a triangle with
the following specifications:
a. One side is 2cm
b. One side is 3cm
c. One side is 4cm
Side-Side-Side (SSS) Postulate
If three sides of one triangle are congruent to three
sides of another triangle, then the two triangles are
congruent.
Triangles
Use a ruler and protractor to construct a triangle with
the following specifications:
a. One side is 5cm
b. One side is 8cm
c. One angle is 40˚
Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one triangle
are congruent to two sides and the included angle of
another triangle, then the two triangles are
congruent.
Triangles
Use a ruler and protractor to construct a triangle with
the following specifications:
a. One side is 9cm
b. One angle is 50˚
c. One angle is 30˚
Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle
are congruent to two angles and the included side of
another triangle, then the two triangles are
congruent.
Three Ways to Prove Triangles
Congruent
1. SSS Postulate
2. SAS Postulate
3. ASA Postulate
Cannot Use
SSA
or
AAA
Example
Name the congruent angles or sides necessary to
prove the triangles congruent by the specified
method.
a. SSS
b. SAS
Example
Name the congruent angles or sides necessary to
prove the triangles congruent by the specified
method.
a. SAS
b. ASA
Example
Name the congruent angles or sides necessary to
prove the triangles congruent by the specified
method.
a. SAS
b. ASA
Example
Name the postulate (if any) that can be used to prove
the triangles congruent.
Proof
Given: Diagram
Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐾𝐽
SSS Postulate or SAS Postulate or ASA Postulate
Proof
Given: Diagram
Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐾𝐽
Statements
1.
2.
3.
4.
𝐺𝐻 ≅ 𝐺𝐾
𝐻𝐽 ≅ 𝐾𝐽
𝐺𝐽 ≅ 𝐺𝐽
∆𝐺𝐻𝐽 ≅ ∆𝐺𝐾𝐽
Reasons
1.
2.
3.
4.
Given (indicated in diagram)
Given (indicated in diagram)
Reflexive Property
SSS Postulate
Flow Proof
Given: Diagram
Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐾𝐽
Proof
Given: 𝐴𝐷 ≅ 𝐴𝐸
𝐴𝐵 ≅ 𝐴𝐶
Prove: ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐸𝐶
SSS Postulate or SAS Postulate or ASA Postulate
Proof
Given: 𝐴𝐷 ≅ 𝐴𝐸
𝐴𝐵 ≅ 𝐴𝐶
Prove: ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐸𝐶
Statements
1. 𝐴𝐷 ≅ 𝐴𝐸
2. 𝐴𝐵 ≅ 𝐴𝐶
3. ∠ 𝐴 ≅ ∠ 𝐴
4. ∆𝐴𝐷𝐵 ≅ ∆𝐴𝐸𝐶
Reasons
1. Given
2. Given
3. Reflexive Property
4. SAS Postulate
Assignment
Section 3.2 Problem Set A #1-10
Proof
Given: 𝐴𝐷 ≅ 𝐶𝐷
B is the midpoint of 𝐴𝐶
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
SSS Postulate or SAS Postulate or ASA Postulate
Proof
Given: 𝐴𝐷 ≅ 𝐶𝐷
B is the midpoint of 𝐴𝐶
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
Statements
Reasons
4. 𝐷𝐵 ≅ 𝐷𝐵
5. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
1. Given
2. Given
3. If a point is a midpoint, then it
divides the segment into two
congruent segments.
4. Reflexive Property
5. SSS Postulate
1. 𝐴𝐷 ≅ 𝐶𝐷
2. B is the midpoint of 𝐴𝐶
3. 𝐴𝐵 ≅ 𝐶𝐵
Proof
Given: ∠3 ≅ ∠6
𝐾𝑅 ≅ 𝑃𝑅
∠𝐾𝑅𝑂 ≅ ∠𝑃𝑅𝑀
Prove: ∆𝐾𝑅𝑀 ≅ ∆𝑃𝑅𝑂
SSS Postulate or SAS Postulate or ASA Postulate
Proof
Given: ∠3 ≅ ∠6
𝐾𝑅 ≅ 𝑃𝑅
∠𝐾𝑅𝑂 ≅ ∠𝑃𝑅𝑀
Prove: ∆𝐾𝑅𝑀 ≅ ∆𝑃𝑅𝑂
Statements
Reasons
1. ∠3 ≅ ∠6
2. ∠𝐾𝑀𝑂 is straight
3. ∠4 is supp. to ∠3
4. ∠5 is supp. to ∠6
5. ∠4 ≅ ∠5
6. 𝐾𝑅 ≅ 𝑃𝑅
7. ∠𝐾𝑅𝑂 ≅ ∠𝑃𝑅𝑀
8. ∠𝐾𝑅𝑀 ≅ ∠𝑃𝑅𝑂
9. ∆𝐾𝑅𝑀 ≅ ∆𝑃𝑅𝑂
1.
2.
3.
4.
5.
6.
7.
8.
9.
Given
Given
Given
ASA Postulate
Proof
Given: ∠3 ≅ ∠6
𝐾𝑅 ≅ 𝑃𝑅
∠𝐾𝑅𝑂 ≅ ∠𝑃𝑅𝑀
Prove: ∆𝐾𝑅𝑀 ≅ ∆𝑃𝑅𝑂
Statements
Reasons
1. ∠3 ≅ ∠6
2. ∠𝐾𝑀𝑂 is straight
3. ∠4 is supp. to ∠3
4. ∠5 is supp. to ∠6
5. ∠4 ≅ ∠5
6. 𝐾𝑅 ≅ 𝑃𝑅
7. ∠𝐾𝑅𝑂 ≅ ∠𝑃𝑅𝑀
8. ∠𝐾𝑅𝑀 ≅ ∠𝑃𝑅𝑂
9. ∆𝐾𝑅𝑀 ≅ ∆𝑃𝑅𝑂
1. Given
2. Assumed from diagram
3. If the sum of two angles is a straight angle, then the
angles are supplementary.
4. If the sum of two angles is a straight angle, then the
angles are supplementary.
5. If two angles are supplementary to congruent angles,
then they are congruent.
6. Given
7. Given
8. If an angle is subtracted from two congruent angles,
then the differences are congruent.
9. ASA Postulate
Assignment
Section 3.2 Problem Set A #12-16
Problem Set B #17