Proving congruence of tw0 triangles Proving congruence of tw0 triangles Proving congruence of tw0 triangles

2. Step 1. Identify what the given are,
and what is to be proved. Mark the
given information on the diagram.
Step 2. Identify the congruence
theorem to be used and the additional
information needed and why.

3. Example 1. Prove that Δ𝑅𝐴𝑊≅Δ𝑅𝑆𝐷 in figure 6.
 Step 1. Identify what the given are and what
is to be proved.
Given: 𝐴𝑅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅≅𝑆𝑅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ , 𝑊𝑅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
≅𝐷𝑅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Prove: Δ𝑅𝐴𝑊≅Δ𝑅𝑆𝐷
Step 2. Identify the congruence theorem to be
used and the additional information needed
and why.
∠𝑊𝑅𝐴 and ∠𝐷𝑅𝑆 are vertical angles so they are
congruent by Vertical Angle Theorem, hence
SAS congruence postulate can be used to
prove Δ𝑅𝐴𝑊≅Δ𝑅𝑆𝐷.

4. Example 1. Prove that Δ𝑅𝐴𝑊≅Δ𝑅𝑆𝐷 in figure 6.
 Step 3. Write down the statements and the
reasons in a two-column proof. Make sure the
last statement contains what should be
proved.

5. Example 2. Based on Figure 7 at the right, prove that Δ𝑅𝐸𝐶
≅Δ𝐶𝐴𝑅.
 Step 1. Identify what the given are and what
is to be proved.
Given: 𝑅𝐸̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅≅𝐶𝐴̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ , 𝐸𝐶̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅≅𝐴𝑅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Prove: Δ𝑅𝐸𝐶≅Δ𝐶𝐴𝑅
Step 2. Identify the congruence theorem to be
used and the additional information needed
and why.
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅CR is the common side of Δ𝑅𝐸𝐶 𝑎𝑛𝑑 Δ𝐶𝐴𝑅, so by
reflexive property. Hence, SSS postulate can
be used to prove Δ𝑅𝐸𝐶≅Δ𝐶𝐴𝑅 because each of
the three sides of ΔREC is congruent
respectively to the three sides of ΔCAR.

6. Example 2. Based on Figure 7 at the right, prove that Δ𝑅𝐸𝐶
≅Δ𝐶𝐴𝑅.
 Step 3. Write down the statements and the
reasons in a two-column proof. Make sure the
last statement contains what should be
proved.
 .

7. Example 3: In the right triangles in Figure 9 at the right,
prove that Δ𝐴𝑅𝐺≅Δ𝐴𝑀𝑆.
 Step 1. Identify what the given are and what is to
be proved.
Given: 𝑅𝐴̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅≅𝑀𝐴̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ , ∠𝐺 and ∠𝑆 are right angles
Prove: Δ𝐴𝑅𝐺≅Δ𝐴𝑀𝑆
Step 2. Identify the congruence theorem to be used
and the additional information needed and why.
Since 𝑅𝑀̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
intersects 𝐺𝑆̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅, vertical angles are
formed, and vertical angles are congruent.
Thus, AAS congruence theorem can be used
to prove Δ𝐴𝑅𝐺≅Δ𝐴𝑀𝑆.

8. Example 3: In the right triangles in Figure 9 at the right,
prove that Δ𝐴𝑅𝐺≅Δ𝐴𝑀𝑆.
 Step 3. Write down the statements and the
reasons in a two-column proof. Make sure the last
statement contains what should be proved.

9. A
B
C
D
E
Given: 𝐴𝐶 ≅ 𝐸𝐶, 𝐶𝐵 ≅ 𝐶𝐷
Prove: ΔC𝐴B≅ΔCE𝐷

10. M A
T
H
Prove: ΔTHM≅ΔMAT

11. M
E
A
N
T
Prove: Δ𝐴ME≅Δ𝐴TN

13. Triangles are classified according to sides
and angles. Some triangles have
characteristics that come from both of
these classifications like scalene right
triangle and isosceles right triangle.

14. No two sides
are congruent

15. Legs are
congruent

16. After proving
that the two
triangles are
congruent, we
can also say
that the other
corresponding
parts of the
congruent
triangles are
congruent
(CPCTC).

17.  Definition of
Perpendicular Lines
 Right Angle Theorem
 Definition of Right
Triangle
 Reflexive Property
 CPCTC
 HyL Congruence
Theorem
Definition of Perpendicular Lines
Definition of Perpendicular Lines
Right Angle Theorem
Definition of Right Triangle
Reflexive Property
HyL Congruence Theorem
CPCTC

18.  Definition of
Perpendicular Lines
 Right Angle Theorem
 Given
 Definition of Midpoint
 Reflexive Property
 CPCTC
 SAS Congruence Postulate
or LL Congruence
Theorem
 Definition of Isosceles
Triangle

19.  Definition of
Perpendicular Lines
 Right Angle Theorem
 Given
 Definition of Midpoint
 Reflexive Property
 CPCTC
 SAS Congruence Postulate
or LL Congruence
Theorem
 Definition of Isosceles
Triangle

20. Reflexive Property
Given
CPCTC
AAS Congruence
Theorem