Theorems and Postulates

1. Pop Quiz: Identify
1. Two angles whose measures have a sum of 90deg
2. A statement that can be proven
3. Triangles in which corresponding parts (sides and angles) are equal in
measure
4. If a = b and b = c then a = c
5. a(b + c) = ab + ac
6. If a=b, then a+c=b+c
7. If a=b, then a-c=b-c
8. If a=b, then a/c=b/c
9. A point is an _________ term.
10. An angle is a __________ term
11. _________ are statements that are considered true without proof or validation.

2. Pop Quiz: Identify
1. Complementary Angles
2. Theorem
3. Congruent Triangles
4. Transitive Property of Equality
5. Distributive Property of Equality
6. Addition Property of Equality
7. Subtraction Property of Equality
8. Division Property of Equality
9. Undefined
10. Defined
11. Postulates

3. Math 8 – Triangle
Congruence, Postulates, and
Proving
Ms. Andi Fullido
© Quipper

4. Objective
•At the end of this lesson, you should be
able to identify the conditions for which
two triangles are congruent.

5. Geometric figures are said to be congruent if
they have the same size and shape.
• Two triangles are congruent when all corresponding sides and
interior angles are congruent.
Consider Triangle QRS and IHG:

6. Triangles:
• The tick marks show that the corresponding sides are
congruent:
• 𝑄𝑅 ≅ 𝐼𝐻
• 𝑅𝑆 ≅ 𝐻𝐺
• 𝑆𝑄 ≅ 𝐺𝐼
• 𝑄𝑅 ≅ 𝐼𝐻

7. Triangles:
The angle arcs show that the corresponding
interior angles are congruent:
•∠Q≅∠I
•∠R≅∠H
•∠S≅∠G
•When all six statements are stated, then we can
conclude that △QRS≅△IHG.

8. If △ABC≅△DEF, what is the value
of x and y?

9. Try it! Solution
•Since the two triangles are congruent,
then the corresponding sides are
congruent:
•AB≅DE
•BC≅EF
•AC≅DF

10. Try it! Solution
• Find x:
• x=AB, but from the congruence statement, AB=DE. This
means x=DE, too. From the figure, DE=14. Thus, x=14.
• Find y:
• y=FD, but from the congruence statement, FD=AC. This
means y=AC, too. From the figure, AC=12. Thus, y=12.

11. Keypoints:
•Two triangles are congruent when all
corresponding sides and interior
angles are congruent.

12. Objectives
• At the end of this lesson, you should be able to:
• enumerate the triangle congruence postulates;
• define the SSS, SAS, ASA postulates; and
• recognize congruent triangles using the triangle
congruence postulates.

13. There are three postulates for proving that
two triangles are congruent:
•Side-Side-Side (SSS) Congruence
Postulate
•Side-Angle-Side (SAS) Congruence
Postulate
•Angle-Side-Angle (ASA) Congruence
Postulate

14. SSS Congruence Postulate
• Postulate: The Side-Side-Side Congruence Postulate. If three
sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.

15. SSS Congruence Postulate
• With reference to triangles ABC and XYZ:
• If 𝐴𝐵 ≅ 𝑋𝑌, 𝐵𝐶 ≅ 𝑌𝑍, and 𝐴𝐶 ≅ 𝑋𝑍, then △ABC≅△XYZ.

16. SAS Congruence Postulate
• Postulate: The Side-Angle-Side Congruence Postulate. If two sides
and the included angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two triangles are
congruent.
• Note: The included angle is any angle between two sides.

17. SAS Congruence Postulate
• With reference to triangles ABC and UVW:
• If 𝐴𝐶 ≅ 𝑈𝑊, ∠A≅∠U, and 𝐴𝐵 ≅ 𝑈𝑉, then △ABC≅△UVW.

18. ASA Congruence Postulate
• Postulate: The Angle-Side-Angle Congruence Postulate. If two angles
and the included side of one triangle are congruent to two angles and
the included side of a second triangle, then the two triangles are
congruent.
• Note: The included side is any side between two angles.

22. Study the two congruent triangles in the
picture below. Which statement is true about
the second triangle?

23. • a) a = 7 cm, b = 8 cm, c = 9 cm
• b) a = 8 cm, b = 7 cm, c = 9 cm
• c) a = 8 cm, b = 9 cm, c = 7 cm
• d) a = 9 cm, b = 7 cm, c = 8 cm

24. Tip:
• You can determine which side/angle of one triangle corresponds
with which side/angle of another triangle by using the order of the
triangle names.
• Example: △ABC and △DEF.
• The first letters are A and D. This means ∠A is to ∠D. The second
letters are B and E, which means ∠B is to ∠E. The third set of
angles can be derived from the third letters.
• The name of sides uses two letters. The first two letters are AB and
DE, which means (𝐴𝐵 𝑖𝑠 𝑡𝑜 𝐷𝐸). The last two letters are BC and
EF (𝐵𝐶 𝑖𝑠 𝑡𝑜 𝐸𝐹), while the first and last letters are AC and
DF (𝐴𝐶 𝑖𝑠 𝑡𝑜 𝐷𝐹)respectively.

25. Keypoints
• There are three congruence postulates that can be used to tell if two
triangles are congruent:
• The Side-Side-Side (SSS) Congruence Postulate. If three sides of one
triangle are congruent to three sides of a second triangle, then the two
triangles are congruent.
• The Side-Angle-Side (SAS) Congruence Postulate. If two sides and the
included angle of one triangle are congruent to two sides and the included
angle of a second triangle, then the two triangles are congruent.
• The Angle-Side-Angle (ASA) Congruence Postulate. If two angles and
the included side of one triangle are congruent to two angles and the
included side of a second triangle, then the two triangles are congruent.

26. Theorem: Corresponding parts of congruent
triangles are congruent (CPCTC).

27. If m∠B=2x−5 and m∠Y=3x−65, what is
the exact measure of ∠B?