1. Geometry – Chapter 4
Congruent Triangles

2. 4.1 – Apply Angle Sum Properties
Triangle
Polygon with three sides & three vertices
Triangles can be classified by side and angles
Classifying Triangles by Sides
Scalene
Iscoceles
Equilateral
No congruent sides
Two congruent sides
All sides congruent
Classifying Triangles by Angles
Acute
Right
Obtuse
Equiangular
3 acute angles (< 90)
1 right angle (= 90)
1 obtuse angle (> 90)
3 congruent angles

3. Example 2
Classify ∆PQO by its sides, then determine if the triangle is right.
Points are:
P (-1, 2)
Q (6, 3)
O (0, 0)
GP: #1-2

4. Angles Interior Angles Exterior Angles
Angles on the inside of the triangle (there are three)
Exterior Angles
Angles that form linear pairs with interior angles (there are 6)

5. Theorems 4.1 – Triangle Sum Theorem 4.2 – Exterior Angle Theorem
The sum of the measure of the interior angles of a triangle is 180°
4.2 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles
Example #3 p. 209

6. Corollary to a theorem Corollary to a theorem
Statement that can be proved easily using the theorem
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary
Example 4
A tiled staircase forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle
GP #3 & 5 p. 210

7. 4.2 – Apply congruence & triangles
Two geometric figures are congruent if they have exactly the same size and shape
Congruent figures
All parts of one figure are congruent to the corresponding parts of the other figure (corresponding sides & corresponding angles)
Congruence Statements
Be sure to name figures by their corresponding vertices!

8. examples Example 1 Example 2 GP #1-3 p. 216
Writing a congruence statement and identifying all congruent parts
Example 2
Using properties of congruent figures
𝐷𝐸𝐹𝐺≅𝑆𝑃𝑄𝑅
GP #1-3 p. 216

9. Third angles theorem Theorem 4.3 – Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent

10. Using third angles theorem
Example 4
Find m< BDC A B
45° N
30° C D
GP #4-5 p. 217

11. Properties of congruent triangles
The properties of congruence that are true for segments and angles are also true for triangles
Theorem 4.4 – Properties of Congruent Triangles
Reflexive property
∆𝐴𝐵𝐶≅∆𝐴𝐵𝐶
Symmetric property
∆𝐴𝐵𝐶≅∆𝐷𝐸𝐹, 𝑡ℎ𝑒𝑛 ∆𝐷𝐸𝐹≅∆𝐴𝐵𝐶
Transitive property
∆𝐴𝐵𝐶≅∆𝐷𝐸𝐹 𝑎𝑛𝑑 ∆𝐷𝐸𝐹≅∆𝑋𝑌𝑍 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶≅∆𝑋𝑌𝑍

12. 4.3 – relate transformations & congruence
Rigid motion
Transformation that preserves length, angle measure, and area
Examples of rigid motions (isometry): translations, reflections, rotations
Congruent figures and Transformations
Two figures are congruent if and only if one or more rigid motions can be used to move one figure onto the other. If any combination of translations, reflections, and rotations can be used to move one shape onto the other, the figures are congruent

13. 4.4 – Prove triangles congruent by SSS
Postulate 19 – 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

14. Example 1
Use the SSS congruence postulate
GP #1-3 p. 232

15. 4.5 – congruence by SAS and HL
Postulate 20 – 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

16. Right triangles
In a right triangle, the sides adjacent to the right angles are called the legs
The side opposite the right angle is called the hypotenuse
Theorem 4.5 – Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent

17. 4.6 – Prove using ASA & AAS
Postulate 21 – 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

18. AAS theorem Theorem 4.6 – Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent

19. Triangles postulates &theorems
5 methods for proving that triangles are congruent
SSS
SAS
HL (right angles only)
ASA
AAS
All 3 sides are congruent
Two sides and the included angle are congruent
Hypotenuse and one of the legs are congruent
Two angles and the included side are congruent
Two angles and a non-included side are congruent

20. 4.7 – use congruent triangles
Congruent triangles have congruent corresponding parts
If two triangles are congruent, their corresponding parts must be congruent as well

21. Euclid’s river example

22. 4.8 – use isosceles and equilateral triangles
Legs
Two congruent sides of an isosceles triangle
Vertex angle
Angle formed by the legs
Base
Third side of an isosceles triangle
Base angles
Angles adjacent to the base (opposite the legs)

23. Isosceles triangles theorem
Theorem 4.7 – Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent
Theorem 4.8 – Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent

24. Example 1
Name two congruent angles F D E
GP #1-2 p. 264

25. Corollaries Corollary to the Base Angles Theorem
If a triangle is equilateral, then it is equiangular
Corollary to the Converse of Base Angles Theorem
If a triangle is equiangular, then it is equilateral
Example 2
If a triangle is equilateral, what is the measure of each angle?
Example 3 – on board GP #5 p. 266