1. Proof Geometry 4-4: Perpendiculars, Right Angles, Congruent Angles
4-5: Equivalence Relations

2. 4 Types of Angles Acute Angle: Right Angle: Obtuse Angle:
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4 Types of Angles
Acute Angle:
an angle whose measure is less than 90.
Right Angle:
an angle whose measure is exactly 90 .
Obtuse Angle:
an angle whose measure is between 90 and 180.
Reflex Angle:
an angle whose measure is greater than 180.

3. Complementary Angles Definition: A pair of angles whose sum is 90˚
Examples:
Adjacent Angles
( a common side )
Non-Adjacent Angles

4. Congruent Angles Definition:
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Congruent Angles
Definition:
If two angles have the same measure, then they are congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is  3 5
Example:
3   5.
Note: we write m3 =m 5.

5. Perpendicular
Two lines, rays, or segments that intersect to form a right angle are called perpendicular
L1 ⊥ L2 L1 L2

6. Equivalence Relations
An equivalence relation is a relation satisfying ALL of the following properties:
Reflexive Property: a = a, for every a.
Symmetric Property: If a = b, then b = a.
Transitive Property: If a = b and b = c, then a = c

7. Theorem Congruence (≌) is an equivalence relation.
Reflexive property: A ≌ A, for every A
Symmetric property: If A ≌ B, then B ≌ A
Transitive property: If A ≌ B and B ≌ C, then A ≌ C

8. Other examples / non-examples of equivalence relations
Is parallel to
Approximately equal (can fail the transitive property)
Is similar to (triangles)
Greater or equal, less than or equal (fails the symmetric property)

9. Homework 4-4, 4-5
4-4 p : #6-8, 10, 11, 13.   For 13, consider a system of two linear equations over x (acute angle) and y (obtuse)
4-5 p : # 1, 3-6, 8, 9