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

4 Types of Angles Acute Angle: Right Angle: Obtuse Angle: 4/9/2019 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.

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

Congruent Angles Definition: 4/9/2019 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.

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

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

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

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)

Homework 4-4, 4-5 4-4 p. 96-97: #6-8, 10, 11, 13.   For 13, consider a system of two linear equations over x (acute angle) and y (obtuse) 4-5 p. 98-100: # 1, 3-6, 8, 9