Number Operations & Equality
This property states that any number is equal to itself.
What is the Reflexive Property of Equality?
State the Ruler Postulate.
Points on a line can be numbered so the positive difference of coordinates gives distance.
State the Protractor Postulate.200Define the Angle Addition Postulate.
Rays in a half-rotation can be numbered 0–180 so the positive difference gives the angle measure.
Define a bisector.
A figure dividing a segment or angle into two congruent parts.
Define a linear pair.
Two adjacent angles whose non-common sides form opposite rays.
If a = b, then b = a. Which property justifies this reversal?
What is the Symmetric Property of Equality?
If E is between D and F, and DE = EF, what is E?
What is The midpoint of DF
State the Betweenness of Rays Theorem in both directions.
If B is interior to ∠AOC, then m∠AOB + m∠BOC = m∠AOC; conversely, if that sum holds, OB is between OA and OC.
If ray BD bisects ∠ABC, what equation relates the parts?
m∠ABD = m∠DBC
What is true of all vertical angles?
They are congruent.
If a = b and b = c, then a = c.
What is the Transitive Property of Equality?
Describe how to determine the coordinates of a missing endpoint when the midpoint and one endpoint are known.
What is Set midpoint equal to ((x₁ + x₂)/2, (y₁ + y₂)/2) and solve for the missing coordinate.
In your own words, explain how the Protractor Postulate and Betweenness of Rays Theorem together justify angle measurement.
They connect coordinate numbering to additive relationships of adjacent angles.
If m∠1 = 3x – 10° and m∠2 = x + 30° are complementary, find x.
25°
State the Perpendicular Lines Theorem.
If two lines form congruent adjacent angles, they are perpendicular.
This property connects arithmetic and algebra: a(b + c) = ab + ac.
What is the Distributive Property?
True or False: Betweenness implies congruence. Explain.
False — Betweenness is order; congruence requires equal measure.
Given ∠AOB and ∠BOC are adjacent and sum to 180°, what can you conclude about OA and OC?
They are two connected rays.
Complementary angles are what type of angle?
acute
If two lines are perpendicular to the same line, what can you conclude about them?
They are parallel.
Explain which two properties justify this reasoning:
If x = 4 and 4 = 2 + 2, then x = 2 + 2.
What are the Transitive Property and the Substitution Property of Equality?
If three collinear points have coordinates a, b, c with a < b < c, derive an algebraic condition for b to be the midpoint.
What is the equation
2b = a + c
Explain how the Betweenness of Rays Theorem mirrors the Segment Addition Postulate.
Both express whole = sum of parts when one element lies “between” two others.
Which property justifies: If m∠A = m∠B, then ∠A ≅ ∠B?
Definition of congruent angles
If ∠ABC ≅ ∠CBD and together form a straight line, prove AB ⟂ CD.
Use definition of congruent adjacent angles → Perpendicular Lines Theorem.