Honors Geometry Unit 4 Notes

The goal is to develop students ability to work with segments and angles and properties that arise from these ideas. Students should be able to further develop basic proof techniques, work with parallel and perpendicular lines, both geometrically and algebraically.

Proofs

• vertical angles are congruent
• when a transversal crosses parallel lines
• alternate interior angles are congruent
• corresponding angles are congruent
• the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints

Day 1

• Below are formal‑style definitions that rely only on the four primitive notions you listed—point, line, distance along a line, and distance around a circular arc. Each definition builds directly on those primitives (or on terms that have already been defined earlier in the list), so the whole set remains grounded in the same foundation. ----

1. Line segment

A line segment is the set of all points (P) lying on a given line (l) such that the distance along (l) from a fixed endpoint (A) to (P) plus the distance from (P) to the other fixed endpoint (B) equals the distance from (A) to (B).

Formally:

<math> \text{seg}(A,B)={,P\in l \mid d_{\text{line}}(A,P)+d_{\text{line}}(P,B)=d_{\text{line}}(A,B),}. </math>

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2. Angle

Let three distinct points (A), (O), and (B) be given, with (O) common to the two lines (\overleftrightarrow{OA}) and (\overleftrightarrow{OB}). An angle (\angle AOB) is the ordered pair of rays ((\overrightarrow{OA},\overrightarrow{OB})) together with the measure defined as the distance around a circular arc that subtends the two rays on any circle centered at (O).

In symbols: choose any radius (r>0); let (C) and (D) be the intersection points of the circle of radius (r) centred at (O) with the rays (\overrightarrow{OA}) and (\overrightarrow{OB}). Then the measure of (\angle AOB) is the distance around the circular arc from (C) to (D). The angle itself is the equivalence class of such ordered ray pairs under congruent arc‑length.

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3. Circle

A circle with centre (O) and radius (r) (where (r) is a positive distance along a line) is the set of all points (P) such that the distance around a circular arc from a fixed reference point on the circle back to itself after traversing the entire curve equals the circumference corresponding to radius (r). Equivalently, using only the primitives:

[ \text{circ}(O,r)={,P\mid \exists\ \text{arc } \alpha \text{ with endpoints } P\text{ and }P\text{ on a closed path, } d_{\text{arc}}(\alpha)=2\pi r,}. ]

Because the only primitive that mentions a curved notion is “distance around a circular arc,” we define the circle as the locus of points that can be reached by moving a distance (r) along a straight line from (O) and then completing a closed arc of total length (2\pi r).

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4. Perpendicular lines

Two distinct lines (l_1) and (l_2) are perpendicular if there exists a point (O) that lies on both lines (their intersection) and for any positive distance (r) the two points (A) on (l_1) and (B) on (l_2) satisfying (d_{\text{line}}(O,A)=d_{\text{line}}(O,B)=r) form an angle (\angle AOB) whose measured arc distance equals one‑quarter of the full circular arc of radius (r) (i.e., ( \frac{1}{4}\times 2\pi r = \frac{\pi r}{2})).

Thus the defining condition is: the angle determined by the two intersecting rays is a right angle, expressed solely through the primitive “distance around a circular arc.”

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5. Parallel lines

Two distinct lines (l_1) and (l_2) are parallel if they do not intersect (no point belongs to both) and, for any point (P) on (l_1) and any positive distance (r), there exists a point (Q) on (l_2) such that the distance along a line from (P) to (Q) equals (r) and the angle formed by the segment (PQ) with each line is the same acute angle measured via the circular‑arc distance. In other words, the transversal distance between the lines is constant and the directional relationship (angle) is invariant, all expressed through the primitives of line distance and arc‑based angle measurement.

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How the definitions fit together

1. Line segment gives us a finite piece of a line.
2. Angle supplies a way to talk about the opening between two rays using the arc‑distance primitive.
3. Circle provides the geometric object whose arc‑distance defines the angle’s measure.
4. Perpendicular and parallel are relational properties of lines that are captured by the angle definition (right angle for perpendicular, equal acute angle for parallel) and the non‑intersection condition for parallelism.

All five definitions are thus rooted exclusively in the four primitives you specified, establishing a coherent, axiomatic vocabulary for elementary geometry.

• Introduce slope, parallel and perpendicular lines algebraically.
• Transition to geometric ideas.
  • Remind the ideas of point, line, segment, and plane.
  • Introduce parallel, perpendicular, skew, and coplanar on a plane.
  • Introduce (Remind)
    • Transversal
    • Ray
    • Angle
    • Vertex
    • Collinear
    • Complementary
    • Supplementary
    • Congruent
    • Linear Pair
    • Adjacent angles
    • Vertical angles
    • Alternate interior angles
    • Same side interior angle
    • Consecutive interior angle
    • Corresponding angles

Day 2

• Angle chase practice
• Introduce basic constructions
  • Straightedge for lines
    • Perpendicular Bisector
  • Compass for circles
    • Circles are the set of points equidistant from some point called the center.
  • Construct an equilateral triangle

Day 3

• Prove vertical angles are congruent.
• Prove Alternate interior angles are congruent.
• etc.