geometry unit 0

Fundamentals of Point, Line, and Plane

• Collinear Points: Points are collinear if they lie on the exact same straight line. 180 degrees

• Coplanar Points/Lines: Points or lines are coplanar if they lie within the same two-dimensional plane.

• Naming Geometric Figures:

  • Line: Exactly $2$ points are required to name a line (for example, line $\overleftrightarrow{AB}$).

  • Plane: At least $3$ non-collinear points are required to name a plane (for example, plane $\text{TLC}$), or a single capital script letter.

• Intersections:

  • Two Lines: Two distinct lines intersect at exactly one Point.

  • Line and Plane: A line and a plane intersect at either a Point (if the line passes through the plane) or a Line (if the line lies completely within the plane).

  • Two Planes: Two distinct planes intersect at a Line.

Geometric Notation and Symbols

• Point Notation ($A$): Represents a specific location in space named by a single capital letter.

• Line Notation ($m$ or $\overleftrightarrow{AG}$): Represents a line designated by a lowercase script letter or by two points with a double-sided arrow over them.

• Plane Notation (Plane $\text{TLC}$): Represents a plane designated by three non-collinear points or a capital script letter.

• Segment Notation ($\overline{KR}$): Represents a line segment connecting endpoints $K$ and $R$.

• Ray Notation ($\overrightarrow{ME}$): Represents a ray starting at initial point $M$ and extending infinitely through point E$.\n- **Distance Notation** (SE$): Represents the numerical distance or length between point$S$and point$E$.

• Relational Symbols:

  • Congruence: $\cong$

  • Perpendicularity: $\perp$

  • Parallelism: $\parallel$

Segment Addition Postulate

• Postulate Statement: If point $B$ lies between point $A$ and point $C$ on a straight line segment, then the lengths of the smaller segments sum to the total length of the segment:

$AB + BC = AC$

Basic Measurement Problems

• Problem 9: Given $AB = 5$, $BC = ?$, and $AC = 19$:

  • $AB + BC = AC$

  • $5 + BC = 19$

  • $BC = 19 - 5$

  • $BC = 14$

• Problem 10: Given $AB = ?$, $BC = 12.5$, and $AC = 53.5$:

  • $AB + BC = AC$

  • $AB + 12.5 = 53.5$

  • $AB = 53.5 - 12.5$

  • $AB = 41$

Algebraic Segment Problems

• Problem 11: Given $AB = 2x$, $BC = x - 3$, and $AC = 45$:

  • $AB + BC = AC$

  • $(2x) + (x - 3) = 45$

  • $3x - 3 = 45$

  • $3x = 48$

  • $x = 16$

  • $AB = 2(16) = 32$

  • $BC = 16 - 3 = 13$

  • $AC = 32 + 13 = 45$

• Problem 12: Given $AB = 4$, $BC = 3(x - 1)$, and $AC = 7x - 15$:

  • $AB + BC = AC$

  • $4 + 3(x - 1) = 7x - 15$

  • $4 + 3x - 3 = 7x - 15$

  • $3x + 1 = 7x - 15$

  • $16 = 4x$

  • $x = 4$

  • $BC = 3(4 - 1) = 9$

  • $AC = 7(4) - 15 = 13$

Angle Addition Postulate

• Postulate Statement: If point $D$ lies in the interior of angle $\angle ABC$, then:

$m\angle ABD + m\angle DBC = m\angle ABC$

• Angle Bisector Definition: A ray that divides an angle into two congruent adjacent angles.

• Problem 13: Given straight line $DG$ with vertex $H$, $m\angle EHF = 61^\circ$, and $m\angle EHG = 133^\circ$:

  • Finding $m\angle FHG$:

  • $m\angle EHF + m\angle FHG = m\angle EHG$

  • $61^\circ + m\angle FHG = 133^\circ$

  • $m\angle FHG = 133^\circ - 61^\circ = 72^\circ$

  • Finding $m\angle DHE$:

  • Since points $D$, $H$, and $G$ form a straight line, $m\angle DHG = 180^\circ$

  • $m\angle DHE + m\angle EHG = 180^\circ$

  • $m\angle DHE + 133^\circ = 180^\circ$

  • $m\angle DHE = 180^\circ - 133^\circ = 47^\circ$

• Problem 14: Given $m\angle PMQ = 79^\circ$, $m\angle QML = 3x + 12$, and $m\angle PML = 125^\circ$:

  • $m\angle PMQ + m\angle QML = m\angle PML$

  • $79 + (3x + 12) = 125$

  • $3x + 91 = 125$

  • $3x = 34$

  • $x = \frac{34}{3} \approx 11.33$

  • $m\angle QML = 3\left(\frac{34}{3}\right) + 12 = 46^\circ$

• Problem 15: Given $m\angle PMR = 3x - 23$, $m\angle RMQ = 4x + 6$, and $m\angle PMQ = 5x + 8$:

  • $m\angle PMR + m\angle RMQ = m\angle PMQ$

  • $(3x - 23) + (4x + 6) = 5x + 8$

  • $7x - 17 = 5x + 8$

  • $2x = 25$

  • $x = 12.5$

  • $m\angle PMR = 3(12.5) - 23 = 14.5^\circ$

• Problem 16: Given that ray $\overrightarrow{MR}$ bisects $\angle PMQ$, $m\angle PMR = 2(x - 12)$, and $m\angle RMQ = x + 11$:

  • By definition of an angle bisector:

  • $m\angle PMR = m\angle RMQ$

  • $2(x - 12) = x + 11$

  • $2x - 24 = x + 11$

  • $x = 35$

  • $m\angle PMR = 2(35 - 12) = 46^\circ$

  • $m\angle RMQ = 35 + 11 = 46^\circ$

  • $m\angle PMQ = 46^\circ + 46^\circ = 92^\circ$

Angle Relationships in Intersecting Lines

• Complementary Angles: Two angles whose measures sum to $90^\circ$.

• Supplementary Angles: Two angles whose measures sum to $180^\circ$.

• Linear Pair: Two adjacent angles whose non-common sides form a straight line (their sum is $180^\circ$).

• Vertical Angles: Two non-adjacent angles formed by two intersecting lines; they are opposite each other and congruent.

Angle Calculation Problems

• Problem 17: Complementary Pair identification: $\angle 1$ and $\angle 4$ (or $\angle 1$ and $\angle 5$).

• Problem 18: Linear Pair identification: $\angle 3$ and $\angle 4$ (or $\angle 2$ and $\angle 3$).

• Problem 19: Vertical Angle pair: $\angle 4$ and $\angle 2$.

• Problem 20: If $m\angle 4 = 43^\circ$, find $m\angle 2$:

  • $\angle 4$ and $\angle 2$ are vertical angles, so $m\angle 2 = 43^\circ$.

• Problem 21: If $m\angle 2 = 33^\circ$, find $m\angle 5$:

  • $m\angle 4 = m\angle 2 = 33^\circ$ (vertical angles).

  • Since $\angle 1 = 90^\circ$, $\angle 4$ and $\angle 5$ are complementary:

  • $m\angle 5 = 90^\circ - 33^\circ = 57^\circ$.

• Problem 22: If $m\angle 2 = 37^\circ$, find $m\angle 3$:

  • $\angle 2$ and $\angle 3$ form a linear pair:

  • $m\angle 3 = 180^\circ - 37^\circ = 143^\circ$.

Complementary and Supplementary Angle Problems

• Problem 23: Right angle $\angle ABD = 90^\circ$ divided by ray $\overrightarrow{BC}$, with $m\angle ABC = 3x + 12$ and $m\angle CBD = x - 8$:

  • $(3x + 12) + (x - 8) = 90$

  • $4x + 4 = 90$

  • $4x = 86$

  • $x = 21.5$

• Problem 24: If $m\angle CBD = 4x$, then $m\angle ABC = 90^\circ - 4x$.

• Problem 25: Straight line $AD$ with ray $\overrightarrow{BC}$ forming a linear pair. If $m\angle ABC = 119^\circ$:

  • $m\angle ABC + m\angle CBD = 180^\circ$

  • $m\angle CBD = 180^\circ - 119^\circ = 61^\circ$

• Problem 26: Given $m\angle ABC = 7x - 3$ and $m\angle CBD = 2x + 12$ forming a linear pair:

  • $(7x - 3) + (2x + 12) = 180$

  • $9x + 9 = 180$

  • $9x = 171$

  • $x = 19$

  • $m\angle ABC = 7(19) - 3 = 130^\circ$

  • $m\angle CBD = 2(19) + 12 = 50^\circ$

Multi-Step Intersecting Line Systems

• Problem 27: Analyzing intersecting lines at vertex $E$:

  • Vertical Angles Relationship: $5x + 10 = 2x + 46$

  • $3x = 36$

  • $x = 12$

  • Linear Pair Relationship: $(8x + 14) + (5x + 10) = 180$

  • $13x + 24 = 180$

  • $13x = 156$

  • $x = 12$

  • Calculating Angle Measure:

  • $m\angle DEF = 2(12) + 46 = 70^\circ$

Midpoint and Distance Formulas

• Midpoint Formula:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

• Problem 28: Find the midpoint between points $(3, 8)$ and $(-11, 4)$:

  • $M = \left(\frac{3 + (-11)}{2}, \frac{8 + 4}{2}\right)$

  • $M = \left(\frac{-8}{2}, \frac{12}{2}\right)$

  • $M = (-4, 6)$

• Distance Formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Problem 29: Given $K(0, 0)$ and $E(3, 3)$, calculate distance $KE$:

  • $KE = \sqrt{(3 - 0)^2 + (3 - 0)^2}$

  • $KE = \sqrt{3^2 + 3^2}$

  • $KE = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24\,\text{units}$