Honors Geometry Unit 1 Review: Foundations of Geometry

Geometry rests on three undefined terms, which are point, line, and plane. A point is an exact location with no size, so it has zero dimensions. A line is a straight path with no thickness that extends forever in both directions, so it has one dimension. A plane is a flat surface that extends forever in all directions, so it has two dimensions. Solids like prisms and spheres add a third dimension.

Everything else gets defined from these. A segment is the part of a line between two endpoints, and a ray starts at one endpoint and extends forever in one direction. An angle is two rays that share an endpoint called the vertex.

Notation carries meaning. $\overleftrightarrow{AB}$ is a line, $\overrightarrow{AB}$ is a ray starting at $A$, and $\overline{AB}$ is a segment. Plain $AB$ with no symbol means the length of the segment, which is a number. Mixing these up costs points on proofs.

You also need the difference between a postulate and a theorem. A postulate is accepted without proof, like "through any two distinct points there is exactly one line." A theorem is a statement you prove using definitions, postulates, and earlier theorems. Euclid organized geometry this way in his Elements around 300 BC, and your course still follows that structure.

The coordinate plane is two perpendicular number lines, the horizontal x-axis and the vertical y-axis, crossing at the origin $(0, 0)$. Every point is an ordered pair $(x, y)$. The axes split the plane into four quadrants numbered counterclockwise from the upper right, so Quadrant II holds points like $(-3, 5)$. A point on an axis, like $(0, 4)$, is in no quadrant.

Three formulas do most of the work. Distance is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, which is the Pythagorean theorem with legs equal to the horizontal and vertical changes. Midpoint is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$, the average of each coordinate. For $x_2 \ne x_1$, slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Vertical lines have undefined slope. Distinct nonvertical parallel lines have equal slopes, while nonvertical perpendicular lines have slopes whose product is $-1$. A vertical line is perpendicular to a horizontal line.

Worked example. Take $A(-2, 1)$ and $B(4, 9)$. The changes are $4 - (-2) = 6$ and $9 - 1 = 8$, so $AB = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$. The midpoint is $\left(\frac{-2 + 4}{2}, \frac{1 + 9}{2}\right) = (1, 5)$. The slope is $\frac{8}{6} = \frac{4}{3}$, so any line perpendicular to $\overleftrightarrow{AB}$ has slope $-\frac{3}{4}$.

A few postulates describe how the building blocks fit together. Two distinct points determine exactly one line. Three non-collinear points determine exactly one plane, and so do a line plus a point not on it. Two distinct planes that intersect meet in a line, not a point. Two distinct lines that intersect meet in exactly one point.

Collinear points lie on one line, and coplanar points lie in one plane. Any two points are collinear and any three points are coplanar, so these words only become a real test with three or four points. Lines that never meet are parallel if they are coplanar and skew if they are not. Skew lines only show up in 3D figures, like an edge on the top of a box and a non-parallel edge on the bottom.

Angles are classified as acute (between $0°$ and $90°$), right ($90°$), obtuse (between $90°$ and $180°$), or straight ($180°$). Angle pairs matter more than single angles. Vertical angles, the opposite angles formed by two intersecting lines, are always congruent. A linear pair is two adjacent angles whose outer sides form a line, so they add to $180°$.

Worked example. Two lines cross, forming vertical angles of $(3x + 10)°$ and $(5x - 30)°$. Set them equal, $3x + 10 = 5x - 30$, so $40 = 2x$ and $x = 20$. Each angle is $70°$, and the angle forming a linear pair with either one is $180° - 70° = 110°$.

The ruler postulate says points on a line can be matched with real numbers, and the distance between two points is the absolute value of the difference. If $A$ is at $-3$ and $B$ is at $8$, then $AB = |8 - (-3)| = 11$. The protractor postulate assigns readings to rays in a plane from a common vertex. An angle's measure is the absolute value of the difference between its rays' readings. Always read from the scale that starts at $0$ on the ray you lined up.

The segment addition postulate says that if $B$ is between $A$ and $C$, then $AB + BC = AC$. The angle addition postulate says that if $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$. A midpoint splits a segment into two congruent halves, and an angle bisector splits an angle into two congruent angles.

Use $=$ to compare measures and $\cong$ to show that figures are congruent. For segments of equal length, write $AB = CD$ or $\overline{AB} \cong \overline{CD}$. An equals sign between the segments would say they are the same figure, not merely congruent.

Worked example. Point $D$ is in the interior of $\angle ABC$, with $m\angle ABD = (2x + 5)°$, $m\angle DBC = (3x - 10)°$, and $m\angle ABC = 95°$. By angle addition, $(2x + 5) + (3x - 10) = 95$, so $5x - 5 = 95$ and $x = 20$. Then $m\angle ABD = 45°$ and $m\angle DBC = 50°$, and $45 + 50 = 95$ checks out.