Notes
Building blocks
- Point, line and plane are undefined terms. Collinear points share a line; coplanar points share a plane.
- Two points determine one line. Three noncollinear points determine one plane. Two planes meet in a line.
Segments
- Segment Addition: if $B$ is between $A$ and $C$, then $AB + BC = AC$.
- Congruent segments have equal lengths. A midpoint splits a segment into two congruent parts.
| Formula | Rule |
|---|---|
| Midpoint | $\displaystyle \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ |
| Distance | $\displaystyle \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ |
Angles
- Acute $< 90^\circ$, right $= 90^\circ$, obtuse between $90^\circ$ and $180^\circ$, straight $= 180^\circ$.
- Angle Addition: if $D$ is inside $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
- Complementary angles sum to $90^\circ$; supplementary angles sum to $180^\circ$.
- A linear pair is supplementary. Vertical angles are congruent.
Constructions
Use only a compass and straightedge. Points on the perpendicular bisector of a segment are equidistant from its endpoints.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \sqrt{6^2 + 8^2} = \sqrt{100} = 10$.
$4x - 3 = 2x + 9$ gives $x = 6$, so $AM = 21$ and $AB = 42$.
$8x + 20 = 180$, so $x = 20$ and $m\angle 1 = 110^\circ$.
Vertical angles are equal: $4x + 6 = 6x - 20$, so $2x = 26$.
$180 - x = 3(90 - x)$ gives $180 - x = 270 - 3x$, so $2x = 90$.
Any three points lie in at least one plane, but they need not lie on one line. Two points are always collinear.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A park map uses a coordinate grid in which each unit is 100 meters. The fountain is at $F(-2, 1)$ and the snack stand is at $S(6, 7)$.
(a) Find the straight-line distance from the fountain to the snack stand, in meters.
Show the answer and scoring guide
$\displaystyle d = \sqrt{(6 - (-2))^2 + (7 - 1)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ units, which is 1,000 meters.
Check off the points you earned:
(b) A bench will be placed exactly halfway between the fountain and the snack stand. Find its coordinates.
Show the answer and scoring guide
$\displaystyle \left(\frac{-2 + 6}{2}, \frac{1 + 7}{2}\right) = (2, 4)$
Check off the points you earned:
(c) Two straight paths cross at the bench. One angle between the paths is $(3x + 15)^\circ$ and the angle next to it along the same path is $(2x + 15)^\circ$. Find both angles.
Show the answer and scoring guide
They form a linear pair: $5x + 30 = 180$, so $x = 30$. The angles are $105^\circ$ and $75^\circ$.
Check off the points you earned: