Unit 0 Review — Foundations of Geometry
This cumulative review connects all seven Unit 0 lessons: geometric notation, segment and angle measurement, angle relationships, compass-and-straightedge constructions, and coordinate geometry. Every worked example and extended problem blends at least two concepts.
Mastering these foundational tools — measurement, angle reasoning, constructions, and coordinate methods — is the prerequisite for every proof and theorem in the rest of the course.
This review covers all seven lessons of Unit 0. Each worked example and extended problem blends at least two Unit 0 concepts so you practice connecting ideas, not just recalling isolated procedures.
Key Vocabulary Review
Collinear points
Points that lie on the same line.
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
Distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Complementary angles
Two angles whose measures sum to 90°.
Supplementary angles
Two angles whose measures sum to 180°.
Vertical angles
Non-adjacent angles formed by two intersecting lines; always congruent.
Perpendicular bisector
A line that bisects a segment at a right angle.
Slope
m = rise/run = (y₂−y₁)/(x₂−x₁)
Parallel lines
Lines in the same plane that never intersect; equal slopes.
Perpendicular lines
Lines that intersect at 90°; slopes are negative reciprocals.
Angle bisector
A ray that divides an angle into two congruent angles.
Worked Examples
Points A, B, and C are collinear with B between A and C. AB = 2x + 3, BC = x + 7, and AC = 31. Find x and each segment length.
P(1, 2) and Q(7, 6). Find the distance PQ and the midpoint M.
Two angles are complementary. One measures (3x + 5)° and the other (x + 15)°. Find x, each angle measure, and classify each angle.
Two lines intersect. One pair of vertical angles measures (4x + 10)° and (6x − 20)°. Find x and all four angle measures.
A student constructs the perpendicular bisector of segment AB using compass and straightedge. The two arc intersection points are P (above AB) and Q (below AB). Explain why PQ is perpendicular to AB and why it bisects AB.
A(1, 1), B(5, 4), C(9, 7). (a) Find the midpoint M of AB. (b) Find the slope of AB and the slope of BC. (c) Are A, B, C collinear? Justify.
Multiple Choice
Circle the best answer for each question.
B is between A and C. AB = 3x − 1, BC = 2x + 4, AC = 28. What is x?
What is the distance between R(−2, 3) and S(4, −5)?
The midpoint of segment PQ is M(3, −1). P is at (−1, 5). What are the coordinates of Q?
An angle measures 127°. Which classification is correct?
Two angles are complementary. One measures (2x + 10)°. The other measures (3x − 5)°. What is x?
Two lines intersect. One angle measures (5x + 8)°. Its vertical angle measures (7x − 12)°. What is the measure of each angle?
∠JKL and ∠LKM form a linear pair. ∠JKL = (4x + 6)°. ∠LKM = (2x + 12)°. What is m∠JKL?
Which construction produces a line through a given point that is perpendicular to a given line?
Point P lies on the perpendicular bisector of segment AB. Which statement must be true?
What is the slope of the line through (−3, 5) and (1, −3)?
Line j has slope 3/4. Which slope would make line k perpendicular to line j?
Are points D(0, 0), E(2, 3), and F(4, 6) collinear?
Guided Practice
Show all work. Each problem blends two or more Unit 0 skills.
X is between W and Z. WX = 4x − 2, XZ = 2x + 8, WZ = 42. Find x and both segment lengths.
Find the distance between A(−3, 4) and B(5, −2). Then find the midpoint M of AB.
∠PQR and ∠RQS are supplementary. m∠PQR = (5x + 20)°. m∠RQS = (3x − 4)°. Find x and both angle measures.
Two lines intersect forming angles of (8x − 5)° and (6x + 15)°. If these are vertical angles, find x and the measure of all four angles.
Describe the steps to construct an angle bisector of ∠ABC using only a compass and straightedge. What property guarantees the construction is exact?
Line p passes through (0, 2) and (4, 5). Line q passes through (1, 0) and (5, 3). Are lines p and q parallel, perpendicular, or neither? Justify.
The midpoint of segment CD is M(4, 1). One endpoint is C(1, −3). Find D.
Points G(0, 0), H(3, 4), and K(6, 8) are given. Test whether G, H, K are collinear using slope. Then find the distance GK.
Extended Mixed Problems
Each problem requires connecting multiple Unit 0 concepts in a single coherent task.
Segment Addition + Coordinate Distance
On a number line, A is at −4, B is at 3x − 1, and C is at 14. B is between A and C, and AB = BC. (a) Find x. (b) Find the coordinate of B. (c) Now treat A and C as points in the coordinate plane at A(−4, 0) and C(14, 0). Verify that the distance formula gives AC = 18.
Angle Bisector Construction + Angle Algebra
∠ABC measures (6x + 4)°. Ray BD bisects ∠ABC so that ∠ABD = (2x + 18)°. (a) Find x and m∠ABC. (b) Classify ∠ABC. (c) Describe the compass-and-straightedge steps to construct BD.
Midpoint, Slope, and Perpendicular Bisector
Segment JK has endpoints J(2, 6) and K(8, 2). (a) Find the midpoint M of JK. (b) Find the slope of JK. (c) What is the slope of the perpendicular bisector of JK? (d) Does the point P(5, 4) lie on the perpendicular bisector? Justify.
Collinearity Test + Geometric Notation
Points R(1, 2), S(4, 5), and T(7, 9) are given. (a) Test whether R, S, T are collinear using slope. (b) If they are not collinear, name the three distinct segments using correct notation. (c) If they are collinear, write the correct notation for the line through all three.
Angle Relationships + Parallel Line Verification
Line ℓ passes through (0, 1) and (4, 3). Line m passes through (0, −2) and (4, 0). (a) Find the slope of each line. (b) Are ℓ and m parallel? (c) A transversal crosses both lines. If one co-interior angle is 65°, what is the other co-interior angle? What relationship do these angles have?
Full Unit 0 Blend: Notation, Measurement, Angle, and Coordinate
In the coordinate plane, A(0, 0), B(6, 0), and C(6, 8). (a) Name the geometric figure formed by A, B, C using correct notation. (b) Find AB, BC, and AC using the distance formula. (c) Find m∠ABC. (d) Classify ∠ABC. (e) Find the midpoint of AC.
Common Unit Mistakes
Common Mistakes
Writing AB + BC = AC without checking that B is between A and C.
Verify B is between A and C before applying the Segment Addition Postulate.
Subtracting coordinates in different orders for rise and run: (y₂−y₁)/(x₁−x₂).
Always subtract in the same order: m = (y₂−y₁)/(x₂−x₁).
Forgetting to square both differences in the distance formula: d = √[(x₂−x₁)+(y₂−y₁)].
Square each difference: d = √[(x₂−x₁)² + (y₂−y₁)²].
Confusing midpoint (average of coordinates) with distance (square root formula).
Midpoint averages the coordinates. Distance uses the Pythagorean relationship.
Calling two angles complementary because they look adjacent in a diagram.
Complementary means their measures sum to 90°, regardless of position.
Assuming vertical angles are supplementary.
Vertical angles are congruent (equal), not supplementary.
Setting up a linear pair equation as ∠1 = ∠2 instead of ∠1 + ∠2 = 180°.
Linear pair angles are supplementary: their sum is 180°.
Changing the compass width between arcs when copying a segment or angle.
Keep the compass width fixed throughout each construction step.
Drawing the perpendicular bisector arcs with a radius smaller than half the segment.
Open the compass to more than half the segment length so the arcs intersect.
Checking only one pair of slopes to test collinearity of three points.
Calculate slopes for two pairs that share a common point; both must be equal.
Concluding lines are perpendicular because their slopes are negative: m₁ = −2, m₂ = −3.
Perpendicular lines have slopes whose product equals −1: m₁ · m₂ = −1.
Using the midpoint formula to find distance: M = √[(x₁+x₂)² + (y₁+y₂)²].
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2). Distance uses subtraction, not addition.
Naming an angle by its vertex alone when three angles share that vertex: ∠B.
Use three letters to name the angle unambiguously: ∠ABC.
Treating the angle bisector as the perpendicular bisector of the angle's sides.
The angle bisector divides the angle into two equal angles; it does not bisect the sides.
Check Your Understanding
1. A student says: "I can find the midpoint of a segment by using the distance formula and dividing by 2." Is this correct? Explain why or why not.
2. Two lines have slopes m₁ and m₂. Describe all three possible relationships (parallel, perpendicular, neither) in terms of m₁ and m₂, including the special cases of horizontal and vertical lines.
3. Explain why a compass-and-straightedge construction is considered more rigorous than a drawing made with a ruler and protractor.