Module 01 — Geometry Foundations · quiz worth 100 pts · 2026-09-04
The test mixes construction marks, angle-pair position, and proof reasons. Sorting the stem first keeps the lookalike choices from blending together.
Before calculating, name the bucket and say what evidence the diagram or proof line gives you.
Read arcs, endpoints, chord marks, and circle ticks.
Read opposite/adjacent or inside/outside plus same/opposite side.
Say what changed on the line: theorem, substitution, algebra, transitive link, or definition.
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This page teaches the ideas before it tests you. Move through each card in order. On every diagram, first name what you see; then say the rule; then do any math.
This page is only for Geometry Foundations Exam Part One. Its 15 original multiple-choice questions use material from lessons 01.01–01.09. It does not include triangle-congruence shortcuts or anything from Module 02.
Point, line, and plane are undefined terms. A point has no dimension. A line has one dimension and continues forever in both directions. A plane is flat, has two dimensions, and continues forever.
A segment has two endpoints. A ray has one endpoint and continues in one direction. Opposite rays share an endpoint and form a line.
A definition tells exactly what a term means. A postulate is accepted as true without proof. A theorem has been proven from definitions, postulates, and earlier results.
Example: “Through any two points, there is exactly one line” is the Points Postulate. “Vertical angles are congruent” is the Vertical Angles Theorem.
Draw a new ray. Open the compass to the full length of the given segment. Without changing that width, place the compass point on the new ray’s endpoint and swing an arc across the ray. The endpoint and the arc intersection form the congruent segment.
Draw a new ray. From the original vertex, draw an arc that crosses both sides of the original angle. Keep that radius and draw the same arc from the new vertex. Now copy the distance between the original arc’s two crossing points onto the new arc. Draw the second ray through the new intersection.
An angle copy transfers two distances, not a degree number.
Draw matching arcs from the old and new vertices without changing the compass.
Measure between the two places where the old arc hits the angle rays.
Transfer that chord to the new arc, then draw through the new crossing.
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Segment: use the same compass width—more than half the segment—from both endpoints. The paired arcs cross above and below. The line through those crossings cuts the segment in half.
Angle: make one arc from the vertex that hits both rays. From those two hits, draw equal interior arcs. The ray from the vertex through their crossing splits the angle in half.
Both constructions use equal paired arcs, but they begin in different places.
Start from its two endpoints; the compass must be wider than half the segment.
First mark both rays with one vertex arc, then start the equal interior arcs at those marks.
The line or ray through the paired-arc crossings creates the equal split.
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Parallel through a point: draw a transversal and copy the angle at its first intersection to the new point. Equal corresponding angles make the new line parallel.
Perpendicular: create two equal points on the given line, draw equal arcs from those points, and connect the arc crossings. The new line meets the original at 90°.
Ask whether the construction copies an angle or creates equal points around a line.
Copy a corresponding angle along a transversal.
Use equal arcs to locate a line that meets the original at 90 degrees.
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Keep the compass set to the circle’s radius and step around the circle to make six equal marks. Connect every other mark for an equilateral triangle. Connect all six neighboring marks for a regular hexagon.
For a square, construct two perpendicular diameters and connect the four places where the diameters meet the circle.
Vertical angles are opposite each other and are congruent. Adjacent angles share a side. A linear pair is adjacent and forms a straight line, so its measures add to 180°. Complementary angles add to 90°; supplementary angles add to 180°.
First ask: opposite or touching? Then ask: equal, 90°, or 180°?
When a transversal crosses two parallel lines, classify the pair by position:
Corresponding: same corner at each intersection. Alternate interior: inside, opposite sides. Alternate exterior: outside, opposite sides. Same-side interior: inside, same side.
Use the picture’s position before looking at any answer choices.
Do not memorize the angle numbers; use position.
Interior lies between the parallel lines. Exterior lies beyond them.
Alternate means opposite sides of the transversal. Same-side means the same side.
Matching corner positions at the two intersections are corresponding.
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If the lines are parallel, corresponding, alternate interior, and alternate exterior angles are congruent. Same-side interior angles are supplementary.
Memory check: three relationships give equal measures; the inside-and-same-side pair gives 180°.
Suppose same-side interior angles measure (6x + 12)° and (4x + 18)°. Same-side interior angles are supplementary, so begin with:
Combine like terms: 10x + 30 = 180. Subtract 30: 10x = 150. Divide by 10: x = 15. Substitute back to get 102° and 78°. Check: 102 + 78 = 180.
The geometry relationship comes before the algebra.
Same-side interior angles are supplementary when the lines are parallel.
Add the two expressions and set the sum equal to 180.
Substitute x into both expressions and make sure their measures total 180.
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Match the reason to the move you just made. A geometry theorem creates an angle relationship. Substitution replaces something with a known equal value. An Equality Property performs the same operation on both sides. Transitive links A to B and B to C. The definition of congruent angles changes equal measures into congruent angles.
Read the new line and ask what changed from the line before it.
Use the postulate, theorem, or definition that creates it.
Use substitution or the equality property matching the algebra operation.
Use transitive to link matches, or definition of congruence to change = into ≅.
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When the problem asks you to prove the Vertical Angles Theorem, you cannot cite that theorem as the reason. Build it from two straight-angle sums instead. Reveal the chain one line at a time.
For each multiple-choice question: 1. name the topic; 2. point to the diagram clue or proof move; 3. state the rule in your own words; 4. solve only if math is needed; 5. choose the answer that matches your reasoning.
If two choices look possible, compare their exact words: endpoint count, inside versus outside, congruent versus supplementary, or theorem versus equality property.
Open the 15-question Part One practice exam. Retry every miss after reading its explanation. The practice stays entirely inside Geometry Foundations material from lessons 01.01–01.09.
