01.09 Geometry Foundations Review

What 01.09 is for

The Canvas practice test is designed to feel like the module test. This walkthrough and its quiz cover the same skills with different examples.

If proofs still feel new, review 01.06 and 01.07 first. The portal does not reproduce or submit 01.09 or 01.10.

Construction cheat sheet

• Copy a segment: new ray, compass set to the given length, one arc from the new endpoint.
• Copy an angle: new ray, matching vertex arcs, then copy the chord between the original hits.
• Bisect a segment: equal-width arcs above and below from both endpoints; line through the crossings.
• Bisect an angle: vertex arc through both rays, equal inner arcs from those hits, ray through their crossing.
• Parallel: a transversal and a copied corresponding angle.
• Perpendicular on the line: two line hits around the point, then paired arcs above and below.
• Perpendicular off the line: two line hits from the off-line point, then an arc crossing on the far side.

Construction cheat sheet (2)

• Regular hexagon: compass set to the radius, six ticks, connect every tick.
• Equilateral triangle: the same six ticks, connect every other tick.
• Square: one diameter and its perpendicular bisector, then connect the four circle hits.

Copy-an-angle order

1. Draw a ray whose endpoint is the new vertex.
2. From the original vertex, swing an arc through both rays.
3. Keep that width and swing the matching arc from the new vertex.
4. Copy the chord between the two original hits onto the new arc.
5. Draw the second ray through the new chord mark.

A step cannot use a point that has not been constructed yet.

Words in the same choice list

• Complementary add to 90°; supplementary add to 180°. They need not touch.
• Adjacent share a side; vertical are opposite, nonadjacent, and congruent.
• A linear pair is adjacent and supplementary.
• A midpoint splits a segment into congruent pieces; an angle bisector splits an angle into congruent angles.
• Segment Addition joins segment pieces; the Addition Property of Equality changes both sides of an equation.

Postulates, theorems, and properties

Postulates: two points determine one line; two intersecting lines meet in one point; two intersecting planes meet in one line; three noncollinear points determine one plane; Segment Addition; Angle Addition; Linear Pair.

Theorems: vertical angles are congruent. With parallel lines, corresponding, alternate interior, and alternate exterior angles are congruent; same-side interior angles are supplementary.

Postulates, theorems, and properties (2)

Proof reasons: Given; definitions of midpoint, congruence, bisector, and perpendicular; reflexive and transitive properties; substitution; addition, subtraction, multiplication, and division properties; distributive property; combine like terms.

The same proof can appear as two columns, a paragraph, or a flow chart. The logical links do not change.

A sample proof skeleton

Given: JK ∥ LM. A transversal cuts them at O and P. m∠KOP = 7x and m∠OPL = 110°. The angles are same-side interior.

7x + 110 = 180 by the Same-Side Interior Angles Theorem and substitution.
7x = 70 by the Subtraction Property of Equality.
x = 10 by the Division Property of Equality.

If the question asks for the number sentence, choose 7x + 110 = 180. If it asks for the reason, choose Same-Side Interior, not Corresponding.