Geometric Reasoning and Constructions: ACT Math Lesson

Work with converses, inverses, contrapositives, counterexamples, two-column proofs, and compass-and-straightedge constructions.

Conditional statements and their relatives

A conditional statement, "If $p$ , then $q$ ," has three related forms. The converse swaps the two parts: "If $q$ , then $p$ ." The inverse negates both parts without swapping: "If not $p$ , then not $q$ ." The contrapositive both negates and swaps: "If not $q$ , then not $p$ ." A conditional statement and its contrapositive always share the same truth value; the converse and inverse do not automatically share that guarantee.

Form	Wording	Always same truth value as the original?
Original	If $p$ , then $q$	(itself)
Converse	If $q$ , then $p$	No
Inverse	If not $p$ , then not $q$	No
Contrapositive	If not $q$ , then not $p$	Yes

Counterexamples

A counterexample disproves a statement by satisfying its hypothesis while violating its conclusion. Testing whether a converse is true means looking for exactly this kind of case: a figure or number that meets the "if" condition without meeting the "then" condition. One counterexample is enough to prove a statement false, but no number of examples proves a statement true.

Two-column proofs

A two-column proof pairs each statement with its justification, a definition, postulate, theorem, or given fact, in a separate column. The reflexive property (a quantity equals itself), the symmetric property (an equation can be reversed), and the Segment or Angle Addition Postulate (parts sum to a whole) are common justifications on this section.

Compass-and-straightedge constructions

A handful of constructions come up repeatedly: bisecting a segment or an angle, constructing a perpendicular through a point, copying a segment or angle, and building an equilateral triangle from one given side. Each relies on drawing arcs of equal radius to locate points that are exactly equidistant, which is what makes the construction exact instead of a visual estimate.

Bisect an angle: draw an arc centered at the vertex crossing both sides, then draw equal arcs from those two crossing points to find a point on the bisector.
Perpendicular bisector (and midpoint): draw equal-radius arcs centered at each endpoint so that they intersect above and below the segment, then connect the intersections.
Equilateral triangle: draw two arcs with radius equal to the given side, centered at each endpoint, and connect their intersection to both endpoints.
Copy an angle: preserve the arc's radius and the distance between the two points where the arc crosses the angle's sides.

Do not trust the picture

A diagram is evidence only for what is actually marked or stated, not for how it happens to look. Do not assume two lines are parallel because they look parallel, two sides are equal because they look equal, or an angle is $90^\circ$ because it looks square. Reasoning problems on this section are built around exactly this assumption: use only the labels, tick marks, and relationships the problem actually gives you, and treat everything else in the figure as undetermined.

Where this appears as a model

Consider a rule enforced through its logical opposite. A city ordinance stating that a building over 50 feet requires a sprinkler system has the contrapositive that a building without a sprinkler system does not exceed 50 feet, letting an inspector confirm compliance without measuring height directly.

Working with conditional statements

Worked example. Write the contrapositive of "If a polygon is a triangle, then it has three sides."

The contrapositive negates and swaps both parts of the original statement.

"If a polygon does not have three sides, then it is not a triangle."

Testing a converse with a counterexample

Worked example. Is the converse of "All squares are rectangles" true? If not, give a counterexample.

The converse is "All rectangles are squares." Look for a rectangle that is not a square.

A 4-by-8 rectangle satisfies the hypothesis (it is a rectangle) but not the conclusion (it is not a square). The converse is false.

Worked example. The statement "If two angles are vertical angles, then they are congruent" is true. Is its converse also true?

Recognize. Testing a converse means swapping the hypothesis and conclusion, then checking whether the new statement always holds or has a counterexample.

Set up. Converse: "If two angles are congruent, then they are vertical angles."

Solve. Counterexample: two angles of an equilateral triangle are congruent ( $60^\circ$ each) but are not vertical angles.

Check. The counterexample satisfies the converse's hypothesis (the angles are congruent) but not its conclusion (they are not vertical), which is exactly what a valid counterexample requires.

Best route. Pure reasoning. No calculation is needed once a counterexample is found.

Common error. Assuming that because the original conditional statement is true, its converse must be true as well.

Strategy summary

Label $p$ and $q$ before writing any related statement, then apply "swap," "negate," or both.
Only the contrapositive is guaranteed to match the original's truth.
To disprove something, find one case that makes the "if" true and the "then" false.
In constructions and proofs, ask what makes each step exact: equal radii, a definition, or a stated fact.

Geometric Reasoning and Constructions