ACT Math study guide · Chapter 29
Geometric Reasoning and Constructions
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.
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.
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.
A handful of compass-and-straightedge 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.
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° 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: 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° 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.
▸ Practice
Question 1
- AA. If a shape is not a square, then it does not have four right angles.
- BB. If a shape has four right angles, then it is a square.
- CC. If a shape has four right angles, then it is a rectangle.
- DD. If a shape does not have four right angles, then it is not a square.
Question 2
- FF. If the ground is not wet, then it is not raining.
- GG. If it is raining, then the ground is not wet.
- HH. If it is not raining, then the ground is not wet.
- JJ. If the ground is wet, then it is raining.
Question 3
- AA. If a number is not even, then it is not divisible by 2.
- BB. If a number is not divisible by 2, then it is not even.
- CC. If a number is divisible by 2, then it is even.
- DD. If a number is odd, then it is divisible by 2.
4. A conditional statement “If p, then q” is true. Which related statement must also be true?
F) If q, then p. G) If not p, then not q. H) If p, then not q. J) If not q, then not p.
Question 5
- AA. Constructing a parallel line through an endpoint
- BB. Copying the segment length twice in a row
- CC. Constructing the perpendicular bisector of the segment
- DD. Bisecting one of the segment's endpoint angles
Question 6
- FF. Draw two arcs of different radii from each side
- GG. Draw a straight line connecting the two sides of the angle
- HH. Draw an arc centered at the vertex that crosses both sides of the angle
- JJ. Measure the angle directly with a protractor
Question 7
- AA. 40
- BB. 3
- CC. 1.6
- DD. 13
Question 8
- FF. Yes, because every parallelogram has four sides.
- GG. No; a parallelogram with no right angles is a counterexample.
- HH. Yes, both statements are always true together.
- JJ. No; a rectangle is a counterexample.
Question 9
- AA. Using the compass to mark two points on the line equidistant from the given point, then constructing the perpendicular bisector of the segment between them
- BB. Measuring 90° from the point using a protractor
- CC. Drawing an arc from the point that crosses the line at exactly one spot
- DD. Drawing a line parallel to the given line through the point
Question 10
- FF. A diagram illustrating each step
- GG. A restatement of the given information
- HH. The final answer to the problem
- JJ. The justification, such as a definition, postulate, or theorem, for each statement
Question 11
- AA. The triangle must be equilateral.
- BB. The triangle is isosceles, but not necessarily equilateral.
- CC. No conclusion can be drawn about the triangle.
- DD. The triangle must be a right triangle.
Question 12
- FF. The exact degree measure read from a protractor
- GG. The distance between the two points where an arc crosses the angle's two sides
- HH. The distance from the vertex to the arc's center
- JJ. The total length of both sides of the angle
Question 13
- AA. If AB = CD, then CD = AB
- BB. AB + BC = AC
- CC. AB = AB
- DD. AB = CD (given)
Question 14
- FF. The definition of supplementary angles, which sum to 180°
- GG. The Angle Addition Postulate
- HH. The definition of vertical angles, which are equal
- JJ. The definition of complementary angles, which sum to 90°
Question 15
- AA. Bisecting the given side and drawing a perpendicular through its midpoint
- BB. Drawing a single arc centered at one endpoint only
- CC. Copying the angle at only one endpoint of the side
- DD. Drawing two arcs of the same radius as the given side, centered at each endpoint, and connecting their intersection to both endpoints
Question 16
- FF. A rectangle
- GG. A square
- HH. A rhombus with no right angles
- JJ. An equilateral triangle
▸ Answers and Explanations
Show the answers and explanations
1: B. The converse of “If p, then q” swaps the hypothesis and conclusion: “If a shape has four right angles, then it is a square.” Choice A states the inverse instead, negating both parts of the original statement instead of swapping them.
2: F. The contrapositive of “If p, then q” negates and swaps both parts: “If the ground is not wet, then it is not raining.” Choice J states the converse instead, swapping the two parts without negating them.
3: A. The inverse of “If p, then q” negates both parts without swapping them: “If a number is not even, then it is not divisible by 2.” Choice B states the contrapositive instead, both negating and swapping the two parts.
4: J. A conditional statement and its contrapositive always share the same truth value, so if “If p, then q” is true, “If not q, then not p” must also be true. Choice F states the converse, which is not guaranteed to share the same truth value as the original statement.
5: C. The perpendicular bisector of a segment crosses that segment at its exact midpoint, which is how this construction locates a midpoint. Choice D describes bisecting an angle at an endpoint, a different construction that does not locate the segment's midpoint.
6: H. Bisecting an angle begins by drawing an arc centered at the vertex that intersects both sides of the angle, creating two points used for the rest of the construction. Choice F describes drawing two arcs of unequal radii, which would not create the symmetric points needed to bisect the angle accurately.
7: D. By the Segment Addition Postulate, AC = AB + BC = 5 + 8 = 13. Choice A multiplies the two given lengths instead of adding them.
8: G. A parallelogram with no right angles satisfies the hypothesis (it is a parallelogram) but not the conclusion (it is not a rectangle), making it a valid counterexample. Choice F assumes the converse must be true simply because both shapes share a side count, without testing it against a counterexample.
9: A. Marking two points on the line equidistant from the given point, then constructing the perpendicular bisector of the segment connecting them, produces a line through the original point that is perpendicular to the given line. Choice D describes constructing a parallel line, which does not produce the required perpendicular relationship.
10: J. A two-column proof pairs each statement with its justification, a definition, postulate, or theorem, in the right-hand column. Choice F describes a diagram, which is not what the right-hand column of a two-column proof contains.
11: B. Two equal angles guarantee that the triangle is isosceles, since the sides opposite equal angles are themselves equal, but nothing about the third angle is established without more information. Choice A assumes two equal angles must extend to all three, which is not guaranteed without further information about the triangle.
12: G. Copying an angle preserves the arc's radius and the distance between the two points where the arc crosses the angle's sides, which together recreate the same angle measure. Choice F assumes a protractor reading is used, but a compass-and-straightedge construction preserves distances, not a measured degree value.
13: C. The reflexive property states that any quantity is equal to itself: AB = AB. Choice A states the symmetric property instead, which allows an equation to be reversed instead of showing a quantity equal to itself.
14: F. Supplementary angles are defined as two angles whose measures sum to 180°, which is what justifies finding ∠B from ∠A in this step. Choice J describes complementary angles, which sum to 90°, not the 180° relationship used here.
15: D. Drawing two arcs with the same radius as the given side, centered at each endpoint, and connecting their intersection point to both endpoints produces a triangle with all three sides equal. Choice A describes bisecting the side, a construction used to find a perpendicular bisector, not to build an equilateral triangle.
16: H. A rhombus with no right angles has four equal sides, satisfying the statement's hypothesis, but is not a square, violating its conclusion, making it a valid counterexample. Choice F selects a rectangle, which does not necessarily have four equal sides and therefore does not even satisfy the statement's hypothesis.