United States
Hong KongClassifying figures and angle relationships
Skill 1 reads "Identify and classify simple two- and three-dimensional figures according to their mathematical properties". Classification is by property, not by name, so the hierarchy is the point: every square is a rectangle and no rectangle has to be a square.
Two words in the skill statement decide how to prepare. Classify and properties. You are not being asked to name a shape from a picture; you are being asked what follows from its properties.
Triangles, two ways
| By sides | Property | By angles | Property |
|---|---|---|---|
| Equilateral | Three equal sides | Acute | All angles less than a right angle |
| Isosceles | At least two equal sides | Right | One right angle |
| Scalene | No equal sides | Obtuse | One angle greater than a right angle |
Every triangle has one label from each column, which is a thing items are built on. A triangle can be isosceles and right at the same time, and an option claiming those are exclusive is wrong.
The angles of any triangle sum to 180 degrees. That single fact settles a surprising number of items, because it lets you find a missing angle from two given ones.
The quadrilateral hierarchy
This is where classification questions live, and it runs downward: quadrilateral, then parallelogram, then rectangle and rhombus, then square.
- A parallelogram has two pairs of parallel sides
- A rectangle is a parallelogram with right angles
- A rhombus is a parallelogram with four equal sides
- A square is both, so it is a rectangle and a rhombus at once
- A trapezoid has exactly one pair of parallel sides, so it is not a parallelogram
Read that list downward and every statement of the form "every X is a Y" is true. Read it upward and every one is false. That asymmetry is the whole question type.
Which statement is true of all rhombuses?
- All four angles are right angles.
- All four sides are equal in length.
- The diagonals are equal in length.
- Exactly one pair of sides is parallel.
Angle relationships
Four pairings cover almost everything at this level.
| Relationship | What it says |
|---|---|
| Complementary | Two angles that add to a right angle |
| Supplementary | Two angles that add to a straight angle, 180 degrees |
| Vertical | Opposite angles at a crossing, always equal |
| Corresponding | Matching angles on parallel lines cut by a transversal, equal |
The parallel-lines case is the one worth drawing once. When a transversal crosses two parallel lines, every angle it makes is equal to one of two values, and those two values are supplementary. Knowing that, you can label a whole diagram from a single given angle.
Three-dimensional figures
The rule says simple three-dimensional figures. Cubes, rectangular prisms, cylinders, cones, spheres and pyramids, classified by faces, edges and vertices.
A cube has six faces, twelve edges and eight vertices. That is worth knowing on sight because counting them from a picture under time pressure is where mistakes happen, and because the same counts hold for any rectangular prism.
How much this is worth
One skill of four in a competency we estimate at about seven of the 35 questions. That estimate is our arithmetic on the number of skills the rule states, not a published figure, so treat this as one or two items.
It is still worth an hour, because classification questions are the fastest on the subtest once you know the hierarchy, and speed banked here funds the algebra questions, which are more than twice as numerous on the same estimate.
Common questions
Is a square a rectangle on the FTCE?
Yes. A square is a parallelogram with right angles and four equal sides, so it is both a rectangle and a rhombus. Every square is a rectangle and no rectangle has to be a square, and classification items are built on exactly that asymmetry.
Can a triangle be both isosceles and right?
Yes. Triangles carry one label from the side classification and one from the angle classification, and the two are independent. An option claiming that a right triangle cannot be isosceles is a distractor rather than a rule.
What are complementary and supplementary angles?
Complementary angles add to a right angle. Supplementary angles add to a straight angle, which is 180 degrees. Vertical angles, formed opposite each other where two lines cross, are always equal, and corresponding angles on parallel lines are too.
What three-dimensional shapes are tested?
The rule says simple figures, which means cubes, rectangular prisms, cylinders, cones, spheres and pyramids, classified by faces, edges and vertices. A cube and any rectangular prism both have six faces, twelve edges and eight vertices.