7.3 Geometry

Building Blocks: Points, Lines, and Angles

A point is a location with no size. A line extends forever in two directions. A line segment is part of a line with two endpoints. A ray has one endpoint and extends forever in one direction.

• Parallel lines lie in the same plane and never meet.
• Perpendicular lines meet at a right (90°) angle.
• Intersecting lines cross at any angle.

Angles are classified as acute (< 90°), right (90°), obtuse (between 90° and 180°), straight (180°), or reflex (between 180° and 360°), as described in the previous section.

2-D Shapes and Their Properties

A polygon is a closed figure made of line segments. Polygons are named by side count: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8).

Triangles

Classified two ways. By sides:

• Equilateral — three equal sides (and three 60° angles).
• Isosceles — at least two equal sides (and two equal angles).
• Scalene — no equal sides.

By angles:

• Acute — three angles less than 90°.
• Right — one 90° angle.
• Obtuse — one angle greater than 90°.

Quadrilateral Family

A square is a special rectangle and a special rhombus. The classification is hierarchical: every square is also a rectangle, a parallelogram, and a rhombus, but not every rectangle is a square.

3-D Shapes

Solid figures have faces (flat polygons), edges (where two faces meet), and vertices (where edges meet).

• Prism: two congruent parallel bases; lateral faces are rectangles. A cube is a rectangular prism with six square faces.
• Pyramid: one polygonal base; triangular faces meet at an apex.
• Cylinder: two circular bases; one curved face.
• Cone: one circular base; one curved face ending at an apex.
• Sphere: no faces, edges, or vertices; every surface point is the same distance from the center.

For a rectangular prism, Euler's relationship holds: faces + vertices = edges + 2. A cube: 6 + 8 = 12 + 2.

Nets of 3-D Figures

A net is a 2-D pattern that folds into a 3-D figure. A cube has 11 distinct nets. Students benefit from cutting, folding, and visualizing which nets form which solids. The surface area of a prism is the sum of the face areas in its net.

Symmetry

Line symmetry means a figure can be folded along a line so the two halves match exactly. A rectangle has 2 lines of symmetry; a square has 4; an equilateral triangle has 3; a circle has infinitely many.

Rotational symmetry means a figure maps onto itself after a turn of less than 360°. An equilateral triangle has 120° rotational symmetry (3 positions match in a full turn); a square has 90° rotational symmetry (4 positions).

The Coordinate Plane

The coordinate plane has two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They meet at the origin (0, 0). The axes divide the plane into four quadrants.

In Grade 5, students graph in the first quadrant (positive x and y) using ordered pairs (x, y). To plot the point (3, 4): start at the origin, move 3 units right along the x-axis, then 4 units up parallel to the y-axis.

Real-world tasks include plotting points to make a rectangle, finding the distance between two points with the same x or same y coordinate, and reading a map with a coordinate grid.

Classifying Shapes by Properties

Upper elementary students sort shapes into hierarchies. Examples:

• All squares are rectangles, but only rectangles with equal sides are squares.
• All rhombuses are parallelograms, but only parallelograms with right angles are rectangles.
• A trapezoid (exactly one pair of parallel sides, in the definition used in most U.S. elementary materials) is not a parallelogram (which needs two pairs).

When students justify classifications using properties ("a square must have four right angles AND four equal sides"), they practice geometric reasoning the GACE expects teachers to model.

Worked Example: Plotting and Distance on the Coordinate Plane

A student draws a rectangle on a first-quadrant grid with vertices at (1, 1), (5, 1), (5, 4), and (1, 4). What are the side lengths and the perimeter?

• Bottom side: from (1, 1) to (5, 1) — same y, so the length is the difference of x-values: 5 − 1 = 4 units.
• Right side: from (5, 1) to (5, 4) — same x, so the length is the difference of y-values: 4 − 1 = 3 units.
• Opposite sides of a rectangle are equal, so the top is 4 units and the left is 3 units.
• Perimeter = 4 + 3 + 4 + 3 = 14 units. Area = 4 × 3 = 12 square units.

This example ties coordinate geometry back to perimeter and area: when two points share an x- or y-coordinate, the distance between them is a simple subtraction.

Combining Attributes: Triangle Sort

A single triangle can carry two labels, one from each family. A triangle with sides 5, 5, 8 is isosceles (two equal sides) and obtuse (the angle across from the 8 side is greater than 90°). A triangle with sides 6, 8, 10 is scalene and right (it satisfies the Pythagorean relation 6² + 8² = 10²). Giving students practice sorting the same triangle by both schemes strengthens the idea that classification depends on which attribute is in focus.