Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry Syllabus

Quantitative Reasoning: Geometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Reasoning: Geometry in Graduate Record Examinations (GRE), not a summary of it.

1. 1

   Lines, Angles and Triangles

   3 topics

   • Angle relationships
     • Parallel lines and transversals
     • Vertical and supplementary angles
   • Triangle properties
     • Angle-sum and triangle inequality
     • Similar and congruent triangles
   • Right triangles
     • Pythagorean theorem
     • Special right triangles (45-45-90, 30-60-90)
2. 2

   Polygons and Circles

   3 topics

   • Quadrilaterals and polygons
     • Area and perimeter formulas
     • Interior and exterior angle sums
   • Circle properties
     • Circumference and area
     • Arcs, sectors and chords
     • Inscribed and central angles
   • Composite and shaded-region figures
     • Decomposing complex shapes
     • Finding shaded areas
3. 3

   Three-Dimensional Geometry

   3 topics

   • Surface area and volume of solids
     • Rectangular solids and cubes
     • Cylinders
   • Diagonals and 3D distance
     • Space diagonal of a box
   • Visualising and unfolding solids
     • Nets of common solids
4. 4

   Coordinate Geometry

   3 topics

   • Distance and midpoint
     • Distance formula
     • Midpoint formula
   • Equations of circles and lines
     • Standard circle equation
   • Slope, intercepts and graph interpretation
     • Identifying coordinates from graphs

Quantitative Reasoning: Geometry flashcards for Graduate Record Examinations (GRE)

24 of 50 cards from the Quantitative Reasoning: Geometry deck — real questions with worked answers.

1. When two lines intersect, what is the relationship between the two pairs of opposite (vertical) angles?

   Vertical angles are equal (congruent). The two angles directly across from each other at the intersection have the same measure.

2. What is the sum of two angles that form a linear pair (supplementary angles on a straight line)?

   $180^{\circ}$. Angles that combine to a straight line are supplementary, so they sum to $180^{\circ}$.

3. Define complementary angles and give their sum.

   Two angles are complementary if their measures add to $90^{\circ}$.

4. When a transversal crosses two parallel lines, what is true of alternate interior angles and of corresponding angles?

   Both pairs are equal. Alternate interior angles are congruent and corresponding angles are congruent; co-interior (same-side interior) angles are supplementary, summing to $180^{\circ}$.

5. What is the sum of the interior angles of any triangle?

   $180^{\circ}$.

6. State the Exterior Angle Theorem for a triangle.

   An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

7. In any triangle, how does the length of a side relate to the angle opposite it?

   The largest angle lies opposite the longest side and the smallest angle lies opposite the shortest side; equal sides lie opposite equal angles.

8. State the Triangle Inequality Theorem.

   The sum of the lengths of any two sides of a triangle must be greater than the length of the third side: $a + b > c$ for all sides.

9. What are the side and angle properties of an equilateral triangle?

   All three sides are equal and all three interior angles equal $60^{\circ}$.

10. What defines an isosceles triangle and what follows about its angles?

    It has at least two equal sides; the angles opposite those equal sides (the base angles) are also equal.

11. What is the formula for the area of a triangle with base $b$ and height $h$?

    $$A = \frac{1}{2}bh$$

12. State the Pythagorean Theorem for a right triangle with legs $a$, $b$ and hypotenuse $c$.

    $$a^{2} + b^{2} = c^{2}$$

13. List three common Pythagorean triples used to identify right triangles quickly.

    $(3,4,5)$, $(5,12,13)$, and $(8,15,17)$, along with their multiples such as $(6,8,10)$.

14. What are the side ratios of a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle?

    The sides are in ratio $1 : 1 : \sqrt{2}$ (leg : leg : hypotenuse).

15. What are the side ratios of a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle?

    The sides are in ratio $1 : \sqrt{3} : 2$ (side opposite $30^{\circ}$ : side opposite $60^{\circ}$ : hypotenuse).

16. In a right triangle, what is the relationship between the altitude to the hypotenuse and the two segments it creates?

    The altitude is the geometric mean of the two hypotenuse segments: $h = \sqrt{p \cdot q}$, where $p$ and $q$ are the segment lengths.

17. What is the sum of the interior angles of any quadrilateral?

    $360^{\circ}$.

18. List the defining properties of a parallelogram.

    Opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other.

19. What additional properties does a rectangle have beyond those of a parallelogram?

    All four angles are $90^{\circ}$ and the diagonals are equal in length (and still bisect each other).

20. What distinguishes a rhombus from a general parallelogram?

    All four sides are equal, and its diagonals are perpendicular bisectors of each other and bisect the vertex angles.

21. What is the area of a trapezoid with parallel sides $b_1$ and $b_2$ and height $h$?

    $$A = \frac{1}{2}(b_1 + b_2)h$$

22. What is the formula for the sum of interior angles of a polygon with $n$ sides?

    $$S = (n - 2)\times 180^{\circ}$$

23. What is the measure of each interior angle of a regular polygon with $n$ sides?

    $$\text{Interior angle} = \frac{(n-2)\times 180^{\circ}}{n}$$

24. What is the sum of the exterior angles of any convex polygon, one per vertex?

    $360^{\circ}$, regardless of the number of sides.

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