101 Plane Geometry Terms Every Engineering Board Exam Taker Must Know

INTRODUCTION

Plane geometry is one of those subjects that feels basic until you are staring at a board exam problem and cannot remember which theorem to use. The truth is, most of the plane geometry problems in the Mathematics engineering licensure exam are not conceptually hard. They reward familiarity. If you know the definitions and the key relationships cold, the problems become straightforward.

This list covers 101 essential plane geometry terms from the foundational ones like angle, line, and triangle, all the way to the more specific ones like Ptolemy’s theorem, the Euler line, and Brahmagupta’s formula for cyclic quadrilaterals. Each definition is written with the board exam in mind. You will not just find out what a term means, you will also see why it matters and how it shows up in problems.

Whether you are just starting your review or doing a final sweep before exam day, go through this list carefully. Pay extra attention to the circle theorems (inscribed angle, tangent-chord angle, chord-chord angle), the triangle centers (centroid, incenter, circumcenter, orthocenter), and the area formulas for non-standard shapes. These are the topics where boards questions cluster the most.

Take notes, flag the terms you are shaky on, and revisit them. Plane geometry rewards repetition and pattern recognition above all else.

The 101 Plane Geometry Terms and Definitions

1. Acute Angle

An angle that measures greater than 0° but less than 90°. Acute angles appear constantly in triangle problems, especially in acute triangles where all three angles are acute. On the board exam, knowing whether an angle is acute helps you determine which trigonometric identities and triangle formulas apply.

2. Acute Triangle

A triangle where all three interior angles are less than 90°. The orthocenter of an acute triangle falls inside the triangle. Board problems involving area, altitudes, and circumscribed circles often involve acute triangles, and recognizing the type of triangle early helps you choose the right formula.

3. Adjacent Angles

Two angles that share a common vertex and a common side but do not overlap. Their interiors are on opposite sides of the shared side. Adjacent angles are fundamental in problems involving angle sums, linear pairs, and polygon interior angle calculations.

4. Altitude of a Triangle

A perpendicular segment from a vertex of a triangle to the line containing the opposite side. Every triangle has three altitudes, and they all meet at a single point called the orthocenter. The altitude is used directly in the area formula: A = ½ × base × height.

5. Angle

The figure formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees or radians. In plane geometry, angles are classified as acute, right, obtuse, straight, or reflex. The board exam uses angle relationships extensively in polygon and circle problems.

6. Angle Bisector

A ray that divides an angle into two equal parts. In a triangle, the three angle bisectors meet at the incenter, which is the center of the inscribed circle. The angle bisector theorem states that it divides the opposite side in the ratio of the adjacent sides.

7. Angle Bisector Theorem

In a triangle, the angle bisector from a vertex divides the opposite side into two segments proportional to the adjacent sides. If the bisector from vertex A meets side BC at point D, then BD/DC = AB/AC. This theorem is useful in problems where a side is divided internally in a given ratio.

8. Angle in a Semicircle

Any inscribed angle that subtends a diameter of a circle is always exactly 90°. This is one of the most useful circle theorems on the board exam. If you see a triangle inscribed in a circle with one side as the diameter, the angle opposite that side is a right angle without exception.

9. Apothem

The perpendicular distance from the center of a regular polygon to the midpoint of any one of its sides. The apothem is used directly in the area formula for regular polygons: A = ½ × perimeter × apothem. The apothem is also the inradius of the regular polygon.

10. Arc

A connected portion of a circle’s circumference. A minor arc is less than a semicircle; a major arc is greater than a semicircle. Arc length is calculated as s = rθ, where r is the radius and θ is the central angle in radians. Arc measure in degrees equals the central angle that intercepts it.

11. Arc Length

The actual distance along the curved path of an arc. For a circle with radius r and central angle θ in radians, arc length s = rθ. In degrees, s = (θ/360) × 2πr. Board problems frequently ask for arc length given the radius and central angle, or ask you to find the central angle given the arc length.

12. Area of a Circle

Given by the formula A = πr², where r is the radius. Equivalently, A = πd²/4 in terms of diameter d. This formula is one of the most frequently used in plane geometry board problems, appearing in problems involving sectors, segments, and composite figures.

13. Area of a Rectangle

Computed as A = length × width, or A = lw. The rectangle is the simplest area formula case. Board problems often use rectangles as part of composite figures or in problems involving perimeter and area relationships where one dimension is expressed in terms of the other.

14. Area of a Regular Polygon

Computed as A = ½ × perimeter × apothem, or equivalently A = (ns²)/(4 tan(π/n)), where n is the number of sides and s is the side length. For common polygons, you can also derive the area by dividing the polygon into congruent triangles from the center.

15. Area of a Trapezoid

Given by A = ½(b₁ + b₂)h, where b₁ and b₂ are the two parallel sides (bases) and h is the perpendicular height between them. This formula appears frequently in board exams involving composite figures or problems where a quadrilateral has exactly one pair of parallel sides.

16. Area of a Triangle

The most commonly used triangle area formula is A = ½ × base × height. When the height is unknown, Heron’s formula A = √[s(s−a)(s−b)(s−c)] applies, where s = (a+b+c)/2 is the semi-perimeter. Another form is A = ½ab sin C, which uses two sides and the included angle.

17. Bisector

A line, ray, or segment that divides a geometric figure into two equal parts. An angle bisector divides an angle equally; a segment bisector divides a segment at its midpoint. Bisectors are central to constructions and to theorems involving incenters, circumcenters, and perpendicular bisectors in triangles.

18. Central Angle

An angle whose vertex is at the center of a circle and whose sides are two radii. The measure of a central angle equals the measure of the arc it intercepts. This direct relationship makes central angles the reference point for all other angle-arc relationships in circle geometry.

19. Centroid

The point where the three medians of a triangle intersect. The centroid is located two-thirds of the distance from each vertex to the midpoint of the opposite side. It is the center of mass of a uniform triangular lamina. Board problems often ask for the coordinates of the centroid or the length ratios along the medians.

20. Chord

A line segment with both endpoints lying on a circle. The diameter is the longest chord of a circle. The perpendicular from the center to a chord bisects the chord. Chord-related theorems such as the intersecting chords theorem and the chord-distance relationship are common board exam topics.

21. Chord-Chord Angle

The angle formed by two chords that intersect inside a circle. Its measure equals half the sum of the two arcs intercepted by the angle and its vertical angle. This formula is one of three angle-arc relationships for circles (the others involving angles formed outside the circle by secants or tangents). All three relationships appear regularly on the board exam.

22. Circle

The set of all points in a plane equidistant from a fixed center point. The constant distance is the radius r. Key measurements include circumference C = 2πr and area A = πr². The circle is the most extensively tested figure in plane geometry board problems, with topics ranging from inscribed angles to tangent lines.

23. Circumference

The total length around the boundary of a circle, given by C = 2πr or C = πd. It is the perimeter equivalent for circles. Board problems may ask you to find arc length as a fraction of the circumference, or to find the radius given the circumference.

24. Circumscribed Circle (Circumcircle)

A circle that passes through all the vertices of a polygon, typically a triangle. Its center is the circumcenter, found at the intersection of the perpendicular bisectors of the triangle’s sides. The circumradius R for a triangle is R = abc / (4A), where A is the triangle’s area.

25. Circumscribed Polygon

A polygon whose sides are all tangent to an inscribed circle (the incircle). Every regular polygon can be both inscribed in and circumscribed about a circle. For board problems, circumscribed polygons most often appear in problems involving the relationship between the inradius, area, and perimeter.

26. Collinear Points

Three or more points that all lie on the same straight line. If any set of three points forms a triangle with zero area, those points are collinear. This test, using the area formula or the slope check, is a common collinearity verification method in board exam problems.

27. Complementary Angles

Two angles whose measures add up to exactly 90°. Each angle is said to be the complement of the other. Complementary angles appear frequently in right triangle problems and in problems involving perpendicular lines. If one angle measures θ, its complement is (90° − θ).

28. Concentric Circles

Two or more circles that share the same center but have different radii. The region between two concentric circles is called an annulus or ring. Area of the annulus = π(R² − r²), where R and r are the outer and inner radii. Board problems involving concentric circles often ask for the area of the ring region.

29. Congruent Figures

Figures that have the same shape and the same size. For triangles, congruence is established by the postulates SSS, SAS, ASA, AAS, and HL (for right triangles). Congruent figures can be mapped onto each other by rigid motions:  translations, rotations, and reflections. Board problems use congruence to find unknown sides and angles.

30. Convex Polygon

A polygon where every interior angle is less than 180° and no side, when extended, enters the interior of the polygon. All regular polygons are convex. In a convex polygon, any line segment connecting two interior points lies entirely within the polygon. Convexity is assumed for most polygon problems on the board exam unless stated otherwise.

31. Cyclic Quadrilateral

A quadrilateral inscribed in a circle, meaning all four vertices lie on the circle. The key property: opposite angles are supplementary (add up to 180°). The area is given by Brahmagupta’s formula: A = √[(s−a)(s−b)(s−c)(s−d)], where s is the semi-perimeter. This is a moderately advanced but testable topic.

32. Diagonal of a Polygon

A line segment connecting two non-adjacent vertices of a polygon. A polygon with n sides has n(n−3)/2 diagonals. For a quadrilateral, there are 2 diagonals; for a pentagon, 5; for a hexagon, 9. Board problems sometimes ask for the number of diagonals in a given polygon, which is a straightforward formula application.

33. Diameter

The longest chord of a circle, passing through the center. Its length equals twice the radius: d = 2r. The diameter also defines the inscribed angle theorem: any angle inscribed in a semicircle (subtending the diameter) is a right angle. Diameter problems on the board exam often involve circles inscribed in or circumscribed about other figures.

34. Equilateral Triangle

A triangle with all three sides equal in length and all three angles equal to 60°. It is both equilateral and equiangular. The height of an equilateral triangle with side s is h = (√3/2)s, and its area is A = (√3/4)s². These formulas appear regularly in board exam problems involving regular shapes.

35. Euler Line

The line that passes through three triangle centers: the orthocenter (H), the centroid (G), and the circumcenter (O). The centroid divides the segment from the orthocenter to the circumcenter in a 2:1 ratio. The Euler line is a theoretical result that occasionally appears in advanced board problems and is worth knowing as a connecting concept between triangle centers.

36. Exterior Angle of a Polygon

The angle formed by one side of a polygon and the extension of an adjacent side at each vertex. The sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides. For a regular polygon with n sides, each exterior angle measures 360°/n.

37. Exterior Angle of a Triangle

The angle formed between one side of a triangle and the extension of an adjacent side. An exterior angle equals the sum of the two non-adjacent (remote) interior angles. This theorem is one of the most useful shortcuts in triangle problems and appears frequently in board exams.

38. Geometric Mean

For two positive numbers a and b, the geometric mean is √(ab). In right triangle geometry, the altitude to the hypotenuse creates two smaller triangles each similar to the original. The altitude is the geometric mean of the two segments it creates on the hypotenuse. This relationship is a classic board exam topic.

39. Heron’s Formula

A formula for the area of a triangle given all three side lengths: A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. Named after Hero of Alexandria. Heron’s formula is particularly useful when no height is given or easily computed. Board exams use it for scalene triangle area problems.

40. Hypotenuse

The side of a right triangle opposite the right angle. It is always the longest side of the triangle. In the Pythagorean theorem a² + b² = c², the hypotenuse is c. Problems involving the hypotenuse appear in almost every plane geometry board exam  whether directly through the Pythagorean theorem or through the special triangle ratios.

41. Incenter

The point where the three angle bisectors of a triangle meet. The incenter is equidistant from all three sides and serves as the center of the inscribed circle (incircle). The inradius r can be found from r = A/s, where A is the triangle’s area and s is the semi-perimeter.

42. Inscribed Angle

An angle formed by two chords of a circle that share an endpoint on the circle. The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. Two inscribed angles intercepting the same arc are equal. This theorem is one of the most tested circle theorems in board exams.

43. Inscribed Circle (Incircle)

The largest circle that fits inside a triangle, tangent to all three sides. Its center is the incenter and its radius is the inradius r = A/s, where A is the triangle area and s is the semi-perimeter. For a regular polygon, the inscribed circle has its center at the polygon’s center and its radius equal to the apothem.

44. Inscribed Polygon

A polygon whose vertices all lie on a circle. Every triangle can be inscribed in a circle (circumcircle). For a regular polygon inscribed in a circle of radius R, the side length is s = 2R sin(π/n), where n is the number of sides. Board problems often involve relating the inscribed polygon’s dimensions to the circle’s radius.

45. Interior Angle

An angle formed inside a polygon by two adjacent sides. For any convex polygon with n sides, the sum of all interior angles is (n − 2) × 180°. For a regular polygon, each interior angle measures (n − 2) × 180° / n. Memorizing this formula is essential since interior angle sum problems are standard board exam material.

46. Isosceles Trapezoid

A trapezoid with non-parallel sides (legs) that are equal in length. The base angles are equal, and the diagonals are equal in length. It is symmetric about the perpendicular bisector of the parallel sides. Board problems involving isosceles trapezoids often use these symmetry properties to find unknown lengths.

47. Isosceles Triangle

A triangle with at least two equal sides. The angles opposite the equal sides (base angles) are also equal. The perpendicular from the vertex angle to the base bisects both the base and the vertex angle, making it also the median and altitude from that vertex. Isosceles triangle properties are heavily tested on the board exam.

48. Kite

A quadrilateral with two pairs of consecutive equal sides. One diagonal bisects the other at right angles, and one diagonal bisects the vertex angles. The area of a kite is A = ½d₁d₂, where d₁ and d₂ are the lengths of the diagonals. Kites appear in quadrilateral classification and area problems on the board exam.

49. Law of Cosines

A generalization of the Pythagorean theorem for any triangle: c² = a² + b² − 2ab cos C. It applies when you know two sides and the included angle (SAS) or all three sides (SSS). When angle C = 90°, it reduces to the Pythagorean theorem. Board exams use this law in oblique triangle solving problems.

50. Law of Sines

States that in any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It is used to solve triangles when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). The SSA case can produce zero, one, or two solutions and this ambiguous case is frequently tested on the board exam.

101. Viviani’s Theorem

States that for any point inside an equilateral triangle, the sum of the perpendicular distances from that point to the three sides equals the altitude of the triangle. This is an elegant geometric result that occasionally appears in advanced board exam problems or competition-type questions involving equilateral triangles.

CONCLUSION

You have just gone through 101 of the most important plane geometry terms for the Mathematics engineering board exam. The key takeaway is this: plane geometry is not about memorizing hundreds of isolated facts. It is about understanding a small set of core relationships: angle sums, the Pythagorean theorem, circle angle theorems, and similar triangle ratios, and recognizing which one applies to each problem.

For your final review, prioritize these clusters. For triangles: the area formula in all three forms, Heron’s formula, the law of sines and cosines, and the properties of the four triangle centers. For circles: the inscribed angle theorem, the tangent-perpendicular-radius relationship, the chord-chord angle theorem, and the secant-tangent relationships. For polygons: the interior angle sum formula, diagonal count formula, and the area formula using the apothem for regular polygons.

For practice problems on all these topics, head over to our Plane Geometry Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.

Everything else in this list supports those core topics. If you are solid on the fundamentals, you will be in a strong position to handle whatever plane geometry problems appear on exam day. Good luck, and keep at it.