LET Elementary Mathematics — Plane and Solid GeometryStudy Notes

Angles are measured in degrees and form the building blocks of all geometric reasoning. Understanding angle classifications and relationships is essential for solving triangle, polygon, and parallel-line problems on the LET. **Angle Classifications by Size:** • Acute angle: less than 90° (examples: 35°, 65°, 89°) • Right angle: exactly 90° (marked with a small square) • Obtuse angle: between 90° and 180° (examples: 120°, 145°) • Straight angle: exactly 180° (a straight line) • Reflex angle: between 180° and 360° (examples: 200°, 270°) **Angle Pairs and Their Relationships:** Complementary angles are two angles whose measures sum to 90°. For example, if one angle is 35°, its complement is 90° − 35° = 55°. These pairs appear frequently in right-triangle problems and in Grade 4–6 geometry lessons about perpendicular lines. Supplementary angles are two angles whose measures sum to 180°. If one angle is 120°, its supplement is 180° − 120° = 60°. This relationship is crucial when angles lie on a straight line (a linear pair) or when dealing with parallel lines cut by a transversal. Vertical angles are opposite angles formed when two straight lines intersect. A key property: vertical angles are always equal. If two lines cross and one angle measures 75°, the angle directly opposite it also measures 75°, while the two adjacent angles each measure 180° − 75° = 105°. **Angles Formed by a Transversal Crossing Parallel Lines:** When a transversal (a line that crosses two other lines) cuts two parallel lines, eight angles are created. Three critical relationships hold: 1. Corresponding angles (angles in the same position at each intersection) are equal. If the transversal makes a 65° angle at the first parallel line on the upper-left, it makes a 65° angle on the upper-left at the second parallel line. 2. Alternate interior angles (angles on opposite sides of the transversal, between the parallel lines) are equal. If an interior angle on the left side of the transversal is 65°, the interior angle on the right side is also 65°. 3. Co-interior angles (also called consecutive interior angles or same-side interior angles; angles on the same side of the transversal, between the parallel lines) are supplementary. If one is 65°, the other is 180° − 65° = 115°. These relationships are essential for solving problems about angles in trapezoids and for understanding why the angle-sum rule for polygons works.

Triangles are among the most frequently tested shapes on the LET. Mastering triangle properties, congruence, similarity, and the Pythagorean theorem is non-negotiable for both passing the exam and teaching Grade 4–6 geometry with confidence. **Triangle Classification by Sides:** Equilateral triangles have all three sides equal in length and all three interior angles equal to 60°. Isosceles triangles have two sides of equal length and two equal base angles (the angles opposite the equal sides). Scalene triangles have all three sides of different lengths and all three angles different. **Triangle Classification by Angles:** Acute triangles have all three angles less than 90°. Right triangles have one 90° angle. Obtuse triangles have one angle greater than 90°. **The Triangle Angle-Sum Rule:** The sum of the interior angles of any triangle is always 180°. This is one of the most important rules in geometry and recurs throughout the LET. If a triangle has angles measuring 50° and 70°, the third angle is 180° − 50° − 70° = 60°. **Exterior Angle Theorem:** An exterior angle of a triangle (an angle formed by extending one side beyond a vertex) equals the sum of the two remote interior angles (the two angles not adjacent to the exterior angle). For example, if a triangle has interior angles of 50° and 70°, an exterior angle at the third vertex equals 50° + 70° = 120°. **The Triangle Inequality:** For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This rule quickly eliminates impossible triangles: sides 3, 4, and 8 cannot form a triangle because 3 + 4 = 7, which is not greater than 8. **Congruent Triangles:** Two triangles are congruent if they have the same shape and size (though one may be rotated or reflected). Congruence is proven by five methods: 1. **SSS (Side-Side-Side):** All three sides of one triangle are equal to the three sides of another. 2. **SAS (Side-Angle-Side):** Two sides and the included angle (the angle between those sides) of one triangle equal two sides and the included angle of another. 3. **ASA (Angle-Side-Angle):** Two angles and the included side of one triangle equal two angles and the included side of another. 4. **AAS (Angle-Angle-Side):** Two angles and a non-included side of one triangle equal two angles and the corresponding non-included side of another. 5. **HL (Hypotenuse-Leg):** In two right triangles, the hypotenuse and one leg of one triangle equal the hypotenuse and one leg of the other. **Similar Triangles:** Two triangles are similar if they have the same shape but different sizes. In similar triangles, corresponding angles are equal and corresponding sides are proportional (in the same ratio). Similarity is proven by three methods: 1. **AA (Angle-Angle):** Two angles of one triangle equal two angles of another (the third angles must also be equal by the angle-sum rule). 2. **SSS (proportional):** All three sides of one triangle are proportional to the three sides of another. 3. **SAS (proportional):** Two sides of one triangle are proportional to two sides of another, and the included angles are equal. A key property of similar figures: if the ratio of corresponding sides is k, then the ratio of areas is k². For instance, if one triangle has sides twice as long as another similar triangle, the larger triangle has 2² = 4 times the area. **The Pythagorean Theorem:** In any right triangle, if a and b are the lengths of the legs (the two sides that form the right angle) and c is the length of the hypotenuse (the side opposite the right angle), then a² + b² = c². This formula is used constantly on the LET and in classroom problems about diagonal distances, ladder heights, and map distances. Common Pythagorean triples (sets of three whole numbers that satisfy the theorem) include: • 3-4-5 (and multiples: 6-8-10, 9-12-15, 12-16-20) • 5-12-13 (and multiples: 10-24-26) • 8-15-17 These triples are worth memorizing because they appear in test items and make calculations fast and error-free. The Pythagorean theorem also identifies right triangles: if a² + b² = c², the triangle has a right angle opposite side c.

Polygons are closed figures made of straight line segments. On the LET, most polygon questions focus on angle sums and properties of regular (all sides and angles equal) polygons. **Interior Angle Sum:** For any polygon with n sides, the sum of the interior angles is (n − 2) × 180°. This formula works because any n-sided polygon can be divided into (n − 2) triangles, each contributing 180° to the total. Examples: • Triangle (n = 3): (3 − 2) × 180° = 1 × 180° = 180°. • Quadrilateral (n = 4): (4 − 2) × 180° = 2 × 180° = 360°. • Pentagon (n = 5): (5 − 2) × 180° = 3 × 180° = 540°. • Hexagon (n = 6): (6 − 2) × 180° = 4 × 180° = 720°. • Octagon (n = 8): (8 − 2) × 180° = 6 × 180° = 1080°. **Interior Angle of a Regular Polygon:** A regular polygon has all sides equal and all angles equal. Each interior angle of a regular n-gon measures: Interior angle = [(n − 2) × 180°] / n For a regular hexagon: [(6 − 2) × 180°] / 6 = 720° / 6 = 120° each. For a regular octagon: [(8 − 2) × 180°] / 8 = 1080° / 8 = 135° each. **Exterior Angle Sum:** The sum of the exterior angles of any convex polygon is always 360°. An exterior angle is the angle between one side and the extension of an adjacent side. This is a elegant, universal property and holds regardless of the number of sides. For a regular n-gon, each exterior angle measures 360° / n. For a regular hexagon: 360° / 6 = 60° each. For a regular octagon: 360° / 8 = 45° each. **Special Quadrilaterals:** Quadrilaterals are 4-sided polygons. The interior angles sum to 360°. Common types include: • **Square:** All four sides equal; all four angles 90°. Diagonals are equal and perpendicular. • **Rectangle:** Opposite sides equal; all four angles 90°. Diagonals are equal but not necessarily perpendicular. • **Rhombus:** All four sides equal; opposite angles equal. Diagonals are perpendicular but not necessarily equal. • **Parallelogram:** Opposite sides parallel and equal; opposite angles equal. Diagonals bisect each other. • **Trapezoid:** Exactly one pair of parallel sides (in the Philippine K-12 definition). Co-interior angles on the same leg sum to 180°. Relationships: Every square is a rectangle and a rhombus; every rectangle and rhombus is a parallelogram. Recognizing these inclusions quickly answers classification questions.

Perimeter is the distance around a figure (measured in linear units like cm, m); area is the space inside a figure (measured in square units like cm², m²). On the LET, questions ask whether you need to find perimeter (for fencing, border, or trim) or area (for flooring, painting, or tiling). Reading the question carefully is essential. **Squares and Rectangles:** For a square with side length s: • Perimeter = 4s (four equal sides) • Area = s² For a rectangle with length l and width w: • Perimeter = 2(l + w) or 2l + 2w • Area = l × w Common error: Students sometimes multiply all four sides (l + w + l + w is correct, but multiplying pairs causes confusion). Use 2(l + w) as a clear formula. **Triangles:** For a triangle with base b and height h: • Area = (1/2) × b × h (the height is perpendicular to the base) • Perimeter = sum of all three sides The height must be perpendicular to the base; if it is not given directly, you may need to use the Pythagorean theorem or other properties to find it. **Parallelograms:** For a parallelogram with base b, height h, and side length a: • Area = b × h (not base × side; the height must be perpendicular) • Perimeter = 2(a + b) (sum of two adjacent sides, doubled) Common error: Using the slanted side instead of the perpendicular height leads to wrong area. Emphasize that height is always perpendicular. **Trapezoids:** For a trapezoid with parallel sides a and b, height h, and non-parallel sides c and d: • Area = (1/2) × (a + b) × h (average of parallel sides, times height) • Perimeter = a + b + c + d (sum of all four sides) The height is perpendicular to the parallel sides; the non-parallel sides are often slanted. **Circles:** For a circle with radius r and diameter d = 2r: • Circumference = 2πr or πd • Area = πr² Using π ≈ 3.1416 is standard, but when the radius is a multiple of 7 (e.g., r = 7, 14, 21), use π = 22/7 because the 7s cancel and calculations are cleaner: Area of circle with r = 7: A = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 cm². **Composite Figures:** Many test items feature composite figures (an L-shaped lot, a rectangle topped with a triangle, a circle inside a square). Strategy: 1. Identify the component shapes. 2. Calculate the area or perimeter of each. 3. For area: add areas if combining, subtract if removing a piece. 4. For perimeter: trace the outer boundary carefully (internal edges are not part of the perimeter). **Unit Consistency:** A critical error on the LET: mixing units. If a figure's dimensions are in meters and centimeters, convert all to one unit before calculating. For example, a rectangle with length 2 m and width 50 cm must have both converted to 200 cm and 50 cm (or 2 m and 0.5 m) before multiplying. Forgetting this step leads to answers off by factors of 10 or 100.

Three-dimensional solids—cubes, prisms, cylinders, cones, spheres, and pyramids—are common on the LET. Surface area is the total area of all outer faces (measured in square units); volume is the space inside (measured in cubic units). Confusing these two is a frequent error. **Cubes:** For a cube with edge length s: • Volume = s³ (length × width × height, all equal) • Surface area = 6s² (six square faces, each with area s²) **Rectangular Prisms:** For a rectangular prism (box) with length l, width w, and height h: • Volume = l × w × h • Surface area = 2(lw + lh + wh) (sum of areas of three pairs of opposite rectangular faces) Alternatively, surface area = 2lw + 2lh + 2wh. The two expressions are equivalent. **Cylinders:** For a cylinder with radius r and height h: • Volume = πr²h (area of circular base, times height) • Surface area = 2πr² + 2πrh (two circular bases plus lateral—the curved—surface) The lateral surface area is 2πrh; think of it as the circumference (2πr) times the height (h). **Cones:** For a cone with base radius r, height h, and slant height l (the distance from the apex along the surface to a point on the base circle): • Volume = (1/3)πr²h (one-third of a cylinder with the same base and height) • Surface area = πr² + πrl (circular base plus lateral surface) Note: The slant height l is not the height h; they satisfy l² = h² + r² (a Pythagorean relationship). **Spheres:** For a sphere with radius r: • Volume = (4/3)πr³ • Surface area = 4πr² **Pyramids:** For a pyramid with base area B and height h: • Volume = (1/3)Bh (one-third of a prism with the same base and height) • Surface area = B + lateral faces (area of base plus sum of areas of triangular faces) The lateral surface area depends on the shape of the base and the slant heights of the triangular faces. For a square pyramid with base edge a and slant height l, the lateral area is 4 × (1/2) × a × l = 2al. **Key Relationship: Cone and Pyramid Volumes:** Both cones and pyramids have volumes equal to (1/3) times the volume of a cylinder or prism with the same base and height. This is a central concept on the LET: a cone is to a cylinder as a pyramid is to a prism, with a factor of 1/3. **Unit Handling:** Volume requires cubic units; surface area requires square units. If a dimension is given in meters, volume is in m³ (cubic meters) and surface area is in m² (square meters). Mixing units leads to errors; always convert before calculating. **Common Formula Pitfalls:** 1. Using radius instead of diameter (or vice versa) in formulas. 2. Forgetting the (1/3) factor in cone and pyramid volumes. 3. Confusing slant height with height in cone and pyramid problems. 4. Mixing square units (area) with cubic units (volume). 5. Forgetting to convert units before calculating.

Circles appear frequently on the LET, both as standalone questions and as components of composite figures. Understanding the parts of a circle and the relationships between them is essential. **Circle Terminology:** The **center** is the fixed point from which all points on the circle are equidistant. The **radius** (r) is the distance from the center to any point on the circle. The **diameter** (d) passes through the center and equals 2r. A **chord** is any line segment connecting two points on the circle; the diameter is the longest chord. An **arc** is a part of the circumference. A **central angle** has its vertex at the center and its sides are radii; the arc it subtends is called a minor arc (if less than 180°) or major arc (if more than 180°). A **tangent** is a line that touches the circle at exactly one point. A key property: a tangent is perpendicular to the radius at the point of tangency. A **secant** is a line that intersects the circle at two points. **Circumference and Area:** Circumference (the distance around the circle) is C = 2πr or C = πd. Area is A = πr². The constant π ≈ 3.1416 (or 22/7 when convenient) is the ratio of any circle's circumference to its diameter. From circumference, you can find radius: r = C / (2π). From area, you can find radius: r = √(A / π). These inverse relationships are tested on the LET. **Arc Length and Sector Area:** An arc is part of the circumference. If a central angle is θ (in degrees), the arc length is: Arc length = (θ / 360°) × 2πr = (θ / 360°) × C A sector is a pie-shaped region bounded by two radii and an arc. Its area is: Sector area = (θ / 360°) × πr² = (θ / 360°) × A Note: The same fraction θ / 360° scales both the circumference and the area. **Inscribed and Circumscribed Figures:** A polygon is inscribed in a circle if all its vertices lie on the circle. A polygon is circumscribed about a circle if all its sides are tangent to the circle. These concepts appear in advanced geometry but may be tested in context (e.g., "A square is inscribed in a circle of radius 5 cm; find the area of the square"). **Angles in a Circle:** An inscribed angle (vertex on the circle, sides are chords) is half the central angle that subtends the same arc. This relationship is key to several advanced problems but is less common on the elementary LET. **Practical Applications:** Circle problems in the LET often involve real-world contexts: the circumference of a wheel, the area of a circular garden, the arc length of a section of a track, or the sector area of a pizza slice.

Coordinate geometry uses the Cartesian plane (with x and y axes) to represent and analyze geometric figures. On the LET, coordinate questions typically involve the distance formula and sometimes the midpoint formula. **The Distance Formula:** The distance between two points (x₁, y₁) and (x₂, y₂) is: d = √[(x₂ − x₁)² + (y₂ − y₁)²] This formula is derived from the Pythagorean theorem: the horizontal distance is |x₂ − x₁| and the vertical distance is |y₂ − y₁|; the straight-line distance (hypotenuse) is the square root of the sum of their squares. **The Midpoint Formula:** The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) is: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) The midpoint is simply the average of the x-coordinates and the average of the y-coordinates. **Slope of a Line:** The slope m of the line through (x₁, y₁) and (x₂, y₂) is: m = (y₂ − y₁) / (x₂ − x₁) Slope measures the steepness of a line. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (if one has slope m, the other has slope −1/m). **Applications:** Coordinate geometry questions on the LET often ask for the length of a segment, the coordinates of a midpoint, or whether two line segments are perpendicular or parallel. These questions reinforce the Pythagorean theorem and angle concepts in a coordinate setting.

On the LET, geometry questions test both formula recall and reasoning. Success requires strategy and vigilance about common errors. **Strategy 1: Read the Question Carefully** Does the question ask for perimeter or area? Length or volume? The penalty for answering the wrong quantity is automatic failure on that item. Underline or highlight key words: "fencing" (perimeter), "flooring" (area), "filling" (volume), "around" (perimeter), "inside" (area or volume). **Strategy 2: Check Units** If dimensions are in mixed units (5 m and 200 cm), convert all to one unit before calculating. Area must be in square units, volume in cubic units. If a dimension is in meters, area is in m², not cm² or km². **Strategy 3: Use Known Formulas; Do Not Derive Them** On a timed exam, deriving a formula wastes time. Memorize the core formulas (angle sums, Pythagorean theorem, perimeter and area of standard shapes, volume and surface area of solids) and recall them directly. Practice using them under time pressure. **Strategy 4: Recognize Pythagorean Triples** The 3-4-5, 5-12-13, and 8-15-17 triples (and their multiples) appear frequently. If you see sides 6, 8, and 10, immediately recognize the 3-4-5 triple scaled by 2; the hypotenuse is 10 (no calculation needed). This saves time and reduces arithmetic errors. **Strategy 5: Draw a Diagram** For word problems, sketch the figure (even roughly). Label all given information and identify what you're solving for. A diagram prevents misinterpretation and often suggests which formula to use. **Strategy 6: Verify Your Answer** After solving, check your answer against the original question. Does the magnitude make sense? (A triangle's area of 10,000 m² might be unreasonably large for a described lot.) Does the unit match the question's expectation? Substituting your answer back into the formula is a final check. **Common Pitfalls:** 1. **Confusing radius and diameter.** If a circle has radius 5 cm, its area is πr² = π(5)² = 25π cm², NOT π(10)² = 100π cm². 2. **Using side length instead of height for area.** For a triangle or trapezoid, the height must be perpendicular to the base, not the slanted side. 3. **Forgetting the (1/3) factor for cones and pyramids.** A cone's volume is (1/3)πr²h, not πr²h. This factor is non-negotiable. 4. **Mixing square and cubic units.** Surface area is in square units (m²), volume in cubic units (m³). Answering "50 m²" when the question asks for volume is incorrect. 5. **Arithmetic errors with π.** Use π ≈ 3.14 or 22/7 consistently. Avoid rounding π to 3 unless explicitly told; this introduces unacceptable error. 6. **Not converting units.** A field 2 km long and 500 m wide must have both converted (2,000 m and 500 m) before multiplying. Mixing leads to answers off by factors of 10 or 100. 7. **Misidentifying which polygon property to use.** Recall: interior angle sum for all polygons, exterior angle sum (always 360°) for specific questions, and each interior angle of a regular polygon requires dividing the sum by the number of sides. 8. **Overlooking composite figures.** If a figure is L-shaped or has a cut-out, split it into standard shapes, compute each, and add (or subtract for cut-outs). 9. **Confusing height and slant height in cones.** Height h is perpendicular from apex to base; slant height l is along the surface. They relate by l² = h² + r². 10. **Assuming all quadrilaterals are special.** Not all four-sided figures are squares or rectangles. Use the general interior angle sum of 360° for any quadrilateral.