Practical Geometry - Advanced applications of quadrilateral construction

In lower classes, geometry revolves around understanding basic shapes, line segments, and angles. As we progress to Class 8 Mathematics, geometry transforms from a descriptive study into a constructive, operational science. Practical Geometry focuses on translating geometric properties into precise physical drawings using standard mathematical instruments: a ruler, a pair of compasses, and a protractor.

While a triangle can be uniquely determined using just 3 independent measurements (such as SSS, SAS, or ASA criteria), a quadrilateral is a four-sided polygon that possesses greater flexibility. A set of 4 side lengths alone cannot lock a quadrilateral into a fixed, rigid shape—it can flex into infinitely many configurations. Consequently, five independent measurements are mathematically necessary to construct a unique quadrilateral.

In advanced applications of quadrilateral construction, the given measurements are not always straightforward. Often, problem statements provide fewer than five explicit numerical values. In such scenarios, students must act as mathematical detectives, applying the inherent geometric properties of special quadrilaterals—such as parallelograms, rhombuses, rectangles, squares, and kites—or utilizing the Angle Sum Property to uncover the missing implicit measurements required for construction.

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In-Depth Conceptual Breakdown

1. The Principle of Unique Determination

Why do we need exactly 5 independent measurements to construct a unique quadrilateral?

Consider four rigid rods hinged together at their endpoints to form a four-sided frame. If you press on two opposite corners, the frame easily deforms without changing the lengths of any of its four sides. To make this structure rigid, you must insert a diagonal brace across two opposite vertices.

This diagonal divides the quadrilateral into two distinct triangles. Because a single triangle requires 3 independent elements to be uniquely constructed, the first triangle uses 3 measurements (e.g., two sides and one diagonal). The second triangle, sharing that diagonal as a common base, requires 2 additional measurements (e.g., the remaining two sides).

$\text{Total Measurements Required} = 3 \text{ (First Triangle)} + 2 \text{ (Second Triangle)} = 5 \text{ Measurements}$

        C
       / \
      /   \
     /     \
    D-------B
     \     /
      \   /
       \ /
        A

Figure Conceptualization: Quadrilateral $ABCD$ split into $\triangle ABD$ and $\triangle BCD$ by diagonal $BD$.

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2. Standard Quadrilateral Construction Cases

Before mastering advanced applications, let us review the primary scenarios where 5 measurements are directly provided:

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3. Advanced Application I: Exploiting Inherent Geometric Properties

In advanced textbook problems, you will encounter questions like: "Construct a rhombus whose diagonals are $6\text{ cm}$ and $8\text{ cm}$." At first glance, only two numbers are given! However, the word rhombus carries hidden geometric information.

By applying the mathematical properties of special quadrilaterals, we extract the remaining required measurements:

+-------------------+----------------------------------------------------+---------------------------------------------------+
| Special Polygon   | Inherent Geometric Properties                      | Hidden Measurements Unlocked                      |
+-------------------+----------------------------------------------------+---------------------------------------------------+
| Parallelogram     | - Opposite sides are equal and parallel.           | - Giving 2 adjacent sides defines all 4 sides.    |
|                   | - Opposite angles are equal.                       | - Adjacent angles are supplementary.              |
|                   | - Diagonals bisect each other.                     |                                                   |
+-------------------+----------------------------------------------------+---------------------------------------------------+
| Rhombus           | - All 4 sides are equal.                           | - Giving 1 side defines all 4 sides.              |
|                   | - Diagonals bisect each other at right angles      | - Diagonals form 4 right-angled triangles at the  |
|                   |   ($90^\circ$).                                    |   intersection point (midpoint).                  |
+-------------------+----------------------------------------------------+---------------------------------------------------+
| Rectangle         | - Opposite sides are equal and parallel.           | - Giving 2 adjacent sides defines all 4 sides.    |
|                   | - All interior angles equal $90^\circ$.            | - All 4 interior angles are known ($90^\circ$).   |
|                   | - Diagonals are equal and bisect each other.       |                                                   |
+-------------------+----------------------------------------------------+---------------------------------------------------+
| Square            | - All 4 sides are equal.                           | - Giving 1 side or 1 diagonal is sufficient to    |
|                   | - All interior angles equal $90^\circ$.            |   deduce all sides, angles, and diagonals.        |
|                   | - Diagonals are equal and bisect at $90^\circ$.    |                                                   |
+-------------------+----------------------------------------------------+---------------------------------------------------+
| Kite              | - Two pairs of equal adjacent sides.               | - Diagonals intersect perpendicularly.            |
|                   | - One diagonal perpendicularly bisects the other.  | - One diagonal bisects opposite vertex angles.   |
+-------------------+----------------------------------------------------+---------------------------------------------------+

Key Technique: Constructing a Rhombus Using Perpendicular Bisectors

When only two diagonal lengths ($d_1$ and $d_2$) of a rhombus are given:

1. Draw line segment $AC = d_1$.
2. Construct the perpendicular bisector of $AC$, intersecting $AC$ at midpoint $O$.
3. Along the perpendicular bisector, cut off arcs of length $\frac{d_2}{2}$ both above and below $AC$ to locate vertices $B$ and $D$.
4. Connect $A, B, C, D$ to complete the rhombus.

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4. Advanced Application II: Using the Angle Sum Property

When a problem provides 2 adjacent sides and 3 angles, but one of the given angles is not adjacent to the given sides, direct construction becomes impossible without prior calculation.

Recall the Angle Sum Property of a Quadrilateral: $\sum \angle = \angle A + \angle B + \angle C + \angle D = 360^\circ$

Deductive Step:

If you are given side $AB$, side $BC$, $\angle A$, $\angle C$, and $\angle D$, you cannot directly build $\angle D$ because vertex $D$'s position in space is initially unknown.

To solve this:

1. Calculate the missing angle $\angle B$: $\angle B = 360^\circ - (\angle A + \angle C + \angle D)$
2. Draw base $AB$.
3. Construct $\angle A$ at vertex $A$ and $\angle B$ at vertex $B$.
4. Mark vertex $C$ along the ray of $\angle B$ using distance $BC$.
5. Construct $\angle C$ at vertex $C$. The ray of $\angle C$ will intersect the ray of $\angle A$ precisely at vertex $D$.

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Real-World Applications

1. Structural Truss Engineering and Architecture

In civil engineering, structures made of four-sided components (like rectangular building frames or quadrilateral bridges) are naturally unstable against shear stress (wind or earthquakes). Engineers convert flexible quadrilaterals into rigid frameworks by installing diagonal cross-beams. Understanding quadrilateral construction helps engineers calculate exact structural lengths, joint angles, and load distribution paths.

       UNSTABLE FRAME                    RIGID TRUSS FRAME
       +--------------+                  +--------------+
       |              |                  | \            |
       |              |   --------->     |  \  Diagonal |
       |              |                  |   \ Brace    |
       +--------------+                  +--------------+

2. Land Surveying and Plot Boundary Mapping

Surveyors routinely map irregular land plots bounded by four non-parallel sides. Since physical obstructions (trees, ponds, structures) often prevent direct measurement across every diagonal, surveyors measure two convenient boundary lengths and three accessible internal/external angles using a transit or modern total station. They then use the Angle Sum Property and geometric construction principles to generate accurate scaled land deeds and cadastral maps.

3. Computer Graphics and CAD Software Systems

Computer-Aided Design (CAD) software and 2D vector graphics engines (like Adobe Illustrator or AutoCAD) rely on parametric geometry algorithms. When a designer inputs dynamic constraints—such as making two line segments parallel, forcing a $90^\circ$ corner, or fixing diagonal lengths—the software uses the exact geometric construction algorithms discussed in this chapter to render 2D quadrilateral meshes in real time.

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Step-by-Step Solved Textbook Examples

Example 1: Rhombus Construction from Diagonals

Problem: Construct a rhombus $ABCD$ whose diagonals are $AC = 6.4\text{ cm}$ and $BD = 5.2\text{ cm}$.

Mathematical Reasoning:

• In a rhombus, diagonals bisect each other at right angles ($90^\circ$).
• Midpoint $O$ divides $AC$ into $AO = OC = \frac{6.4}{2} = 3.2\text{ cm}$.
• Midpoint $O$ divides $BD$ into $BO = OD = \frac{5.2}{2} = 2.6\text{ cm}$.

Step-by-Step Construction Procedure:

1. Rough Sketch: Draw a quick quadrilateral labeled $ABCD$, showing diagonals intersecting at $O$ at $90^\circ$. Mark $AC = 6.4\text{ cm}$ and $BD = 5.2\text{ cm}$.
2. Step 1: Using a ruler, draw line segment $AC = 6.4\text{ cm}$.
3. Step 2: With $A$ as center and a compass radius greater than half of $AC$ ($> 3.2\text{ cm}$), draw arcs above and below $AC$. With $C$ as center and the same radius, draw intersecting arcs. Draw the line passing through these arc intersections. This line $XY$ is the perpendicular bisector of $AC$, intersecting $AC$ at midpoint $O$.
4. Step 3: Calculate half of diagonal $BD$: $\frac{BD}{2} = \frac{5.2\text{ cm}}{2} = 2.6\text{ cm}$
5. Step 4: Set compass radius to $2.6\text{ cm}$. Place the compass point at midpoint $O$ and draw an arc intersecting the perpendicular bisector $XY$ above $AC$ at point $B$.
6. Step 5: Keeping the compass radius at $2.6\text{ cm}$, place the compass point at $O$ and draw an arc intersecting $XY$ below $AC$ at point $D$.
7. Step 6: Join line segments $AB$, $BC$, $CD$, and $DA$.

                  B (Top Vertex)
                  |
                  |
        A --------+-------- C   (Diagonal AC = 6.4 cm)
                  | O (Midpoint)
                  |
                  D (Bottom Vertex)

Final Answer Statement:

$\text{Rhombus } ABCD \text{ is the required figure, with diagonals } AC = 6.4\text{ cm} \text{ and } BD = 5.2\text{ cm}.$

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Example 2: Advanced Parallelogram Construction

Problem: Construct a parallelogram $MORE$ where $MO = 6\text{ cm}$, $OR = 4.5\text{ cm}$, and $\angle M = 70^\circ$.

Mathematical Reasoning:

• In a parallelogram, opposite sides are equal: $ER = MO = 6\text{ cm}$ $ME = OR = 4.5\text{ cm}$
• Adjacent angles are supplementary: $\angle O = 180^\circ - \angle M = 180^\circ - 70^\circ = 110^\circ$

Step-by-Step Construction Procedure:

1. Rough Sketch: Draw a four-sided figure $MORE$. Mark $MO = 6\text{ cm}$, $OR = 4.5\text{ cm}$, $RE = 6\text{ cm}$, $EM = 4.5\text{ cm}$, and $\angle M = 70^\circ$.
2. Step 1: Draw base line segment $MO = 6\text{ cm}$ using a ruler.
3. Step 2: At point $M$, construct an angle of $70^\circ$ using a protractor. Draw the ray $MX$.
4. Step 3: At point $O$, construct an angle of $110^\circ$ using a protractor. Draw the ray $OY$.
5. Step 4: Set your compass to a radius of $4.5\text{ cm}$. With $M$ as center, cut an arc on ray $MX$ to locate vertex $E$. Thus, $ME = 4.5\text{ cm}$.
6. Step 5: With the same compass radius of $4.5\text{ cm}$ and $O$ as center, cut an arc on ray $OY$ to locate vertex $R$. Thus, $OR = 4.5\text{ cm}$.
7. Step 6: Join point $E$ and point $R$ with a straight line segment.

Verification Check:

Measure segment $ER$ with a ruler. It will equal $6\text{ cm}$. Measure $\angle E$; it will equal $110^\circ$, confirming opposite angles are equal ($\angle M = \angle R = 70^\circ$ and $\angle O = \angle E = 110^\circ$).

Final Answer Statement:

$\text{Parallelogram } MORE \text{ is uniquely constructed with adjacent sides } 6\text{ cm} \text{ and } 4.5\text{ cm} \text{ and interior angle } \angle M = 70^\circ.$

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Example 3: Quadrilateral Construction Using Angle Sum Deduction

Problem: Construct a quadrilateral $HELP$ where $HE = 6\text{ cm}$, $EL = 4.5\text{ cm}$, $\angle H = 60^\circ$, $\angle L = 105^\circ$, and $\angle P = 120^\circ$.

Mathematical Reasoning:

We are given two adjacent sides ($HE$ and $EL$). Therefore, we need the angles at vertices $H$, $E$, and $L$ to build rays from the ends of these sides. However, we are given $\angle P$ instead of $\angle E$.

Apply the Angle Sum Property of a Quadrilateral: $\angle H + \angle E + \angle L + \angle P = 360^\circ$ $60^\circ + \angle E + 105^\circ + 120^\circ = 360^\circ$ $285^\circ + \angle E = 360^\circ$ $\angle E = 360^\circ - 285^\circ = 75^\circ$

Now we have the necessary sequence: side $HE$, angle $\angle E$, side $EL$, angle $\angle L$, and angle $\angle H$.

Step-by-Step Construction Procedure:

1. Rough Sketch: Draw quadrilateral $HELP$. Label $HE = 6\text{ cm}$, $EL = 4.5\text{ cm}$, $\angle H = 60^\circ$, $\angle E = 75^\circ$, $\angle L = 105^\circ$, and $\angle P = 120^\circ$.
2. Step 1: Draw line segment $HE = 6\text{ cm}$.
3. Step 2: At vertex $H$, construct an angle of $60^\circ$ using a ruler and compass (or protractor) and extend ray $HX$.
4. Step 3: At vertex $E$, construct an angle of $75^\circ$ ($90^\circ - 15^\circ$ constructible via compass, or measured via protractor) and extend ray $EY$.
5. Step 4: Set compass radius to $4.5\text{ cm}$. With $E$ as center, mark an arc along ray $EY$ to locate vertex $L$.
6. Step 5: At vertex $L$, construct an angle of $105^\circ$ with respect to segment $EL$. Extend ray $LZ$.
7. Step 6: The point of intersection between ray $HX$ (from vertex $H$) and ray $LZ$ (from vertex $L$) is vertex $P$.

         P (Intersection of rays HX and LZ)
        / \
       /   \
      /     \  L
     /       \ /
    H---------E

Verification Check:

Measure $\angle P$ with a protractor. It will measure exactly $120^\circ$.

Final Answer Statement:

$\text{Quadrilateral } HELP \text{ is constructed with computed angle } \angle E = 75^\circ.$

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Example 4: Constructing a Square Given Only Its Diagonal

Problem: Construct a square $READ$ whose diagonal $RD = 5.4\text{ cm}$.

Mathematical Reasoning:

• A square is a special rhombus with equal diagonals that bisect each other at $90^\circ$.
• Thus, diagonal $EA = RD = 5.4\text{ cm}$.
• The intersection point $O$ of the diagonals divides each diagonal into halves: $RO = OD = EO = OA = \frac{5.4}{2} = 2.7\text{ cm}$

Step-by-Step Construction Procedure:

1. Step 1: Draw line segment $RD = 5.4\text{ cm}$ using a ruler.
2. Step 2: Construct the perpendicular bisector $XY$ of line segment $RD$. Label the midpoint as $O$.
3. Step 3: Set the compass radius to $2.7\text{ cm}$ ($\frac{5.4}{2}$).
4. Step 4: Place the compass point at midpoint $O$. Cut an arc on the upper ray of $XY$ to mark vertex $E$.
5. Step 5: Keeping the same $2.7\text{ cm}$ radius and compass point at $O$, cut an arc on the lower ray of $XY$ to mark vertex $A$.
6. Step 6: Join $R$ to $E$, $E$ to $D$, $D$ to $A$, and $A$ to $R$.

Final Answer Statement:

$\text{Square } READ \text{ is constructed with diagonal } RD = 5.4\text{ cm} \text{ and side lengths } \approx 3.8\text{ cm}.$

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Common Student Mistakes to Avoid

1. Constructing Arcs from the Wrong Reference Point

• The Error: When 3 sides and 2 diagonals are given, students often draw arcs from arbitrary vertices, causing arcs that fail to intersect or create wrong shapes.
• The Correction: Always construct a base triangle first using 3 known values (such as 2 sides and 1 diagonal). Use the endpoints of that base triangle as explicit anchor centers for subsequent arcs.

2. Reading Protractor Scale Misalignments

• The Error: Reading the outer scale instead of the inner scale on a protractor (or vice versa), resulting in constructing an obtuse angle ($120^\circ$) instead of the intended acute angle ($60^\circ$).
• The Correction: Remember that acute angles must visually appear sharper than a $90^\circ$ right angle, while obtuse angles must appear wider. Always double-check your angle visually after marking it.

       ACUTE (< 90°)            OBTUSE (> 90°)
           /                        \
          /                          \
         /____                        \____

3. Misapplying Bisector Cuts for Special Quadrilaterals

• The Error: When constructing a rhombus from two diagonals $d_1$ and $d_2$, students sometimes cut off the full length of $d_2$ on either side of the midpoint $O$ instead of half ($\frac{d_2}{2}$). This doubles the vertical height and results in a non-rhombus shape.
• The Correction: Always explicitly calculate $\frac{d_1}{2}$ and $\frac{d_2}{2}$ in your preliminary rough work before picking up your compass.

4. Omitting the Rough Sketch and Labeling

• The Error: Skipping the rough sketch leads to confusion about which angles are adjacent and which sides are included, frequently leading to restarted drawings or incorrect layouts.
• The Correction: Draw a neat, freehand rough sketch in the margin before every construction. Mark all given dimensions, calculated angles, and diagonal lines directly onto this sketch.

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Practice Questions for Self-Assessment

Question 1

Construct a rectangle $MINE$ where side $MI = 7\text{ cm}$ and diagonal $ME = 8.5\text{ cm}$.

<details> <summary><strong>Click to View Complete Solution</strong></summary>

Solution:

1. Geometric Deductions:

   • In rectangle $MINE$, opposite sides are equal ($NE = MI = 7\text{ cm}$).
   • All interior angles are right angles ($\angle M = \angle I = \angle N = \angle E = 90^\circ$).
   • $\triangle MIE$ forms a right-angled triangle with base $MI = 7\text{ cm}$, angle $\angle I = 90^\circ$, and hypotenuse $ME = 8.5\text{ cm}$.

2. Step-by-Step Construction:

   • Step 1: Draw line segment $MI = 7\text{ cm}$.
   • Step 2: At point $I$, construct an angle of $90^\circ$ using a compass or protractor. Extend ray $IX$.
   • Step 3: Set compass radius to $8.5\text{ cm}$. Place compass point at $M$ and draw an arc intersecting ray $IX$ at vertex $E$.
   • Step 4: At point $M$, construct an angle of $90^\circ$ and extend ray $MY$.
   • Step 5: Set compass radius to length $IE$ (measured from the drawing, or using $7\text{ cm}$ from $E$ parallel to $MI$). Place compass point at $E$ and draw an arc of radius $7\text{ cm}$ intersecting ray $MY$ at vertex $N$.
   • Step 6: Join $N$ to $E$.

3. Final Answer: $\text{Rectangle } MINE \text{ is constructed with sides } 7\text{ cm} \text{ and } \approx 4.8\text{ cm}, \text{ and diagonal } 8.5\text{ cm}.$

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Question 2

Construct a quadrilateral $PQRS$ where $PQ = 4\text{ cm}$, $QR = 5\text{ cm}$, $RS = 4.5\text{ cm}$, $\angle Q = 100^\circ$, and $\angle R = 80^\circ$.

<details> <summary><strong>Click to View Complete Solution</strong></summary>

Solution:

1. Geometric Analysis:

   • This problem falls under Case 4: 3 Sides & 2 Included Angles.
   • Known sides: $PQ = 4\text{ cm}$, $QR = 5\text{ cm}$, $RS = 4.5\text{ cm}$.
   • Included angles: $\angle Q$ (between $PQ$ and $QR$) and $\angle R$ (between $QR$ and $RS$).

2. Step-by-Step Construction:

   • Step 1: Draw the base line segment $QR = 5\text{ cm}$.
   • Step 2: At point $Q$, draw a ray $QX$ making an angle of $100^\circ$ with $QR$ using a protractor.
   • Step 3: At point $R$, draw a ray $RY$ making an angle of $80^\circ$ with $QR$ using a protractor.
   • Step 4: Set compass radius to $4\text{ cm}$. With $Q$ as center, cut an arc on ray $QX$ to locate vertex $P$.
   • Step 5: Set compass radius to $4.5\text{ cm}$. With $R$ as center, cut an arc on ray $RY$ to locate vertex $S$.
   • Step 6: Connect point $P$ and point $S$ with a line segment.

3. Final Answer: $\text{Quadrilateral } PQRS \text{ is constructed with } PS \approx 5.8\text{ cm}.$

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Question 3

Construct a kite $EAGL$ where $EA = AG = 4\text{ cm}$, $EL = GL = 6\text{ cm}$, and diagonal $EG = 5\text{ cm}$.

<details> <summary><strong>Click to View Complete Solution</strong></summary>

Solution:

1. Geometric Deductions:

   • A kite has two distinct pairs of equal adjacent sides ($EA = AG = 4\text{ cm}$ and $EL = GL = 6\text{ cm}$).
   • The main diagonal $EG = 5\text{ cm}$ splits the kite into two triangles: $\triangle EAG$ (isosceles with sides $4, 4, 5$) and $\triangle EGL$ (isosceles with sides $6, 6, 5$).

2. Step-by-Step Construction:

   • Step 1: Draw the common base diagonal segment $EG = 5\text{ cm}$.
   • Step 2: Set compass radius to $4\text{ cm}$. With $E$ as center, draw an arc above $EG$.
   • Step 3: Keeping compass radius at $4\text{ cm}$, place compass at $G$ and draw an arc intersecting the previous arc above $EG$ to locate vertex $A$.
   • Step 4: Set compass radius to $6\text{ cm}$. With $E$ as center, draw an arc below $EG$.
   • Step 5: Keeping compass radius at $6\text{ cm}$, place compass at $G$ and draw an arc intersecting the previous arc below $EG$ to locate vertex $L$.
   • Step 6: Join $E$ to $A$, $A$ to $G$, $G$ to $L$, and $L$ to $E$.

3. Final Answer: $\text{Kite } EAGL \text{ is constructed with equal adjacent sides } 4\text{ cm} \text{ and } 6\text{ cm}.$

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Exam Revision & FAQs

FAQ 1: Why can't we construct a unique quadrilateral if only 4 sides are given?

Answer: 4 side lengths do not provide structural rigidity. Four hinged sides form a flexible mechanism that can deform into infinitely many quadrilateral shapes with different interior angles and diagonal lengths. A 5th measurement (either an angle or a diagonal) is required to fix the shape into a single, unique geometry.

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FAQ 2: What should I do if a problem asks to construct a parallelogram given 2 adjacent sides and 1 diagonal?

Answer: Use the property that opposite sides of a parallelogram are equal. If adjacent sides are $a$ and $b$, and the diagonal is $d$:

1. Construct the base triangle using sides $a$, $b$, and diagonal $d$ (via SSS construction).
2. Locate the 4th vertex by drawing an arc of radius $a$ from the vertex opposite to $a$, and an arc of radius $b$ from the vertex opposite to $b$.
3. Connect the vertices to complete the parallelogram.

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FAQ 3: How can I construct precise angles like $75^\circ$ or $105^\circ$ using only a ruler and compass?

Answer:

• To construct $75^\circ$: Construct a $90^\circ$ angle and a $60^\circ$ angle on the same base point. Bisect the $30^\circ$ region between $60^\circ$ and $90^\circ$: $75^\circ = 60^\circ + \frac{90^\circ - 60^\circ}{2} = 60^\circ + 15^\circ$
• To construct $105^\circ$: Construct a $90^\circ$ angle and a $120^\circ$ angle on the same base point. Bisect the $30^\circ$ region between $90^\circ$ and $120^\circ$: $105^\circ = 90^\circ + \frac{120^\circ - 90^\circ}{2} = 90^\circ + 15^\circ$

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FAQ 4: How accurate do geometric constructions need to be in board examinations?

Answer: Exam standards require precision within $\pm 1\text{ mm}$ for line lengths and $\pm 1^\circ$ for angles. To ensure full marks:

1. Use a hard, finely sharpened pencil ($2H$ or $H$).
2. Keep all construction arcs visible; never erase arc lines, as examiners award marks for showing clear construction steps.
3. Keep compass hinges firm so they do not slip mid-arc.