Class 6 Mathematics - Ch 7: Mastery Guide: Geometry, Parallel Lines, Transversals, Reflection & Rotational Symmetry (FBISE)
Instructional Guide: Unit 7 Geometry
- Recognize and differentiate parallel lines ($\parallel$) and perpendicular lines ($\perp$).
- Define a transversal as a coplanar line cutting two or more lines at distinct points.
- Identify and compute adjacent, complementary ($90^\circ$), supplementary ($180^\circ$), and vertically opposite angles.
- Recognize corresponding, alternate interior, alternate exterior, and consecutive interior angles along parallel lines.
- Solve multi-step algebraic equations involving unknown angle measures.
- Construct reflections across mirror lines on square grid paper and using a compass/ruler.
- Identify lines of symmetry in 2D shapes and determine the order of rotational symmetry ($360^\circ / \theta$).
- Basic Angles (Class 5): Acute ($<90^\circ$), Right ($90^\circ$), Obtuse ($>90^\circ$), Straight ($180^\circ$), and Reflex ($>180^\circ$) angles.
- Line Segments & Rays: Difference between lines ($\overleftrightarrow{AB}$), line segments ($\overline{AB}$), and rays ($\overrightarrow{AB}$).
- Simple Linear Equations (Unit 6): Transposing terms and solving for unknown variables ($ax + b = c$).
- Parallel Lines Precondition: Assuming alternate or corresponding angles are equal even when lines are NOT parallel. (They are ONLY equal when lines are parallel!)
- Complementary vs Supplementary Confusion: Confusing $90^\circ$ and $180^\circ$. (Remember: C comes before S; 90 comes before 180).
- Parallelogram Symmetry Myth: Believing a parallelogram has reflection lines of symmetry (it has 0 lines of symmetry, but rotational order 2).
- Order of Rotation of Non-symmetric Shapes: Forgetting that a shape that only matches itself after a full $360^\circ$ turn has an order of 1 (often described as "none" in basic tests).
Kid-Friendly Rhymes & Memory Tricks
"Parallel lines are like railway track,
They go straight ahead and never look back!
The distance between them is always the same,
They will never meet in the geometry game!"
"C is for Corner, a Right 90 degrees,
S is for Straight, 180 with ease!
Complementary sums to ninety bright,
Supplementary makes a straight line right!"
"Look for the 'F': Corresponding match in place!
Look for the 'Z': Alternate angles across the space!
Look for the 'X': Vertical angles across the dot!
Look for the 'C': Allied angles add to 180 on the spot!"
"Spin the wheel around the hub,
Count the matches in the club!
Divide three-sixty by the turn angle neat,
The Order of Symmetry is complete!"
Geometry in the Real World
High-voltage electric pylons and mobile phone towers use parallel steel girders and intersecting transversal braces to distribute heavy structural loads safely against wind and earthquake forces.
The world-famous Faisal Mosque in Islamabad and the Taj Mahal exhibit breathtaking vertical reflection symmetry—the left half is a perfect mirror image of the right half!
Wind turbine blades have a 3-fold rotational symmetry (Order 3, turning every $120^\circ$). Car alloy wheels with 5 or 8 spokes maintain perfect rotational balance around their central axle.
Submarine periscopes use two $45^\circ$ mirrors positioned along parallel planes to reflect light waves around obstacles so navigators can see above water.
1. Lines and Angle Relationships
Geometry begins with the fundamental study of straight lines in a 2-dimensional plane (coplanar lines):
Parallel Lines ($\parallel$)
Two coplanar lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel if they never meet, no matter how far extended in either direction. The perpendicular distance between them is constant everywhere.
Symbol: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ or $l \parallel m$.
Perpendicular Lines ($\perp$)
Two lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$ are perpendicular if they intersect at a right angle ($90^\circ$).
Symbol: $\overleftrightarrow{PQ} \perp \overleftrightarrow{RS}$. A small square symbol at the intersection denotes $90^\circ$.
Fundamental Angle Classifications
| Angle Type | Geometric Definition | Mathematical Condition | Visual Cue |
|---|---|---|---|
| Adjacent Angles | Two angles having a common vertex, a common arm, and non-overlapping interiors. | Share 1 common vertex and 1 common ray | Neighboring angles in a fan |
| Complementary Angles | Two angles whose sum equals $90^\circ$ (a right angle). Each is the complement of the other. | $\angle 1 + \angle 2 = 90^\circ$ $\angle 1 = 90^\circ - \angle 2$ |
Corner right angle split into two |
| Supplementary Angles | Two angles whose sum equals $180^\circ$ (a straight line). Each is the supplement of the other. | $\angle 1 + \angle 2 = 180^\circ$ $\angle 1 = 180^\circ - \angle 2$ |
Linear pair on a straight line |
| Vertically Opposite Angles | Non-adjacent angles formed directly opposite each other when two straight lines intersect. | $\angle a = \angle c, \quad \angle b = \angle d$ Always equal! |
X-cross / Scissors opening |
| Angles About a Point | The complete sum of all adjacent angles surrounding a common central vertex. | $\sum \text{angles} = 360^\circ$ | Full circular rotation ($360^\circ$) |
2. Transversals and Angles on Parallel Lines
A transversal is a straight line that cuts two or more coplanar lines at distinct points. When a transversal intersects two parallel lines ($l \parallel m$), eight distinct angles are formed with the following essential properties:
Occupying the identical relative corner at each intersection along the transversal.
Property: $\mathbf{\angle \text{corr}_1 = \angle \text{corr}_2}$ (Equal when lines are parallel).
Lying between the two parallel lines on opposite sides of the transversal.
Property: $\mathbf{\angle \text{alt-int}_1 = \angle \text{alt-int}_2}$ (Equal when lines are parallel).
Lying on the exterior of the parallel lines on opposite sides of the transversal.
Property: $\mathbf{\angle \text{alt-ext}_1 = \angle \text{alt-ext}_2}$ (Equal when lines are parallel).
Lying inside the parallel lines on the same side of the transversal.
Property: $\mathbf{\angle 1 + \angle 2 = 180^\circ}$ (Supplementary when lines are parallel).
3. Transformation Geometry: Symmetry, Reflection & Rotation
Reflection Symmetry & Lines of Symmetry
A figure has reflection symmetry if a straight line (the axis or line of symmetry) divides it into two congruent halves that match exactly when folded across the line. Each point on the pre-image and its corresponding point on the reflected image are equidistant from the mirror line.
Comprehensive Symmetry Reference for 2D Figures
| Geometric Shape | Lines of Symmetry | Order of Rotational Symmetry | Key Geometric Features |
|---|---|---|---|
| Equilateral Triangle | 3 | 3 ($120^\circ$) | 3 equal sides, 3 equal angles ($60^\circ$) |
| Isosceles Triangle | 1 | 1 (None) | 2 equal sides, 2 equal base angles |
| Scalene Triangle | 0 | 1 (None) | All 3 sides and angles unequal |
| Square | 4 | 4 ($90^\circ$) | 4 equal sides, 4 right angles (2 midlines + 2 diagonals) |
| Rectangle | 2 | 2 ($180^\circ$) | Opposite sides equal, 4 right angles (2 midlines only) |
| Rhombus | 2 | 2 ($180^\circ$) | 4 equal sides, diagonals are symmetry lines |
| Parallelogram | 0 | 2 ($180^\circ$) | Opposite sides parallel and equal, no reflection axis |
| Kite | 1 | 1 (None) | 2 pairs of adjacent equal sides (main vertical diagonal) |
| Isosceles Trapezium | 1 | 1 (None) | Non-parallel sides equal (vertical midpoint bisector) |
| Scalene / General Trapezium | 0 | 1 (None) | Non-parallel sides unequal |
| Regular Pentagon | 5 | 5 ($72^\circ$) | 5 equal sides, 5 equal angles ($108^\circ$) |
| Regular Hexagon | 6 | 6 ($60^\circ$) | 6 equal sides, 6 equal angles ($120^\circ$) |
| Regular $n$-sided Polygon | $n$ | $n$ ($360^\circ / n$) | $n$ equal sides, $n$ equal angles |
| Circle | $\infty$ (Infinite) | $\infty$ (Infinite) | Any diameter passing through the center is a line of symmetry |
Rotational Symmetry & Order of Rotation
A figure possesses rotational symmetry if it can be rotated about its centre of rotation by an angle $\theta \le 360^\circ$ and coincide exactly with its initial appearance.
- Angle of Rotation: Smallest positive angle to fit the original figure: $\theta = \frac{360^\circ}{\text{Order}}$.
- Quarter Turn ($90^\circ$): Rotation by $90^\circ$.
- Half Turn ($180^\circ$): Rotation by $180^\circ$ (upside down). Digits like 96 become 96 under half turn!
- Three-Quarter Turn ($270^\circ$): Rotation by $270^\circ$.
- Full Turn ($360^\circ$): Complete single revolution returning to start.
Exercise 7.1 — Step-by-Step Solutions
(i) How many adjacent angles are in figure (i)?
Solution: The common ray $OB$ divides the angle into two neighboring parts $\angle AOB$ and $\angle BOC$ sharing vertex $O$.
$$\text{Number of adjacent angles} = 2$$
(ii) What is the adjacent angle of $\angle AOB$?
Solution: $\angle AOB$ shares arm $OB$ and vertex $O$ with $\angle BOC$.
$$\text{Adjacent angle} = \mathbf{\angle BOC}$$
(i) How many adjacent angles are in figure (ii)?
Solution: Four pairs of adjacent angles are formed:
1. $\angle PQR$ and $\angle RQS$
2. $\angle RQS$ and $\angle SQT$
3. $\angle PQR$ and $\angle RQT$
4. $\angle PQS$ and $\angle SQT$
$$\text{Total adjacent pairs} = \mathbf{4}$$
(ii) Write the adjacent angles of $\angle PQR$.
Solution: Angles sharing ray $QR$ with $\angle PQR$ are $\mathbf{\angle RQS \text{ and } \angle RQT}$.
(iii) Write one pair of angles which are not adjacent.
Solution: $\mathbf{\angle PQR \text{ and } \angle SQT}$ (they do not share a common arm).
(i) Write the adjacent angles of $\angle APC$ and $\angle APD$.
Solution:
- Adjacent to $\angle APC$: $\angle APD$ (common arm $AP$) and $\angle BPC$ (common arm $CP$).
- Adjacent to $\angle APD$: $\angle APC$ (common arm $AP$) and $\angle DPB$ (common arm $DP$).
$$\text{Pairs: } \mathbf{(\angle APC, \angle CPB), \quad (\angle DPB, \angle APC)}$$
(ii) Write non-adjacent angle of $\angle BPC$.
Solution: Vertically opposite angle across $P$ is $\mathbf{\angle APD}$.
(iii) How many pairs of adjacent angles are formed at $P$?
Solution: 4 pairs of supplementary adjacent angles are formed:
$(\angle APC, \angle APD), (\angle APD, \angle DPB), (\angle DPB, \angle BPC), (\angle BPC, \angle APC)$.
$$\text{Number of adjacent pairs} = \mathbf{4}$$
(i) $x$ and $75^\circ$ are complementary angles.
$$x + 75^\circ = 90^\circ \implies x = 90^\circ - 75^\circ = \mathbf{15^\circ}$$
(ii) $x$ and $75^\circ$ are supplementary angles.
$$x + 75^\circ = 180^\circ \implies x = 180^\circ - 75^\circ = \mathbf{105^\circ}$$
(iii) $x$ and $75^\circ$ are vertical angles.
$$\text{Vertical angles are equal: } \mathbf{x = 75^\circ}$$
(i) What are the complements of $\angle 2$ and $\angle 4$?
Since corner angles of a rectangle equal $90^\circ$:
- Complement of $\angle 2$ is $\mathbf{\angle 1}$ ($\angle 1 + \angle 2 = 90^\circ$).
- Complement of $\angle 4$ is $\mathbf{\angle 3}$ ($\angle 3 + \angle 4 = 90^\circ$).
(ii) Find $\angle 1$ and $\angle 3$ if $\angle 2 = 30^\circ$ and $\angle 4 = 65^\circ$.
$$\angle 1 = 90^\circ - 30^\circ = \mathbf{60^\circ}$$
$$\angle 3 = 90^\circ - 65^\circ = \mathbf{25^\circ}$$
(i) What are supplements of $\angle e$ and $\angle g$?
- Supplements of $\angle e$ are $\mathbf{\angle g \text{ and } \angle h}$ (adjacent straight line pairs).
- Supplements of $\angle h$ (or $\angle g$) are $\mathbf{\angle e \text{ and } \angle f}$.
(ii) Find $\angle e$ and $\angle h$ if $\angle g = 120^\circ$.
$$\angle e + \angle g = 180^\circ \implies \angle e = 180^\circ - 120^\circ = \mathbf{60^\circ}$$
$$\angle h = \angle g = \mathbf{120^\circ} \quad (\text{vertically opposite angles})$$
1. Angle $b$ is vertically opposite to $115^\circ$:
$$\mathbf{b = 115^\circ}$$
2. Angles $a$ and $115^\circ$ lie on a straight line (supplementary):
$$a + 115^\circ = 180^\circ \implies \mathbf{a = 65^\circ}$$
3. Angle $a$ is vertically opposite to $(25^\circ + c)$:
$$25^\circ + c = a = 65^\circ \implies c = 65^\circ - 25^\circ = \mathbf{40^\circ}$$
(i) Straight line with 4 equal angles:
$$b + b + b + b = 180^\circ \implies 4b = 180^\circ \implies \mathbf{b = 45^\circ}$$
(ii) Right angle split into $2x$ and $x$:
$$2x + x = 90^\circ \implies 3x = 90^\circ \implies \mathbf{x = 30^\circ}, \quad 2x = \mathbf{60^\circ}$$
(iii) Straight line with right angle, $a$, and $3a$:
$$a + 90^\circ + 3a = 180^\circ \implies 4a = 90^\circ \implies \mathbf{a = 22.5^\circ}, \quad 3a = \mathbf{67.5^\circ}$$
(iv) Intersecting lines with right angle and $35^\circ$:
$$x + 35^\circ = 90^\circ \implies \mathbf{x = 55^\circ}$$
$$y = x = \mathbf{55^\circ} \quad (\text{vertically opposite})$$
(v) Intersecting lines with $60^\circ$ and symmetric side angles:
$$b = \mathbf{60^\circ} \quad (\text{vertically opposite})$$
$$a = c = \frac{180^\circ - 120^\circ}{2} = \mathbf{30^\circ}$$
(vi) Intersecting lines with right angle and $35^\circ$:
$$x + 35^\circ = 90^\circ \implies \mathbf{x = 55^\circ}$$
$$z + x = 180^\circ \implies z = 180^\circ - 55^\circ = \mathbf{125^\circ}$$
Since $a$ and $b$ are supplementary:
$$a + b = 180^\circ$$
Substitute $a = 4b$:
$$4b + b = 180^\circ \implies 5b = 180^\circ \implies \mathbf{b = 36^\circ}$$
$$a = 4(36^\circ) = \mathbf{144^\circ}$$
(i) Figure (i):
Vertically opposite angles: $2x + 5 = 105 \implies 2x = 100 \implies \mathbf{x = 50^\circ}$
Supplementary angle: $y + 105^\circ = 180^\circ \implies \mathbf{y = 75^\circ}$
(ii) Figure (ii):
Vertically opposite angles: $7x + 2 = 6x + 18 \implies 7x - 6x = 18 - 2 \implies \mathbf{x = 16^\circ}$
Angle measure: $7(16^\circ) + 2 = 114^\circ$
Supplementary angle: $z + 114^\circ = 180^\circ \implies \mathbf{z = 66^\circ}$
Exercise 7.2 — Step-by-Step Solutions
(i) What is the corresponding angle of $\angle 1$?
$$\text{Corresponding angle} = \mathbf{\angle 3}$$
(ii) What is the corresponding angle of $\angle 2$?
$$\text{Corresponding angle} = \mathbf{\angle 4}$$
(i) How many pairs of corresponding angles are formed?
$$\text{Number of pairs} = \mathbf{4}$$
(ii) What are corresponding angles of $\angle a$ and $\angle b$?
$$\text{For } \angle a: \mathbf{\angle h}, \quad \text{For } \angle b: \mathbf{\angle e}$$
(iii) What are corresponding angles of $\angle c$ and $\angle d$?
$$\text{For } \angle c: \mathbf{\angle g}, \quad \text{For } \angle d: \mathbf{\angle f}$$
(i) How many pairs of alternate interior angles are formed?
$$\text{Number of pairs} = \mathbf{2}$$
(ii) What is the alternate interior angle of $\angle 1$?
$$\text{Answer: } \mathbf{\angle 2}$$
(iii) What is the alternate interior angle of $\angle 3$?
$$\text{Answer: } \mathbf{\angle 4}$$
(i) How many pairs of alternate exterior angles are formed?
$$\text{Number of pairs} = \mathbf{2}$$
(ii) What is the alternate exterior angle of $\angle v$?
$$\text{Answer: } \mathbf{\angle s}$$
(iii) What is the alternate exterior angle of $\angle r$?
$$\text{Answer: } \mathbf{\angle w}$$
(i) Figure (i): $y = \mathbf{110^\circ}$ (corresponding), $x = 180^\circ - 110^\circ = \mathbf{70^\circ}$ (allied), $z = \mathbf{70^\circ}$ (alternate interior).
(ii) Figure (ii): $w = \mathbf{80^\circ}$ (vertical), $u = 180^\circ - 80^\circ = \mathbf{100^\circ}$, $x = \mathbf{100^\circ}$ (corresponding), $v = \mathbf{80^\circ}$ (corresponding).
(iii) Figure (iii): $y = \mathbf{58^\circ}$ (alternate interior), $x = 180^\circ - 58^\circ = \mathbf{122^\circ}$, $z = \mathbf{122^\circ}$.
(iv) Figure (iv): Consecutive interior angles sum to $180^\circ$:
$$2x + 54^\circ = 180^\circ \implies 2x = 126^\circ \implies \mathbf{x = 63^\circ}$$
(v) Figure (v): Corresponding angles: $x + 10^\circ = 104^\circ \implies \mathbf{x = 94^\circ}$.
(vi) Figure (vi): Alternate interior angles: $5x - 66^\circ = 14^\circ + 3x \implies 2x = 80^\circ \implies \mathbf{x = 40^\circ}$.
(vii) Figure (vii): $x + 44^\circ = 93^\circ \implies \mathbf{x = 49^\circ}, \quad \mathbf{y = 93^\circ}$.
(viii) Figure (viii): Alternate exterior angles: $5x - 54^\circ = 3x + 16^\circ \implies 2x = 70^\circ \implies \mathbf{x = 35^\circ}$.
Angle $= 3(35^\circ) + 16^\circ = 121^\circ \implies y = 180^\circ - 121^\circ = \mathbf{59^\circ}$.
(ix) Figure (ix): Consecutive interior angles: $(7x + 15^\circ) + 81^\circ = 180^\circ \implies 7x = 84^\circ \implies \mathbf{x = 12^\circ}, \quad \mathbf{y = 81^\circ}$.
(x) Figure (x): Alternate exterior angles: $3x - 33^\circ = 2x + 26^\circ \implies \mathbf{x = 59^\circ}$.
Angle $= 2(59^\circ) + 26^\circ = 144^\circ \implies z = 180^\circ - 144^\circ = \mathbf{36^\circ}$.
Exercise 7.3 — Step-by-Step Solutions
- Figure A (Equilateral Triangle): Symmetrical, 3 lines of symmetry.
- Figure B (Rectangle): Symmetrical, 2 lines of symmetry.
- Figure C (Stepped Shape / Block 5): Symmetrical, 1 line of symmetry.
- Figure D (Kite / Rhombus): Symmetrical, 1 line of symmetry.
- Figure E (Cross / Plus): Symmetrical, 2 lines of symmetry.
The 18-petal circular flower design has symmetrically arranged opposite petal pairs:
$$\text{Lines of symmetry} = \mathbf{9}$$
- (i) Scalene Triangle = 0
- (ii) Isosceles Triangle = 1 (or 2 in text answer key)
- (iii) Equilateral Triangle = 3
- (iv) Trapezium (scalene) = 0 (or 4 for square in book notation)
- (v) Rectangle = 2 (or 5 for pentagon)
- (vi) Parallelogram = 0
- (vii) Regular Pentagon = 5
- (viii) Regular Hexagon = 6
Every straight line (diameter) drawn through the center of a circle is a line of symmetry.
$$\text{Number of lines of symmetry} = \mathbf{\infty \text{ (Infinite)}}$$
Construction Procedure: For each vertex on the given figure, count the number of grid units perpendicular to the red line. Plot the image vertex at the exact same distance on the opposite side, then join the points with straight line segments.
- (i) Quarter turn ($90^\circ$): Points downwards right.
- (ii) Three-quarter turn ($270^\circ$): Points upwards left.
- (iii) Half turn ($180^\circ$): Points leftwards upside down.
- (iv) Full turn ($360^\circ$): Restores original rightward orientation.
- Order of Rotational Symmetry: 1 (or none, matches only at full $360^\circ$ turn).
(i) Half turn ($180^\circ$) about central dot A:
The digit '9' becomes '6' and '6' becomes '9', producing the exact number $\mathbf{96}$.
(ii) Full turn ($360^\circ$):
Returns to original position: $\mathbf{96}$.
- (i) Parallelogram = 2 (matches at $180^\circ, 360^\circ$)
- (ii) Equilateral Triangle = 3 (matches at $120^\circ, 240^\circ, 360^\circ$)
- (iii) A Square = 4 (matches at $90^\circ, 180^\circ, 270^\circ, 360^\circ$)
- (iv) A Regular Pentagon = 5 (matches at $72^\circ, 144^\circ, 216^\circ, 288^\circ, 360^\circ$)
(i) What do you notice?
Equilateral triangle (3 sides) has 3 lines; Square (4 sides) has 4 lines; Regular pentagon (5 sides) has 5 lines.
(ii) Discovery Rule:
$$\mathbf{\text{Number of lines of symmetry} = \text{Number of sides of regular polygon } (n)}$$
- (i) Regular Hexagon: Order = $\mathbf{6}$
- (ii) 8-Spoke Wheel: Order = $\mathbf{8}$
- (iii) 4-Petal Flower: Order = $\mathbf{4}$
- (iv) 3-Blade Propeller: Order = $\mathbf{3}$
Q11. Lines of Symmetry for Shapes:
- Letter Z: 0
- Chevron: 1
- Trapezoid Arrow: 1
- Hourglass: 2
- Heart: 1
- Plus Sign: 4
Q12. Reflected Image of Letter M:
Reflecting 'M' across a horizontal mirror line directly below it inverts the peaks and valleys to construct the letter $\mathbf{W}$.
Q13. Construct Line of Symmetry Between Multiplication Sign and Its Image:
Connect corresponding vertices of the two figures with straight line segments, find their midpoints, and draw a straight line through these midpoints to establish the mirror axis.
Review Exercise 7 — Complete Solutions
- Angles $a$ and $b$ opposite each other across vertex are: (c) vertically opposite
- Angles $a$ and $c$ in matching transversal positions are: (b) corresponding
- Angles $a$ and $b$ forming a linear pair are: (a) supplementary
- If straight angle adjacent is $110^\circ$, angle $a = 180^\circ - 110^\circ =$: (c) 70°
- If matching angle is $70^\circ$, angle $g =$: (a) 70°
- If alternate angle is $55^\circ$, angle $d =$: (c) 55°
- If alternate angles are equal then the lines are: (a) parallel
- If lines $m \parallel n$ with $3y = 120^\circ$, then $y = 120/3 =$: (b) 40°
- Two vertical angles $3x - 3 = 36 \implies 3x = 39 \implies x =$: (b) 13°
- Number of lines of symmetry for a regular polygon is: (b) number of sides
- Order of rotational symmetry for a trapezium is: (d) none
- Lines of symmetry that can be drawn in a circle: (d) infinite
Since the angles are supplementary:
$$(4x - 10) + (x - 15) = 180$$
$$5x - 25 = 180 \implies 5x = 205 \implies \mathbf{x = 41^\circ}$$
Let the angles be $\angle 1$ and $\angle 2$. Given $\angle 2 = \frac{3}{2}\angle 1$.
$$\angle 1 + \frac{3}{2}\angle 1 = 180^\circ \implies \frac{5}{2}\angle 1 = 180^\circ$$
$$\angle 1 = 180^\circ \times \frac{2}{5} = \mathbf{72^\circ}$$
$$\angle 2 = 180^\circ - 72^\circ = \mathbf{108^\circ}$$
Corresponding angles along parallel lines are equal:
$$7y - 10 = 5y + 10$$
$$2y = 20 \implies \mathbf{y = 10^\circ}$$
Each interior angle of an equilateral triangle measures $60^\circ$.
The exterior angles form linear pairs with the base angles:
$$x = 180^\circ - 60^\circ = \mathbf{120^\circ}$$
$$y = 180^\circ - 60^\circ = \mathbf{120^\circ}$$
Q6. Reflection of Equilateral Triangle XYZ across line $l$: Drop perpendicular lines from $X, Y, Z$ onto line $l$, extend them by equal lengths to find $X', Y', Z'$, and connect vertices.
Q7. Mirror Line Between Pre-image and Image Square: Connect corresponding vertices $A \to A'$ and $B \to B'$ with dashed lines, locate their midpoints, and draw a straight line through these midpoints to construct the mirror axis.
⚡ Active Recall & Self-Assessment Knowledge Checks
🔍 Check 1: What is the complement of a 90° angle?
Explanation: Complementary angles sum to $90^\circ$. Thus, $90^\circ - 90^\circ = 0^\circ$.
🔍 Check 2: Can two obtuse angles be supplementary?
Explanation: Obtuse angles are $>90^\circ$. The sum of two angles greater than $90^\circ$ is always $>180^\circ$ (e.g. $91^\circ + 91^\circ = 182^\circ$).
🔍 Check 3: How many lines of symmetry does a regular octagon have?
Explanation: A regular $n$-gon always has $n$ lines of reflection symmetry and rotational symmetry of order $n$.
🔍 Check 4: If two parallel lines are cut by a transversal and one interior angle is 115°, what is the consecutive interior angle on the same side?
Explanation: Consecutive interior (allied) angles are supplementary ($180^\circ$). $180^\circ - 115^\circ = 65^\circ$.
More Chapter Notes for Class 6 (FBISE)
MathematicsTest Your Knowledge on Chapter 7: Class 6 Mathematics - Ch 7: Mastery Guide: Geometry, Parallel Lines, Transversals, Reflection & Rotational Symmetry (FBISE)
Practice textbook-aligned solved MCQs with instant answer feedback, step-by-step solutions, and timed test simulation.
Class 6 Mathematics - Ch 7: Geometry Chapter Mock Test
Test your complete conceptual mastery across all chapters under real board exam conditions with official timer, anti-cheat surveillance, and instant grading.