Question 1

Q1: Given here are some figures. Classify each of them on the basis of the following. (a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex poly...

Accepted Answer

Analysis of each figure: Figure (1): A closed shape made of line segments with one vertex pointing inwards. - (a) Simple curve: Yes, it does not cross itself. - (b) Simple closed curve: Yes, it is simple and its start and end points are the same. - (c) Polygon: Yes, it is a simple closed curve made...

Question 2

Q2: What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides

Accepted Answer

Definition of a Regular Polygon: A regular polygon is a polygon that is both equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Names of Regular Polygons: (i) 3 sides: A regular polygon with 3 sides is called an equilateral triangle. (ii) 4 sides: A regul...

Question 3

Q1: Find $x$ in the following figures.

Accepted Answer

Property Used: The sum of the measures of the exterior angles of any polygon is $360^\circ$. (a) Figure Description: A triangle with three exterior angles given as $125^\circ$, $125^\circ$, and $x$. Solution: We know that the sum of the exterior angles is $360^\circ$. $$x + 125^\circ + 125^\circ = 3...

Question 4

Q2: Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides

Accepted Answer

Formula: For a regular polygon with $n$ sides, the measure of each exterior angle is given by: $$	ext{Each exterior angle} = \frac{360^\circ}{n}$$ (i) 9 sides Given: Number of sides $n = 9$. Solution: $$	ext{Each exterior angle} = \frac{360^\circ}{9} = 40^\circ$$ Final Answer for (i): The measure...

Question 5

Q3: How many sides does a regular polygon have if the measure of an exterior angle is $24^\circ$?

Accepted Answer

Given: Measure of each exterior angle of a regular polygon = $24^\circ$. To Find: Number of sides ($n$) of the regular polygon. Formula: Number of sides $n = \frac{	ext{Sum of all exterior angles}}{	ext{Measure of each exterior angle}}$ Solution: The sum of all exterior angles of any polygon is $3...

Question 6

Q4: How many sides does a regular polygon have if each of its interior angles is $165^\circ$?

Accepted Answer

Given: Measure of each interior angle of a regular polygon = $165^\circ$. To Find: Number of sides ($n$) of the regular polygon. Solution: First, we find the measure of each exterior angle. An interior angle and its corresponding exterior angle form a linear pair, so their sum is $180^\circ$. $$	ex...

Question 7

Q5: (a) Is it possible to have a regular polygon with measure of each exterior angle as $22^\circ$? (b) Can it be an interior angle of a regular polygon?...

Accepted Answer

(a) Is it possible to have a regular polygon with measure of each exterior angle as $22^\circ$? Condition: For a regular polygon to exist, the number of sides, $n$, must be a whole number. The number of sides is calculated by $n = \frac{360^\circ}{	ext{exterior angle}}$. Calculation: $$n = \frac{36...

Question 8

Q6: (a) What is the minimum interior angle possible for a regular polygon? Why? (b) What is the maximum exterior angle possible for a regular polygon?

Accepted Answer

(a) What is the minimum interior angle possible for a regular polygon? Why? Reasoning: The minimum number of sides a polygon can have is 3, which forms a triangle. A regular polygon with 3 sides is an equilateral triangle. The sum of interior angles of a triangle is $180^\circ$. In an equilateral tr...

Question 9

Q1: Given a parallelogram ABCD. Complete each statement along with the definition or property used. (i) AD = _____  (ii) ∠DCB = _____  (iii) OC = _____  (...

Accepted Answer

Given: A parallelogram ABCD with diagonals AC and BD intersecting at O. (i) AD = _____ Answer: AD = BC Property: In a parallelogram, opposite sides are equal in length. (ii) ∠DCB = _____ Answer: ∠DCB = ∠DAB Property: In a parallelogram, opposite angles are equal in measure. (iii) OC = _____ Answer:...

Question 10

Q10: Explain how this figure is a trapezium. Which of its two sides are parallel?

Accepted Answer

Given: A quadrilateral KLMN with $\angle M = 100^\circ$ and $\angle L = 80^\circ$. To Explain: How the figure is a trapezium and identify the parallel sides. Property: If a transversal intersects two lines such that the sum of the interior angles on the same side of the transversal is $180^\circ$, t...

Question 11

Q11: Find $m\angle C$ in Fig 3.27 if $\overline{AB} \| \overline{DC}$.

Accepted Answer

Given: A trapezium ABCD where side AB is parallel to side DC ($\overline{AB} \| \overline{DC}$). The measure of $\angle B$ is $120^\circ$. To Find: The measure of $\angle C$. Property: When two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal a...

Question 12

Q12: Find the measure of $\angle P$ and $\angle S$ if $\overline{SP} \| \overline{RQ}$ in Fig 3.28. (If you find $m\angle R$, is there more than one method...

Accepted Answer

Given: A quadrilateral PQRS where side SP is parallel to side RQ ($\overline{SP} \| \overline{RQ}$). The measure of $\angle Q$ is $130^\circ$. The measure of $\angle R$ is indicated by a right angle symbol, so $m\angle R = 90^\circ$. To Find: The measures of $\angle P$ and $\angle S$. Property: When...

Question 13

Q2: Consider the following parallelograms. Find the values of the unknowns x, y, z.

Accepted Answer

(i) Figure Description: A parallelogram ABCD with $\angle B = 100^\circ$. The other angles are labeled as $x, y, z$. $\angle D = y$, $\angle C = x$, $\angle A = z$. Solution: - $y = \angle B = 100^\circ$ (Opposite angles of a parallelogram are equal). - $x + \angle B = 180^\circ$ (Adjacent angles of...

Question 14

Q3: Can a quadrilateral ABCD be a parallelogram if (i) $\angle D + \angle B = 180^\circ$? (ii) $AB = DC = 8 	ext{ cm}, AD = 4 	ext{ cm}$ and $BC = 4.4 \...

Accepted Answer

(i) Can a quadrilateral ABCD be a parallelogram if $\angle D + \angle B = 180^\circ$? Answer: It can be, but not necessarily. Reason: In a parallelogram, opposite angles must be equal ($\angle D = \angle B$). If $\angle D = \angle B$, then the condition becomes $2\angle B = 180^\circ$, which means $...

Question 15

Q4: Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Accepted Answer

Description of the figure: A kite is a quadrilateral that fits this description.  Properties of a Kite: - A kite has two pairs of equal-length sides that are adjacent to each other. - It has exactly one pair of opposite angles that are equal. These equal angles are between the sides of unequal lengt...

Question 16

Q5: The measures of two adjacent angles of a parallelogram are in the ratio $3:2$. Find the measure of each of the angles of the parallelogram.

Accepted Answer

Given: The ratio of two adjacent angles of a parallelogram is $3:2$. Let: Let the measures of the two adjacent angles be $3x$ and $2x$. Property: Adjacent angles in a parallelogram are supplementary (their sum is $180^\circ$). Solution: $$3x + 2x = 180^\circ$$ $$5x = 180^\circ$$ $$x = \frac{180^\cir...

Question 17

Q6: Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Accepted Answer

Given: Two adjacent angles of a parallelogram are equal. Let: Let the measure of each of the two adjacent angles be $x$. Property: Adjacent angles in a parallelogram are supplementary. Solution: $$x + x = 180^\circ$$ $$2x = 180^\circ$$ $$x = \frac{180^\circ}{2}$$ $$x = 90^\circ$$ Since adjacent angl...

Question 18

Q7: The adjacent figure HOPE is a parallelogram. Find the angle measures $x, y$ and $z$. State the properties you use to find them.

Accepted Answer

Given: A parallelogram HOPE. The exterior angle at vertex H is $70^\circ$. The angle $\angle PHO$ is $40^\circ$. The angle $\angle EHP$ is $y$. The angle $\angle OPE$ is $z$. The angle $\angle EOP$ is $x$. The diagonal is HP. To Find: The values of $x, y, z$. Solution: Step 1: Find the interior angl...

Question 19

Q8: The following figures GUNS and RUNS are parallelograms. Find $x$ and $y$. (Lengths are in cm)

Accepted Answer

(i) Figure GUNS Given: A parallelogram GUNS. Side GU = $3y - 1$, side SN = 26. Side GS = $3x$, side UN = 18. Property: In a parallelogram, opposite sides are equal in length. Solution: GS = UN $$3x = 18$$ $$x = \frac{18}{3} = 6$$ GU = SN $$3y - 1 = 26$$ $$3y = 27$$ $$y = \frac{27}{3} = 9$$ Final Ans...

Question 20

Q9: In the above figure both RISK and CLUE are parallelograms. Find the value of $x$.

Accepted Answer

Given: Two parallelograms, RISK and CLUE, intersecting. In parallelogram RISK, $\angle K = 120^\circ$. In parallelogram CLUE, $\angle L = 70^\circ$. A triangle is formed at the intersection of the two parallelograms, with one angle being $x$. To Find: The value of $x$. Solution: Step 1: Analyze para...

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