Maharashtra Board Class 8 Maths Chapter 8 Quadrilateral Constructions and Types Set 8.3 Solutions

Quadrilateral: Constructions And Types Class 8 Maths Chapter 8 Practice Set 8.3 Solutions Maharashtra Board

Std 8 Maths Practice Set 8.3 Chapter 8 Solutions Answers

Question 1. Measures of opposite angles of a parallelogram are (3x - 2)° and (50 - x)°. Find the measure of its each angle.
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज PQRS दिखाया गया है, जहाँ कोण Q का माप (3x-2)° और कोण S का माप (50-x)° है।
Answer:
Solution:
Let PQRS be the parallelogram.
\( m\angle Q = (3x - 2)^\circ \text{ and } m\angle S = (50 - x)^\circ \)
\( m\angle Q = m\angle S \)...(i)
[Opposite angles of a parallelogram are congruent]
\( \therefore 3x - 2 = 50 - x \)
\( \implies 3x + x = 50 + 2 \)
\( \implies 4x = 52 \)
\( \implies x = \frac{52}{4} \)
\( \implies x = 13 \)
Now, \( m\angle Q = (3x - 2)^\circ \)
\( = (3 \times 13 - 2)^\circ = (39 - 2)^\circ = 37^\circ \)
\( \therefore m\angle S = m\angle Q = 37^\circ \)...[From(i)]
\( m\angle P + m\angle Q = 180^\circ \)
[Adjacent angles of a parallelogram are supplementary]
\( \therefore m\angle P + 37^\circ = 180^\circ \)
\( \implies m\angle P = 180^\circ - 37^\circ = 143^\circ \)
\( \therefore m\angle R = m\angle P = 143^\circ \)
[Opposite angles of a parallelogram are congruent]
\( \therefore \) The measures of the angles of the parallelogram are 37°, 143°, 37° and 143°.
In simple words: To find the angles of a parallelogram, we use the properties that opposite angles are equal and adjacent angles are supplementary. We set up an equation using the given expressions for opposite angles to solve for 'x', then calculate each angle.

🎯 Exam Tip: Remember to clearly state the properties of parallelograms (opposite angles are congruent, adjacent angles are supplementary) as reasons for your steps. This ensures full marks.

Question 2. Referring the given figure of a parallelogram, write the answers of questions given below.
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज WXYZ दिखाया गया है, जिसके विकर्ण WY और XZ बिंदु O पर प्रतिच्छेद करते हैं।
Answer:
Solution:
(i) If I(WZ) = 4.5 cm, then I(XY) = ?
I(WZ) = 4.5 cm ... [Given]
I(XY) = I(WZ) ....[Opposite sides of a parallelogram are congruent]
\( \therefore \) I(XY) = 4.5 cm
(ii) If I(YZ) = 8.2 cm, then I(XW) = ?
I(YZ) = 8.2 cm ... [Given]
I(XW) = I(YZ)
...[Opposite sides of a parallelogram are congruent]
\( \therefore \) I(XW) = 8.2 cm ... [Given]
(iii) If I(OX) = 2.5 cm, then I(OZ) = ?
I(OX) = 2.5 cm ... [Given]
I(OZ) = I(OX)
....[Diagonals of a parallelogram bisect each other]
\( \therefore \) I(OZ) = 2.5 cm
(iv) If I(WO) = 3.3 cm, then I(WY) = ?
I(WO) = 3.3 cm ... [Given]
\( I(WO) = \frac{1}{2} I(WY) \)
....[Diagonals of a parallelogram bisect each other]
\( \therefore 3.3 = \frac{1}{2} I(WY) \)
\( \implies 3.3 \times 2 = I(WY) \)
\( \implies I(WY) = 6.6 \text{ cm} \)
(v) If \( m\angle WZY = 120^\circ \), then \( m\angle WXY = ? \) and \( m\angle XWZ = ? \)
\( m\angle WZY = 120^\circ \)... [Given]
\( m\angle WXY = m\angle WZY \)
.....[Opposite angles of a parallelogram are congruent]
\( \therefore m\angle WXY = 120^\circ \)...(i)
\( m\angle XWZ + m\angle WXY = 180^\circ \)
....[Adjacent angles of a parallelogram are supplementary]
\( \therefore m\angle XWZ + 120^\circ = 180^\circ \)... [From (i)]
\( \implies m\angle XWZ = 180^\circ - 120^\circ \)
\( \implies m\angle XWZ = 60^\circ \)
In simple words: This question tests the understanding of properties of a parallelogram, specifically that opposite sides are equal, diagonals bisect each other, and opposite angles are equal while adjacent angles are supplementary. We apply these rules to find unknown side lengths and angle measures.

🎯 Exam Tip: When dealing with parallelogram properties, clearly identifying which property is being used for each step (e.g., "opposite sides are congruent" or "diagonals bisect each other") is crucial for a complete and correct solution.

Question 3. Construct a parallelogram ABCD such that I(BC) = 7 cm, \( m\angle ABC = 40^\circ \), I(AB) = 3 cm
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज ABCD की कच्ची आकृति दर्शाई गई है, जहाँ AB = 3 सेमी, BC = 7 सेमी और कोण ABC = 40° है।
Answer:
Solution:
Rough Figure
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज ABCD का निर्माण दिखाया गया है जिसमें AB = 3 सेमी, BC = 7 सेमी, AD = 7 सेमी, CD = 3 सेमी और कोण ABC = 40° है।
Opposite sides of a parallelogram are congruent.
\( \therefore \) I(AB) = I(CD) = 3 cm
I(BC) = I(AD) = 7 cm
In simple words: To construct a parallelogram, first draw a rough sketch to visualize the given dimensions and angles. Then, use a ruler and protractor to accurately draw the base, an adjacent side using the given angle, and complete the parallelogram using the property that opposite sides are equal and parallel.

🎯 Exam Tip: Always start construction problems with a clear rough figure. This helps in planning the steps and ensuring all given information is utilized correctly. Accuracy in measurements and angles is key.

Question 4. Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write with reason, what type of a quadrilateral it is.
Answer:
Solution:
Let PQRS be the quadrilateral.
Ratio of consecutive angles of a quadrilateral is 1:2:3:4.
Let the common multiple be x.
\( \therefore m\angle P = x^\circ, m\angle Q = 2x^\circ, m\angle R = 3x^\circ \text{ and } m\angle S = 4x^\circ \)
In PQRS,
\( m\angle P + m\angle Q + m\angle R + m\angle S = 360^\circ \)
...[Sum of the measures of the angles of a quadrilateral is 360°]
\( \therefore x^\circ + 2x^\circ + 3x^\circ + 4x^\circ = 360^\circ \)
\( \implies 10x^\circ = 360^\circ \)
\( \implies x^\circ = \frac{360}{10} \)
\( \implies x^\circ = 36^\circ \)
\( m\angle P = x^\circ = 36^\circ \)
ℹ️ चित्र व्याख्या (Diagram Explanation): एक चतुर्भुज PQRS दिखाया गया है, जिसके कोणों का माप क्रमशः कोण P = x°, कोण Q = 2x°, कोण R = 3x° और कोण S = 4x° है।
\( m\angle Q = 2x^\circ = 2 \times 36^\circ = 72^\circ \)
\( m\angle R = 3x^\circ = 3 \times 36^\circ = 108^\circ \text{ and} \)
\( m\angle S = 4x^\circ = 4 \times 36^\circ = 144^\circ \)
The measures of the angles of the quadrilateral are 36°, 72°, 108°, 144°.
Here, \( m\angle P + m\angle S = 36^\circ + 144^\circ = 180^\circ \)
Since, interior angles are supplementary,
\( \therefore \) side PQ || side SR
\( m\angle P + m\angle Q = 36^\circ + 72^\circ = 108^\circ \neq 180^\circ \)
\( \therefore \) side PS is not parallel to side QR.
Since, one pair of opposite sides of the given quadrilateral is parallel.
\( \therefore \) The given quadrilateral is a trapezium.
In simple words: We find the individual angle measures by using the given ratio and the fact that the sum of angles in a quadrilateral is 360°. To identify the type of quadrilateral, we check for parallel sides by seeing if consecutive interior angles sum up to 180°. If one pair of opposite sides is parallel, it's a trapezium.

🎯 Exam Tip: When asked to identify the type of quadrilateral, calculate all angles first. Then, systematically check for properties like parallel sides (using supplementary interior angles) or congruent sides/angles to determine the specific classification.

Question 5. Construct BARC such that I(BA) = I(BC) = 4.2 cm, I(AC) = 6.0 cm, I(AR) = I(CR) = 5.6 cm
ℹ️ चित्र व्याख्या (Diagram Explanation): एक चतुर्भुज BARC की कच्ची आकृति दर्शाई गई है, जिसमें BA = BC = 4.2 सेमी, AC = 6.0 सेमी, AR = CR = 5.6 सेमी है। यह एक पतंग आकृति का चतुर्भुज (Kite) है।
Answer:
Solution:
Rough Figure
ℹ️ चित्र व्याख्या (Diagram Explanation): एक चतुर्भुज BARC का निर्माण दिखाया गया है जिसमें BA = 4.2 सेमी, BC = 4.2 सेमी, AC = 6.0 सेमी, AR = 5.6 सेमी और CR = 5.6 सेमी है। यह आकृति एक पतंग का प्रतिनिधित्व करती है।
In simple words: This construction involves drawing a quadrilateral with given side lengths. It's essentially constructing two triangles (ABC and ARC) that share a common side (AC). First, draw AC, then use compasses to mark B and R from A and C with the given lengths, and finally join the points to form the quadrilateral.

🎯 Exam Tip: For complex quadrilateral constructions, break the figure down into simpler shapes like triangles. Construct one triangle first using known sides/angles, then use its sides to construct the next part of the quadrilateral. Accuracy of compass and ruler is crucial.

Question 6. Construct PQRS, such that I(PQ) = 3.5 cm, I(QR) = 5.6 cm, I(RS) = 3.5 cm, \( m\angle Q = 110^\circ \), \( m\angle R = 70^\circ \). If it is given that PQRS is a parallelogram, which of the given information is unnecessary?
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज PQRS की कच्ची आकृति दर्शाई गई है, जिसमें PQ = 3.5 सेमी, QR = 5.6 सेमी, RS = 3.5 सेमी, कोण Q = 110° और कोण R = 70° है।
Answer:
Solution:
Rough Figure
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज PQRS का निर्माण दिखाया गया है जिसमें PQ = 3.5 सेमी, QR = 5.6 सेमी, RS = 3.5 सेमी, कोण Q = 110° और कोण R = 70° है। यह आकृति एक सामान्य समांतर चतुर्भुज को दर्शाती है।
1. Since, the opposite sides of a parallelogram are congruent.
\( \therefore \) Either I(PQ) or I(SR) is required.
2. To construct a parallelogram lengths of adjacent sides and measure of one angle is required.
\( \therefore \) Either I(PQ) and \( m\angle Q \) or I(SR) and \( m\angle R \) is the unnecessary information given in the question.
In simple words: To construct a parallelogram, you typically need two adjacent side lengths and one included angle. Any additional information that can be derived from these fundamental properties (like the length of the opposite side or an adjacent angle) becomes redundant.

🎯 Exam Tip: For parallelogram constructions, the minimum information required is two adjacent sides and one angle. Any other measurement (like an opposite side or a second angle) might be redundant if the figure is already identified as a parallelogram. Always analyze what minimal data defines the shape.

Maharashtra Board Class 8 Maths Chapter 8 Quadrilateral: Constructions And Types Practice Set 8.3 Intext Questions And Activities

Question 1. Draw a parallelogram PQRS. Take two rulers of different widths, place one ruler horizontally and draw lines along its edges. Now place the other ruler in slant position over the lines drawn and draw lines along its edges. We get a parallelogram. Draw the diagonals of it and name the point of intersection as T.
(1) Measure the opposite angles of the parallelogram.
(2) Measure the lengths of opposite sides.
(3) Measure the lengths of diagonals.
(4) Measure the lengths of parts of the diagonals made by point T. (Textbook pg. no. 47)
ℹ️ चित्र व्याख्या (Diagram Explanation): एक समांतर चतुर्भुज PQRS दिखाया गया है, जिसके विकर्ण T बिंदु पर प्रतिच्छेद करते हैं। चित्र में कोणों, भुजाओं और विकर्णों के भागों को मापने के लिए पैमाना दिखाया गया है। यह एक प्रयोग की स्थापना है।
Answer:
Solution:
[Students should attempt the above activities on their own.]
In simple words: This activity guides students to construct a parallelogram using rulers and then measure its properties like opposite angles, opposite sides, and diagonal segments. The goal is to experimentally verify the geometric properties of parallelograms.

🎯 Exam Tip: Practical activities like this reinforce theoretical knowledge. Pay close attention to measurement accuracy. Understanding the properties through hands-on experience helps in remembering them for problem-solving.

Question 2. In the given figure of Rs.ABCD, verify with a divider that seg AB = seg CB and seg AD \( \simeq \) seg CD. Similarly measure \( \angle \)BAD and \( \angle \)BCD and verify that they are congruent. (Textbook pg. no. 48)
ℹ️ चित्र व्याख्या (Diagram Explanation): एक पतंग आकृति का चतुर्भुज ABCD दिखाया गया है, जिसमें विकर्ण AC और BD हैं। बिंदु E विकर्ण BD पर स्थित है। इस आकृति में संलग्न भुजाओं के दो अलग-अलग जोड़े समान लंबाई के होते हैं।
Answer:
Solution:
[Students should attempt the above activities on their own.]
In simple words: This activity asks students to use a divider and protractor to verify the properties of a kite (a quadrilateral where two pairs of adjacent sides are equal). Specifically, they should check if adjacent sides are equal and if certain angles are congruent.

🎯 Exam Tip: Use a divider carefully to compare lengths and a protractor precisely to measure angles. This hands-on verification of geometric properties builds intuition and strengthens understanding of definitions for various quadrilaterals.

Std 8 Maths Digest

• Practice Set 8.1 Class 8 Answers
• Practice Set 8.2 Class 8 Answers
• Practice Set 8.3 Class 8 Answers
• Practice Set 9.1 Class 8 Answers
• Practice Set 9.2 Class 8 Answers
• Practice Set 10.1 Class 8 Answers
• Practice Set 10.2 Class 8 Answers