RBSE Solutions Class 9 Maths Chapter 6 Rectilinear Figures More Ques

Chapter 6 Rectilinear Figures Miscellaneous Exercise

Multiple Choice Questions (Q1 to Q12)

 

Question 1. If two angles of a triangle measures 90° and 30°, the third angle is equal to:
(a) 90°
(b) 30°
(c) 60°
(d) 120°
Answer: (c) 60°
In simple words: The total sum of all angles inside any triangle is always 180 degrees. To find the missing angle, subtract the sum of the two known angles from 180 degrees.

🎯 Exam Tip: Always remember that the sum of interior angles in a triangle is 180°. This is a basic and crucial property in geometry.

 

Question 2. The three angles of a triangle are in the ratio of 2 : 3 : 4, its greatest angle measures:
(a) 80°
(b) 60°
(c) 40°
(d) 180°
Answer: (a) 80°
In simple words: First, add up all the parts of the ratio (2+3+4 = 9). Then, divide the total degrees in a triangle (180°) by this sum (9) to find the value of one part. Finally, multiply this value by the largest ratio part (4) to get the biggest angle.

🎯 Exam Tip: When angles are given in a ratio, always find the sum of the ratio parts and equate it to the total sum of angles (180° for a triangle, 360° for a quadrilateral, etc.).

 

Question 3. Each angle of an equilateral triangle measures:
(a) 90°
(b) 30°
(c) 45°
(d) 60°
Answer: (d) 60°
In simple words: An equilateral triangle has all three of its sides equal in length. Because the sides are equal, all its angles are also equal. Since the total angle sum is 180 degrees, each angle must be 180 divided by 3, which is 60 degrees.

🎯 Exam Tip: Memorize the properties of common triangles: equilateral (all angles 60°), isosceles (two angles equal), and right-angled (one angle 90°).

 

Question 5. In figure, the side BC of a ∆ABC has been extended to D. If ∠A = 55° and ∠B = 60° then ∠ACD is:
(a) 120°
(b) 110°
(c) 115°
(d) 125°
Answer: (c) 115°

In simple words: The exterior angle of a triangle (∠ACD) is equal to the sum of its two opposite interior angles (∠A and ∠B). So, add 55° and 60° together to get the answer.

🎯 Exam Tip: The exterior angle theorem is a very useful shortcut for finding unknown angles in triangles, so remember this property well.

 

Question 6. The sum of interior angles of a hexagon is:
(a) 720°
(b) 360°
(c) 540°
(d) 1080°
Answer: (a) 720°
In simple words: A hexagon is a shape with six sides. You can find the sum of its inner angles by using the formula \((n-2) \times 180^\circ\), where 'n' is the number of sides. For a hexagon, \( (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \).

🎯 Exam Tip: Remember the formula \( (n-2) \times 180^\circ \) to calculate the sum of interior angles for any polygon with 'n' sides. For quick recall, learn the sums for common polygons like triangles (180°), quadrilaterals (360°), pentagons (540°), and hexagons (720°).

 

Question 7. The sum of n exterior angles of an n-sided polygon is:
(a) n right angle
(b) 360°
(c) (n-2) right angle
(d) 2n right angle
Answer: (b) 360°
In simple words: No matter how many sides a convex polygon has (whether it's a triangle, square, or a shape with many sides), if you add up all its exterior angles, the total will always be 360 degrees. This is a special property of all convex polygons.

🎯 Exam Tip: Understand that while interior angle sum changes with the number of sides, the sum of exterior angles for any convex polygon is a constant 360°. This is a key distinction.

 

Question 8. The number of sides in a regular polygon is n. Then measure of each interior angle is:
(a) \( \frac {360 }{ n } \) degree
(b) \( \left( \frac {2n - 4}{n} \right) \) right angle
(c) n right angle
(d) 2n right angle
Answer: (b) \( \left( \frac {2n - 4}{n} \right) \) right angle
In simple words: A regular polygon has all its interior angles equal. The sum of all interior angles is \( (n-2) \times 180^\circ \). To find each angle, divide this sum by 'n' (the number of sides). Since a right angle is 90 degrees, you can write 180 as \( 2 \times 90 \), which simplifies the formula.

🎯 Exam Tip: Remember the formula for the sum of interior angles \( (n-2) \times 180^\circ \). For a regular polygon, divide this by 'n' to find each interior angle. Also, be able to convert between degrees and right angles.

 

Question 9. If one angle of a triangle is equal to the sum of the other two angles then the triangle is:
(a) Isosceles triangle
(b) Obtuse angled triangle
(c) Equilateral triangle
(d) Right angled triangle
Answer: (d) Right angled triangle
In simple words: Let the angles be A, B, and C. If A = B + C, and we know that A + B + C = 180° (sum of angles in a triangle), then we can replace B + C with A. So, A + A = 180°, which means 2A = 180°, making A = 90°. A triangle with one 90-degree angle is a right-angled triangle.

🎯 Exam Tip: This is a classic question. The key is to use the property that the sum of all angles in a triangle is 180° to deduce the value of the angle that is equal to the sum of the other two.

 

Question 10. One of the exterior angle of a triangle is 105° and its two opposite interior angles are equal, then each of the equal angle is:
(a) \( 37\frac {1}{2}^\circ \)
(b) \( 52\frac {1}{2}^\circ \)
(c) \( 72\frac {1}{2}^\circ \)
(d) 75°
Answer: (b) \( 52\frac {1}{2}^\circ \)
In simple words: The exterior angle of a triangle is equal to the sum of the two opposite interior angles. If the exterior angle is 105° and the two opposite interior angles are equal (let's call them 'x'), then \( x + x = 105^\circ \). So \( 2x = 105^\circ \), which means \( x = 52.5^\circ \). Each of the equal angles is \( 52\frac {1}{2}^\circ \).

🎯 Exam Tip: When dealing with exterior angles, always remember that an exterior angle equals the sum of its two remote (non-adjacent) interior angles. Use this theorem to quickly set up equations.

 

Question 11. The angles of a triangle are in the ratio 5 : 3 : 7 then the triangle is:
(a) Acute angled triangle
(b) Obtuse angled triangle
(c) Right angled triangle
(d) Equilateral triangle
Answer: (a) Acute angled triangle
In simple words: First, add the ratio numbers: \(5+3+7=15\). Then, divide 180 degrees (total for a triangle) by 15 to find one unit of the ratio: \(180/15 = 12^\circ\). Now, multiply each ratio number by 12 to get the angles: \(5 \times 12 = 60^\circ\), \(3 \times 12 = 36^\circ\), \(7 \times 12 = 84^\circ\). Since all these angles are less than 90 degrees, it is an acute-angled triangle.

🎯 Exam Tip: To classify a triangle by its angles, first calculate all three angles. If all are less than 90°, it's acute. If one is 90°, it's right-angled. If one is greater than 90°, it's obtuse.

 

Question 12. If one of the angle of a triangle is 130° then the angle between the bisectors of the remaining two angles will be
(a) 50°
(b) 65°
(c) 145°
(d) 155°
Answer: (d) 155°
In simple words: If one angle is 130°, the sum of the other two angles is \(180^\circ - 130^\circ = 50^\circ\). When these two angles are bisected, the angle formed by their bisectors (let's call it x) is given by the formula \( x = 90^\circ + \frac{1}{2} \text{(third angle)} \). So, \( x = 90^\circ + \frac{1}{2} (130^\circ) = 90^\circ + 65^\circ = 155^\circ \).

🎯 Exam Tip: Remember the property that the angle formed by the internal bisectors of two angles of a triangle is \( 90^\circ + \frac{1}{2} \) of the third angle. This shortcut helps solve such problems quickly.

 

Question 13. In figure, find ∠A.

Answer: We are given the exterior angle at C, which is \( \angle ACE = 120^\circ \). Using the linear pair property, the interior angle \( \angle ACB = 180^\circ - \angle ACE \). \( \angle ACB = 180^\circ - 120^\circ = 60^\circ \). The exterior angle at B is given as \( \angle ABD = 112^\circ \). According to the exterior angle theorem, an exterior angle of a triangle is equal to the sum of its two opposite interior angles. So, \( \angle ABD = \angle A + \angle ACB \). Substituting the known values: \( 112^\circ = \angle A + 60^\circ \). To find \( \angle A \), we subtract 60° from 112°.
\( \implies \angle A = 112^\circ - 60^\circ \).
\( \implies \angle A = 52^\circ \). Thus, the measure of angle A is 52 degrees.
In simple words: First, use the straight line property to find the angle inside the triangle at C. Then, use the rule that an exterior angle equals the sum of the two opposite interior angles. This helps you find the missing angle A.

🎯 Exam Tip: Always identify linear pairs and use the exterior angle property correctly. These two concepts are fundamental for solving angle problems in triangles.

 

Question 14. In figure, ∠B = 60° and ∠C = 40°. Find the measure of ∠A.

Answer: We know that the sum of all interior angles in any triangle is \(180^\circ\). Given angles are \( \angle B = 60^\circ \) and \( \angle C = 40^\circ \). So, \( \angle A + \angle B + \angle C = 180^\circ \). Substitute the given values into the equation: \( \angle A + 60^\circ + 40^\circ = 180^\circ \). \( \angle A + 100^\circ = 180^\circ \). To find \( \angle A \), subtract 100° from 180°.
\( \implies \angle A = 180^\circ - 100^\circ \).
\( \implies \angle A = 80^\circ \). Therefore, the measure of angle A is 80 degrees.
In simple words: Just add the two given angles, then take that sum away from 180 degrees. The remaining number is the third angle, because all three angles in a triangle always add up to 180.

🎯 Exam Tip: This is a direct application of the angle sum property of a triangle. Ensure you perform the addition and subtraction carefully to avoid calculation errors.

 

Question 16. In figure, find ∠A.

Answer: We will find the interior angles of the quadrilateral ABCD. At point B, the exterior angle is \( 100^\circ \). Using the linear pair property: \( 100^\circ + \angle ABC = 180^\circ \).
\( \implies \angle ABC = 180^\circ - 100^\circ = 80^\circ \). At point C, the exterior angle is \( 95^\circ \). Using the linear pair property: \( \angle DCB + 95^\circ = 180^\circ \).
\( \implies \angle DCB = 180^\circ - 95^\circ = 85^\circ \). Similarly, at point D, the exterior angle is \( 82^\circ \). Using the linear pair axiom: \( 82^\circ + \angle CDA = 180^\circ \).
\( \implies \angle CDA = 180^\circ - 82^\circ = 98^\circ \). Now, in quadrilateral ABCD, the sum of all interior angles is \( 360^\circ \). \( \angle A + \angle ABC + \angle DCB + \angle CDA = 360^\circ \). Substitute the calculated interior angles: \( \angle A + 80^\circ + 85^\circ + 98^\circ = 360^\circ \). \( \angle A + 263^\circ = 360^\circ \). To find \( \angle A \):
\( \implies \angle A = 360^\circ - 263^\circ \).
\( \implies \angle A = 97^\circ \). Thus, angle A is 97 degrees. Finding interior angles from exterior ones is a common geometry step.
In simple words: First, find the inner angles at B, C, and D by subtracting their outer angles from 180 degrees. Then, add these three inner angles together. Subtract this sum from 360 degrees (because all angles in a four-sided shape add up to 360) to find angle A.

🎯 Exam Tip: For problems involving quadrilaterals with exterior angles, always convert the exterior angles to interior angles first using the linear pair property before applying the angle sum property of quadrilaterals.

 

Question 18. Each exterior angle of a regular polygon is 45°. Find the number of sides in the polygon.
Answer: For any regular polygon, the measure of each exterior angle is found by dividing 360 degrees by the number of sides. Given that each exterior angle is \( 45^\circ \). Let 'n' be the number of sides of the polygon. So, \( \text{Measure of each exterior angle} = \frac{360^\circ}{n} \). \( 45^\circ = \frac{360^\circ}{n} \). To find 'n', rearrange the equation:
\( \implies n = \frac{360^\circ}{45^\circ} \).
\( \implies n = 8 \). Hence, the number of sides in the polygon is 8. A polygon with 8 sides is called an octagon.
In simple words: To find how many sides a regular polygon has when you know its outside angle, just divide 360 degrees by the size of that outside angle.

🎯 Exam Tip: Remember the property that the sum of all exterior angles of any convex polygon is always 360°. For a regular polygon, each exterior angle is \( 360^\circ / n \), where 'n' is the number of sides.

 

Question 19. A regular polygon has 12 sides, find the measure of each of its interior angles.
Answer: For a regular polygon with 'n' sides, the measure of each interior angle is given by the formula: \( \text{Each interior angle} = \left( \frac{2n-4}{n} \right) \times 90^\circ \). Here, the number of sides \( n = 12 \). Substitute \( n=12 \) into the formula: \( \text{Each interior angle} = \left( \frac{2 \times 12 - 4}{12} \right) \times 90^\circ \). \( = \left( \frac{24 - 4}{12} \right) \times 90^\circ \). \( = \left( \frac{20}{12} \right) \times 90^\circ \). Simplify the fraction and calculate: \( = \frac{5}{3} \times 90^\circ \).
\( \implies = 5 \times 30^\circ \).
\( \implies = 150^\circ \). Therefore, each interior angle of the 12-sided regular polygon is 150 degrees. This polygon is also known as a dodecagon.
In simple words: Use the formula for each inner angle of a polygon. Put in the number of sides (12) into the formula, then do the math. The formula helps you find how big each angle is when all the angles are the same.

🎯 Exam Tip: The formula for each interior angle of a regular polygon can also be expressed as \( \frac{(n-2) \times 180^\circ}{n} \). Both formulas are equivalent; choose the one you find easier to remember and apply consistently.

 

Question 20. The sum of the interior angles of a polygon is 10 right angles. Find the number of sides.
Answer: We are given that the sum of the interior angles of a polygon is 10 right angles. One right angle is \( 90^\circ \). So, 10 right angles is \( 10 \times 90^\circ = 900^\circ \). The formula for the sum of the interior angles of a polygon with 'n' sides is \( (n-2) \times 180^\circ \). So, we set the formula equal to the given sum: \( (n-2) \times 180^\circ = 900^\circ \). Divide both sides by \( 180^\circ \): \( n-2 = \frac{900^\circ}{180^\circ} \). \( n-2 = 5 \). Add 2 to both sides to find 'n':
\( \implies n = 5 + 2 \).
\( \implies n = 7 \). Thus, the number of sides of the polygon is 7. A polygon with seven sides is known as a heptagon.
In simple words: First, change "10 right angles" into degrees (10 times 90). Then, use the polygon angle formula, \( (n-2) \times 180 \), and set it equal to your degrees. Solve for 'n' to find the number of sides.

🎯 Exam Tip: Always remember that a "right angle" means 90°. When a sum of angles is given in terms of right angles, convert it to degrees first before using the polygon angle sum formula.

 

Question 21. Examine, if it is possible to have a regular polygon whose each interior angle is 110°.
Answer: If each interior angle of a regular polygon is \( 110^\circ \), we can find its exterior angle. The interior and exterior angles at any vertex sum to \( 180^\circ \) (linear pair). So, measure of each exterior angle \( = 180^\circ - 110^\circ = 70^\circ \). For any regular polygon, the sum of all exterior angles is \( 360^\circ \). If there are 'n' sides, and each exterior angle is \( 70^\circ \), then: \( n \times 70^\circ = 360^\circ \). To find 'n':
\( \implies n = \frac{360^\circ}{70^\circ} \).
\( \implies n = \frac{36}{7} \). This value \( n = \frac{36}{7} \) is equal to \( 5\frac{1}{7} \), which is not a whole number. Since the number of sides of a polygon must be a whole number, a regular polygon with each interior angle measuring \( 110^\circ \) cannot exist. The number of sides has to be an integer.
In simple words: If each inner angle is 110 degrees, then each outer angle must be 70 degrees (180 minus 110). Since all outer angles of a polygon always add up to 360 degrees, you divide 360 by 70. If the answer is not a whole number, then such a polygon cannot exist.

🎯 Exam Tip: When checking if a polygon can exist, always ensure that the calculated number of sides ('n') is a positive integer. Fractional or negative values of 'n' indicate impossibility.

 

Question 22. In a ∆ABC, ∠A + ∠B = ∠C, then find the greatest angle of the triangle ABC.
Answer: We know that the sum of interior angles in any triangle is \( 180^\circ \). So, \( \angle A + \angle B + \angle C = 180^\circ \). We are given the condition \( \angle A + \angle B = \angle C \). Substitute \( \angle C \) for \( (\angle A + \angle B) \) in the angle sum equation: \( \angle C + \angle C = 180^\circ \). \( 2\angle C = 180^\circ \). Divide by 2 to find \( \angle C \):
\( \implies \angle C = \frac{180^\circ}{2} \).
\( \implies \angle C = 90^\circ \). Therefore, the greatest angle in this triangle is \( 90^\circ \). This means it is a right-angled triangle.
In simple words: Since two angles together make the third angle, and all three angles add up to 180, it means the third angle must be half of 180, which is 90 degrees. This 90-degree angle is the biggest one.

🎯 Exam Tip: This is a special case problem where one angle is exactly the sum of the other two. Recognize that this instantly implies one angle is 90°, making it a right-angled triangle.

 

Question 23. Find the sum of the interior angles of an octagon.
Answer: To find the sum of the interior angles of a polygon, we use the formula \( (n-2) \times 180^\circ \), where 'n' is the number of sides. An octagon is a polygon with 8 sides. So, \( n = 8 \). Substitute \( n=8 \) into the formula: Sum of the interior angles \( = (8-2) \times 180^\circ \). \( = 6 \times 180^\circ \). \( = 1080^\circ \). Therefore, the sum of the interior angles of an octagon is 1080 degrees. This formula helps to easily calculate the total angle inside any polygon based on its number of sides.
In simple words: An octagon has 8 sides. Take away 2 from the number of sides (8-2 = 6). Then, multiply that number (6) by 180 degrees. This gives you the total sum of all the inside angles of the octagon.

🎯 Exam Tip: Always correctly identify the number of sides 'n' for the given polygon. The formula \( (n-2) \times 180^\circ \) is universal for the sum of interior angles of any polygon.

 

Question 24. Find the measure of each interior angle of a 10-sided regular polygon.
Answer: For a regular polygon with 'n' sides, the measure of each interior angle is given by: \( \text{Each interior angle} = \left( \frac{n-2}{n} \right) \times 180^\circ \). Alternatively, using the given formula in the source: \( \text{Each interior angle} = \left( \frac{2n-4}{n} \right) \times 90^\circ \). For a 10-sided regular polygon, \( n = 10 \). Substitute \( n=10 \) into the formula: \( \text{Each interior angle} = \left( \frac{2 \times 10 - 4}{10} \right) \times 90^\circ \). \( = \left( \frac{20 - 4}{10} \right) \times 90^\circ \). \( = \left( \frac{16}{10} \right) \times 90^\circ \). Simplify the fraction and calculate: \( = \frac{8}{5} \times 90^\circ \).
\( \implies = 8 \times 18^\circ \).
\( \implies = 144^\circ \). Thus, each interior angle of a 10-sided regular polygon (decagon) is 144 degrees. This shows that as the number of sides increases, the interior angles also generally increase.
In simple words: A 10-sided shape is a decagon. Use the formula for one inner angle of a regular polygon. Put 10 for 'n' and then do the math. This tells you how big each of its equal inside angles is.

🎯 Exam Tip: Be careful with calculations involving fractions and degrees. Double-check your arithmetic, especially when simplifying the fraction before multiplying by 90° or 180°.

 

Question 25. The exterior angles of a triangle obtained by producing, the sides in order are 110°, 130° and x. Find x.
Answer: The sum of the exterior angles of any convex polygon (including a triangle) is always \( 360^\circ \). We are given the three exterior angles of a triangle as \( 110^\circ \), \( 130^\circ \), and \( x \). So, we can set up the equation: \( 110^\circ + 130^\circ + x = 360^\circ \). Add the known angles: \( 240^\circ + x = 360^\circ \). To find \( x \), subtract \( 240^\circ \) from \( 360^\circ \):
\( \implies x = 360^\circ - 240^\circ \).
\( \implies x = 120^\circ \). Therefore, the value of \( x \) is 120 degrees. This property is very useful for all types of polygons, not just triangles.
In simple words: All the outside angles of any triangle always add up to 360 degrees. Add the two given outside angles together. Then, subtract that sum from 360 to find the last outside angle.

🎯 Exam Tip: A common mistake is to confuse interior and exterior angles. Always remember that the sum of exterior angles for any convex polygon is a constant 360°, regardless of the number of sides.

 

Question 26. One interior angle of a hexagon is 165° and the remaining interior angles are x° each. Find x.
Answer: First, find the sum of the interior angles of a hexagon. A hexagon has 6 sides, so \( n=6 \). Sum of interior angles \( = (n-2) \times 180^\circ \). \( = (6-2) \times 180^\circ \). \( = 4 \times 180^\circ = 720^\circ \). We are given that one interior angle is \( 165^\circ \), and the remaining \( (6-1) = 5 \) angles are \( x^\circ \) each. So, the sum of all interior angles can also be written as: \( 165^\circ + x + x + x + x + x = 720^\circ \). \( 165^\circ + 5x = 720^\circ \). Subtract \( 165^\circ \) from both sides:
\( \implies 5x = 720^\circ - 165^\circ \).
\( \implies 5x = 555^\circ \). Divide by 5 to find \( x \):
\( \implies x = \frac{555^\circ}{5} \).
\( \implies x = 111^\circ \). Thus, each of the remaining interior angles is 111 degrees. This problem combines the general formula with specific angle information.
In simple words: First, find out what all the inside angles of a 6-sided shape (hexagon) add up to (it's 720 degrees). Then, take away the one angle that is given (165 degrees). The rest of the angles (which are 5 of them, all 'x') add up to the remaining amount. Divide that amount by 5 to find 'x'.

🎯 Exam Tip: Be careful with the number of 'x' terms in the sum. If one angle is specified, subtract 1 from 'n' to find how many remaining angles are equal to 'x'.

 

Question 28. In figure, if ∠x – ∠y = 10°, then find the measure of ∠x and ∠y.

Answer: From the figure, we can see that \( 120^\circ \) is an exterior angle to the triangle. According to the exterior angle property, an exterior angle of a triangle is equal to the sum of its two opposite interior angles. So, \( \angle x + \angle y = 120^\circ \) ...(i). We are also given the condition: \( \angle x - \angle y = 10^\circ \) ...(ii). Now we have a system of two linear equations. Add equation (i) and equation (ii): \( (\angle x + \angle y) + (\angle x - \angle y) = 120^\circ + 10^\circ \). \( 2\angle x = 130^\circ \). Divide by 2 to find \( \angle x \):
\( \implies \angle x = \frac{130^\circ}{2} \).
\( \implies \angle x = 65^\circ \). Substitute the value of \( \angle x \) into equation (i): \( 65^\circ + \angle y = 120^\circ \). Subtract \( 65^\circ \) from both sides to find \( \angle y \):
\( \implies \angle y = 120^\circ - 65^\circ \).
\( \implies \angle y = 55^\circ \). Therefore, \( \angle x = 65^\circ \) and \( \angle y = 55^\circ \). This method of solving simultaneous equations is common in geometry problems.
In simple words: You have two rules about angles x and y. One says they add up to 120, and the other says their difference is 10. Add these two rules together to find x. Then, use that x value in the first rule to find y.

🎯 Exam Tip: This problem combines the exterior angle theorem with solving simultaneous equations. Ensure you correctly set up both equations and solve them carefully, either by substitution or elimination.

 

Question 30. From the given figure, prove that ∠x + ∠y = ∠A + ∠C.

Answer: Construction: Join points A and C to form a diagonal AC. Consider \( \triangle ABC \). The exterior angle \( \angle x \) (at vertex B) is equal to the sum of its two opposite interior angles. So, \( \angle x = \angle BAC + \angle BCA \). In the figure, \( \angle BAC \) is labelled \( \angle 1 \) and \( \angle BCA \) is labelled \( \angle 2 \). Therefore, \( \angle x = \angle 1 + \angle 2 \) ...(i). Now consider \( \triangle ADC \). The exterior angle \( \angle y \) (at vertex D) is equal to the sum of its two opposite interior angles. So, \( \angle y = \angle DAC + \angle DCA \). In the figure, \( \angle DAC \) is labelled \( \angle 3 \) and \( \angle DCA \) is labelled \( \angle 4 \). Therefore, \( \angle y = \angle 3 + \angle 4 \) ...(ii). Now, add equation (i) and equation (ii): \( \angle x + \angle y = (\angle 1 + \angle 2) + (\angle 3 + \angle 4) \). Rearrange the terms:
\( \implies \angle x + \angle y = (\angle 1 + \angle 3) + (\angle 2 + \angle 4) \). From the figure, the total angle at vertex A is \( \angle A = \angle 1 + \angle 3 \). And the total angle at vertex C is \( \angle C = \angle 2 + \angle 4 \). Substitute these into the equation:
\( \implies \angle x + \angle y = \angle A + \angle C \). Hence proved. This demonstrates a useful property of angles in quadrilaterals by breaking them down into triangles.
In simple words: Draw a line inside the shape from A to C. This splits the shape into two triangles. For each triangle, the outside angle (x or y) is equal to the two opposite inside angles added together. Add these two results. Group the small angles back into the big angles A and C, and you'll see they match.

🎯 Exam Tip: When proving angle relationships in complex figures, breaking them down into simpler triangles and applying basic theorems like the exterior angle property often simplifies the problem. Clearly label all angles for better understanding.

 

Question 31. From figure, find ∠x. Here BO and CO are bisectors of ∠B and ∠C respectively.

Answer: We know a property that when angle bisectors of two angles (∠B and ∠C) of a triangle meet at a point O, the angle formed at O (∠BOC or ∠x) is related to the third angle (∠A) by the formula: \( \angle BOC = 90^\circ + \frac {1}{2} \angle A \). From the information provided, we assume that \( \angle A = 80^\circ \). Substitute this value into the formula: \( \angle x = 90^\circ + \frac {1}{2} \times 80^\circ \). \( = 90^\circ + 40^\circ \).
\( \implies \angle x = 130^\circ \). Therefore, the measure of angle x is 130 degrees. This property is very useful for quickly solving problems involving angle bisectors in triangles.
In simple words: There's a special rule for when two angle lines inside a triangle meet. The angle they make (x) is found by adding 90 degrees to half of the third angle (angle A). Since angle A is 80, half of it is 40. So 90 plus 40 gives you x.

🎯 Exam Tip: Memorize the formula for the angle formed by internal angle bisectors: \( \angle BOC = 90^\circ + \frac{1}{2} \angle A \). This is a frequently tested concept in geometry.

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