SUMMARY: The chapter "Lines and Angles" in Class 9 Mathematics introduces students to the basic concepts of lines, angles, and their properties. KEY TOPICS: types of angles, complementary angles, supplementary angles, adjacent angles, linear pair, vertically opposite angles, parallel lines, transversal, angle sum property of a triangle, properties of parallel lines with transversal
If two lines intersect, what is the relationship between the vertically opposite angles?
AThey are equal
BThey are supplementary
CThey are complementary
DThey are adjacent
Check answerHide answer
Correct answer: Option 1 — They are equal
Q31 Mark
Which of the following pairs of angles are complementary?
A30 degrees and 60 degrees
B45 degrees and 45 degrees
C90 degrees and 90 degrees
D120 degrees and 30 degrees
Check answerHide answer
Correct answer: Option 1 — 30 degrees and 60 degrees
Q41 Mark
Two parallel lines are cut by a transversal. If one of the alternate interior angles is 75 degrees, what is the measure of the other alternate interior angle?
A75 degrees
B105 degrees
C90 degrees
D180 degrees
Check answerHide answer
Correct answer: Option 1 — 75 degrees
Q51 Mark
In a pair of supplementary angles, if one angle measures 2x and the other measures 3x, what is the value of x?
A18 degrees
B36 degrees
C12 degrees
D24 degrees
Check answerHide answer
Correct answer: Option 2 — 36 degrees
Short Answer Questions5 questions
Q63 Marks
Define complementary angles and provide an example.
View sample solutionHide solution
Complementary angles are two angles whose measures add up to 90 degrees. For example, if one angle measures 30 degrees, the other angle must measure 60 degrees to be complementary.
Q73 Marks
What is the relationship between parallel lines and a transversal?
View sample solutionHide solution
When a transversal intersects two parallel lines, several angles are formed. The corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.
Q83 Marks
Explain the concept of vertically opposite angles with an example.
View sample solutionHide solution
Vertically opposite angles are the angles opposite each other when two lines intersect. They are always equal. For instance, if two intersecting lines form angles of 40 degrees and 140 degrees, the angles opposite to these will also be 40 degrees and 140 degrees, respectively.
Q93 Marks
If two angles are supplementary, what is the sum of their measures?
View sample solutionHide solution
If two angles are supplementary, the sum of their measures is 180 degrees. For example, if one angle measures 110 degrees, the other must measure 70 degrees to satisfy this condition.
Q103 Marks
Prove that the sum of the angles in a triangle is 180 degrees using the concept of parallel lines.
View sample solutionHide solution
To prove that the sum of the angles in a triangle is 180 degrees, draw a line parallel to one side of the triangle through the opposite vertex. The angles formed at the intersection with the other two sides are equal to the interior angles of the triangle, showing that their sum is 180 degrees.
Long Answer Questions5 questions
Q116 Marks
Explain the concept of complementary angles and provide an example. How can you use this concept to solve problems involving angles in a triangle?
View sample solutionHide solution
Complementary angles are two angles whose measures add up to 90 degrees. For example, if one angle measures 30 degrees, the other angle must measure 60 degrees to be complementary. In the context of a triangle, if one angle is known to be complementary to another, you can find the measure of the third angle by using the fact that the sum of all angles in a triangle is always 180 degrees. Thus, knowing two angles allows you to easily calculate the third angle by subtracting the sum of the known angles from 180 degrees.
Q126 Marks
Describe the properties of parallel lines cut by a transversal. How can these properties be applied to find unknown angle measures?
View sample solutionHide solution
When two parallel lines are cut by a transversal, several angle relationships are formed, including corresponding angles, alternate interior angles, and consecutive interior angles. Corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (add up to 180 degrees). By using these properties, you can find unknown angle measures by setting up equations based on the relationships. For instance, if you know one angle is 70 degrees, you can conclude that the corresponding angle is also 70 degrees, while the consecutive interior angle would be 110 degrees.
Q136 Marks
Prove that the sum of the angles in a triangle is 180 degrees using the properties of parallel lines and transversals.
View sample solutionHide solution
To prove that the sum of the angles in a triangle is 180 degrees, consider a triangle ABC. Extend one side, say BC, and draw a line parallel to AC through point B. The angle at A (angle CAB) and the angle formed at B (angle ABC) are alternate interior angles with respect to the transversal AB and the parallel line through B. Similarly, the angle at C (angle ACB) is corresponding to the angle formed at B. By the properties of parallel lines, we know that the sum of these angles (angle CAB + angle ABC + angle ACB) equals the angle formed on a straight line, which is 180 degrees. Thus, we have proved that the sum of the angles in triangle ABC is 180 degrees.
Q146 Marks
What are vertically opposite angles? Provide a detailed explanation along with a diagram to illustrate your answer.
View sample solutionHide solution
Vertically opposite angles are the angles that are opposite each other when two lines intersect. When two lines cross, they form two pairs of vertically opposite angles. For example, if two lines intersect at point O, creating angles AOB, AOC, BOD, and DOC, then angle AOB is vertically opposite to angle COD, and angle AOC is vertically opposite to angle BOD. These angles are always equal. To illustrate this, you can draw two intersecting lines and label the angles. By measuring or calculating, you can show that the vertically opposite angles are equal, reinforcing the concept.
Q156 Marks
Using the concept of angle bisectors, explain how to construct the angle bisector of a given angle and prove that the angles formed are equal.
View sample solutionHide solution
To construct the angle bisector of a given angle, start by drawing the angle and labeling its vertex as A and its arms as AB and AC. Using a compass, place the pointer at point A and draw an arc that intersects both arms of the angle at points D and E. Without changing the compass width, draw arcs from points D and E, creating intersection point F inside the angle. Draw a line from point A through point F; this line is the angle bisector. To prove that the angles formed are equal, observe that triangles ADF and AEF are congruent by the Side-Angle-Side postulate, as AD = AE (radii of the same arc), AF is common, and angle DAF = angle EAF. Therefore, angle DAB = angle EAC, proving that the angle bisector divides the angle into two equal parts.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): If two lines intersect, the vertically opposite angles are equal.
Reason (R): This property holds true for all intersecting lines.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): If two parallel lines are cut by a transversal, then the alternate interior angles are equal.
Reason (R): This is a property of parallel lines and transversals.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): If two angles are complementary, then their sum is 90 degrees.
Reason (R): Complementary angles can also be adjacent.
Show explanationHide explanation
Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The sum of the angles in a triangle is always 180 degrees.
Reason (R): This is true only for isosceles triangles.
Show explanationHide explanation
Correct answer: Option 4 —
A is false, but R is true.
Q201 Mark
Assertion (A): A pair of adjacent angles formed by two intersecting lines are always supplementary.
Reason (R): Adjacent angles can sometimes be complementary.
Show explanationHide explanation
Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: If two lines intersect, the vertically opposite angles are equal.
Statement 2: If two parallel lines are cut by a transversal, the corresponding angles are supplementary.
Show answerHide answer
Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: The sum of the interior angles of a triangle is 180 degrees.
Statement 2: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: If two angles are complementary, their sum is 90 degrees.
Statement 2: If two angles are supplementary, their sum is 180 degrees.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: Two lines are parallel if they intersect at a right angle.
Statement 2: The alternate interior angles formed by a transversal are equal if the lines are parallel.
Show answerHide answer
Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: A transversal intersects two parallel lines creating alternate exterior angles that are equal.
Statement 2: If two angles are adjacent, they cannot be supplementary.
Show answerHide answer
Correct answer: Option 4 —
Both statements are false.
