RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
Two lines crossing create a small family of angles that are locked to one another in fixed ways. Once you can name them — vertically opposite, corresponding, alternate interior, co-interior — most geometry problems reduce to spotting a pair and writing one equation. Adding a transversal to a pair of parallel lines is where this chapter really earns its marks.
Two angles form a linear pair. If one of them measures 47°, the other measures:
Answer
The other measures 133°. Angles in a linear pair are supplementary, so they add up to 180°. Therefore the second angle is 180° - 47° = 133°. The value 43° would be correct only if the angles were complementary, adding to 90°.
A transversal cuts two parallel lines. If one pair of co-interior angles is in the ratio 4 : 5, the smaller angle is:
Answer
The smaller angle is 80°. Co-interior angles between parallel lines are supplementary, so their sum is 180°. Let them be 4x and 5x. Then 4x + 5x = 180°, so 9x = 180° and x = 20°. The angles are 4 × 20° = 80° and 5 × 20° = 100°, so the smaller is 80°.
In a triangle, an exterior angle measures 118° and one of the interior opposite angles measures 55°. The other interior opposite angle is:
Answer
The other interior opposite angle is 63°. By the exterior angle property, an exterior angle equals the sum of the two interior opposite angles. So 118° = 55° + the unknown angle, giving 118° - 55° = 63°.
If two lines intersect and one of the four angles formed is a right angle, then:
Answer
All four angles are right angles. The angle vertically opposite the given right angle is also 90°, since vertically opposite angles are equal. Each of the remaining two angles forms a linear pair with a 90° angle, so each equals 180° - 90° = 90°. Hence the lines are perpendicular and all four angles measure 90°.
Assertion (A): A triangle cannot have two obtuse angles. Reason (R): The sum of the three interior angles of a triangle is 180°.
Answer
Both A and R are true and R is the correct explanation of A. The angle sum property in R is a true and standard result. If a triangle had two obtuse angles, each greater than 90°, those two alone would already exceed 180°, leaving nothing for the third angle, which must itself be positive. This contradiction proves A, and it follows directly from the angle sum property, so R correctly explains A.
Two supplementary angles are such that one is 24° more than three times the other. Find both angles.
Answer
Let the smaller angle be x°. Then the other is (3x + 24)°. Since they are supplementary, x + 3x + 24 = 180. So 4x = 156, giving x = 39. The other angle is 3(39) + 24 = 117 + 24 = 141°. The two angles are 39° and 141°, and their sum checks out as 180°.
Lines AB and CD intersect at O. If ∠AOC = (2x + 15)° and ∠BOD = (3x - 25)°, find x and hence the measure of ∠AOD.
Answer
∠AOC and ∠BOD are vertically opposite angles, so they are equal. 2x + 15 = 3x - 25 Rearranging, 15 + 25 = 3x - 2x, so x = 40. Then ∠AOC = 2(40) + 15 = 95°. ∠AOD and ∠AOC form a linear pair along line CD, so ∠AOD = 180° - 95° = 85°. Hence x = 40 and ∠AOD = 85°.
The angles of a triangle are in the ratio 2 : 3 : 7. Find each angle and classify the triangle by its angles.
Answer
Let the angles be 2x, 3x and 7x. By the angle sum property, 2x + 3x + 7x = 180°, so 12x = 180° and x = 15°. The angles are 30°, 45° and 105°. Since one angle exceeds 90°, the triangle is obtuse-angled.
Prove that if a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. Then use the result: lines PQ and RS are parallel, a transversal cuts PQ at A and RS at B, and one alternate interior angle at A is (5y - 10)° while the alternate interior angle at B is (3y + 30)°. Find y and both angles.
Answer
Proof: Let the transversal cut the parallel lines PQ and RS at A and B respectively. Consider an alternate interior angle pair, say ∠1 at A and ∠2 at B, and let ∠3 at A be the corresponding angle to ∠2. Since PQ is parallel to RS, corresponding angles are equal, so ∠3 = ∠2. Also ∠1 and ∠3 are vertically opposite angles at the point A, so ∠1 = ∠3. By Euclid's axiom that things equal to the same thing are equal to one another, ∠1 = ∠2. Hence alternate interior angles are equal. Application: Since the two given angles are alternate interior angles between parallel lines, they are equal. 5y - 10 = 3y + 30 2y = 40, so y = 20. Each angle is 5(20) - 10 = 90°, and checking the other expression, 3(20) + 30 = 90°. Both angles measure 90°.
In triangle XYZ, side YZ is produced to a point W. The bisectors of ∠XYZ and ∠XZW meet at point T. (a) If ∠YXZ = 68°, find ∠YTZ. (b) State the general relation between ∠YTZ and ∠YXZ and prove it.
Answer
(b) General relation first, since (a) follows from it. Let ∠XYZ = 2b and ∠YXZ = 2a, so the exterior angle ∠XZW = 2a + 2b by the exterior angle property. TY bisects ∠XYZ, so ∠TYZ = b. TZ bisects ∠XZW, so ∠TZW = a + b. In triangle TYZ, ∠TZW is an exterior angle at Z, so ∠TZW = ∠TYZ + ∠YTZ. Therefore a + b = b + ∠YTZ, which gives ∠YTZ = a. Since 2a = ∠YXZ, we get a = (1/2)∠YXZ. So ∠YTZ = (1/2)∠YXZ: the angle between the internal bisector of one base angle and the external bisector of the other is half the vertex angle. (a) With ∠YXZ = 68°, ∠YTZ = (1/2)(68°) = 34°.
Read the following and answer the questions that follow: A road crosses two parallel railway tracks. A surveyor marks the crossing points and measures one of the angles the road makes with the first track as 65°, taking the road as a transversal cutting the two parallel tracks. (a) Find the corresponding angle at the second track and justify your answer. (b) Find the co-interior angle on the same side of the road between the two tracks. (c) Find the angle vertically opposite the 65° angle. (d) The surveyor checks a third track and finds it parallel to the first. What can be concluded about the third and second tracks, and why?
Answer
(a) The corresponding angle at the second track is 65°. When a transversal cuts two parallel lines, corresponding angles are equal, so the angle in the matching position at the second crossing has the same measure. (b) Co-interior angles between parallel lines are supplementary, so the required angle is 180° - 65° = 115°. (c) Vertically opposite angles are equal, so the angle vertically opposite the 65° angle is also 65°. (d) The third track is parallel to the second track. Both the second and third tracks are parallel to the first, and lines parallel to the same line are parallel to one another.
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