CBSE Class 7 Mathematics A Tale of Three Intersecting Lines MCQs Set 11

Answer: (c) ∠C

Explanation:
1. In short form, an angle of a triangle is named by the vertex at which it is formed.
2. In ∠BCA the middle letter is C, so the vertex is C.
3. So ∠BCA is written as ∠C.

Teacher's Note:
1. The middle letter of a three-letter angle name is always the vertex.
2. The short form keeps only that vertex letter.

Answer: (d) an equilateral triangle

Explanation:
1. A triangle with all three sides equal is an equilateral triangle.
2. An isosceles triangle has two equal sides and a scalene triangle has all sides different.

Teacher's Note:
1. 'Equi' means equal and 'lateral' means side.
2. Equilateral, isosceles and scalene describe sides; right-angled describes an angle.

Answer: (b) a compass

Explanation:
1. With a compass we draw arcs of exactly the required radii from the two ends of the base.
2. The point where the arcs cross is the third vertex, found in one step.
3. With only a ruler we would have to shift the third point again and again. So the answer is a compass.

Teacher's Note:
1. A compass draws all points at a fixed distance from a centre.
2. A protractor measures angles, not lengths.

Answer: (b) perpendicular to the opposite side

Explanation:
1. An altitude is the perpendicular line segment from a vertex to the opposite side.
2. Its length is the height of that vertex above the side taken as base.
3. So it is perpendicular to the opposite side.

Teacher's Note:
1. Altitude means height, and height is always measured at 90°.
2. Every triangle has three altitudes, one from each vertex.

Answer: (c) 180°

Explanation:
1. By the angle sum property, ∠A + ∠B + ∠C = 180° in every triangle.
2. This is proved by drawing a line through one vertex parallel to the opposite side.
3. So the sum is 180°.

Teacher's Note:
1. The three angles together make a straight angle.
2. 360° is the sum for a quadrilateral, not a triangle.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 3 + 4 = 7, and 7 is less than 8.
2. So the longest side is more than the sum of the other two; the triangle inequality fails and A is true.
3. R is the triangle inequality itself, which is true and is exactly why A holds. So R correctly explains A.

Teacher's Note:
1. Check only the longest side against the sum of the other two.
2. Here the arcs from the ends of the 8 cm base would never meet.

Answer: (d) A is false, but R is true.

Explanation:
1. If two angles added up to 180°, the third angle would be 180° − 180° = 0°, and no triangle would form. So A is false.
2. R is the angle sum property, which is true.
3. So A is false but R is true.

Teacher's Note:
1. Every angle of a triangle must be more than 0°.
2. So any two angles must add up to less than 180°.

Answer: (a) Only Statement I is true

Explanation:
1. Statement I is the correct meaning of a scalene triangle, so it is true.
2. An acute-angled triangle must have all three angles acute. Every triangle already has at least two acute angles, so Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. 'Acute-angled' means all three angles are acute.
2. No triangle has only one acute angle.

Answer: (c) 70°

Explanation:
1. By the angle sum property, third angle = 180° − (65° + 45°).
2. = 180° − 110° = 70°.

Teacher's Note:
1. First add the two given angles, then subtract from 180°.
2. Option (b) is just the sum of the two given angles, a common trap.

Answer: (c) 6 cm, 7 cm, 12 cm

Explanation:
1. Compare the longest length with the sum of the other two.
2. (a) 2 + 3 = 5 < 6, fails. (b) 4 + 5 = 9 < 10, fails. (d) 3 + 3 = 6 < 7, fails.
3. (c) 6 + 7 = 13 > 12, so the triangle inequality holds. The answer is 6 cm, 7 cm, 12 cm.

Teacher's Note:
1. Arrange the three lengths in increasing order first.
2. Only the longest length needs to be checked.

Answer: (d) (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q)

Explanation:
1. All three sides equal: equilateral, so (i)-(r).
2. Exactly two sides equal: isosceles, so (ii)-(s).
3. All sides different: scalene, so (iii)-(p).
4. One angle of 90°: right-angled, so (iv)-(q). The answer is (i)-(r), (ii)-(s), (iii)-(p), (iv)-(q).

Teacher's Note:
1. Three names come from sides and one from an angle.
2. Match the easiest pair first and then remove options.

Answer: (b) an isosceles triangle

Explanation:
1. From the figure, AB = AC = 5 cm and BC = 7 cm.
2. Exactly two sides are equal.
3. So △ABC is an isosceles triangle.

Teacher's Note:
1. Read every length marked in the figure.
2. 'Right-angled' is a type by angles, not by sides.

Answer: (b) the point where the two arcs cross each other

Explanation:
1. The arc from one end of the base has all points at the first given distance from that end.
2. The arc from the other end has all points at the second given distance.
3. Only the crossing point is at both correct distances, so the third vertex is the point where the two arcs cross.

Teacher's Note:
1. A point on both arcs satisfies both length conditions at once.
2. Midpoints and perpendiculars play no part here.

Answer: (a) the sum of the two given angles is less than 180°

Explanation:
1. The two arms meet only when the two given angles add up to less than 180°.
2. If the sum is 180° the arms are parallel; if it is more, they move apart. Either way they never meet.
3. The length of the included side makes no difference. So the condition is sum less than 180°.

Teacher's Note:
1. One of the two angles may even be obtuse, as long as the sum is below 180°.
2. The base length never decides whether the arms meet.

Answer: (c) obtuse-angled triangle

Explanation:
1. In an obtuse-angled triangle, the foot of the perpendicular from a vertex next to the obtuse angle falls beyond the opposite side.
2. The base has to be extended first, and the altitude lies outside the triangle.
3. So the answer is an obtuse-angled triangle.

Teacher's Note:
1. In acute-angled triangles all three altitudes lie inside.
2. In a right-angled triangle two altitudes are sides of the triangle.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. The longest length is 100, and 1 + 100 = 101 > 100. The other comparisons (100 < 101 and 1 < 200) also hold.
2. So the triangle inequality is satisfied and a long, thin triangle can be drawn. A is true.
3. R states exactly this condition, so R is true and correctly explains A.

Teacher's Note:
1. A triangle may be very thin and still exist.
2. Even 1 cm is enough to beat equality here.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Three equal lengths always satisfy the triangle inequality (a < a + a), so A is true.
2. By definition an altitude is perpendicular to the side it meets, so R is also true.
3. But altitudes have nothing to do with why an equilateral triangle can be drawn, so R does not explain A.

Teacher's Note:
1. Check whether R is really the reason for A, not just a true fact.
2. Here R is about altitudes, while A is about side lengths.

Answer: (c) Both statements are true

Explanation:
1. When the two radii add up to exactly the base, the circles meet at just one point, on the base itself. No third vertex off the base, so no triangle. Statement I is true.
2. When the radii add up to less than the base, the circles cannot reach each other. Statement II is true.
3. So both statements are true.

Teacher's Note:
1. Sum equal to base: circles touch. Sum less: circles apart. Sum more: circles cross.
2. A triangle needs the circles to cross.

Answer: (b) Only Statement II is true

Explanation:
1. Two right angles already make 90° + 90° = 180°, leaving 0° for the third angle. So Statement I is false.
2. Statement II is true; in fact every triangle has at least two acute angles.
3. So only Statement II is true.

Teacher's Note:
1. A triangle can have at most one angle of 90° or more.
2. Use the angle sum property to test such claims.

Answer: (d) 70°

Explanation:
1. By the angle sum property, ∠B + ∠C = 180° − 40° = 140°.
2. Since ∠B = ∠C, each is 140° ÷ 2 = 70°.

Teacher's Note:
1. Equal angles share the remaining amount equally.
2. Check: 40° + 70° + 70° = 180°.

Answer: (c) 105°

Explanation:
1. First, ∠PRQ = 180° − (48° + 57°) = 180° − 105° = 75°.
2. ∠PRS and ∠PRQ form a straight angle, so ∠PRS = 180° − 75° = 105°.
3. Check: 105° = 48° + 57°, the sum of the two opposite interior angles.

Teacher's Note:
1. An exterior angle equals the sum of the two opposite interior angles.
2. 75° is the interior angle at R, a common wrong choice.

Answer: (d) (i)-(r), (ii)-(s), (iii)-(q), (iv)-(p)

Explanation:
1. (i) 5 + 12 = 17 > 13: the circles cross and a triangle with three unequal sides forms, so (i)-(r).
2. (ii) 4 + 4 = 8 < 9: the circles do not meet, so (ii)-(s).
3. (iii) 6, 6, 6: an equilateral triangle, so (iii)-(q).
4. (iv) 2 + 9 = 11, equal to the longest: the circles only touch, so (iv)-(p). The answer is (i)-(r), (ii)-(s), (iii)-(q), (iv)-(p).

Teacher's Note:
1. Compare the sum of the two smaller lengths with the longest one.
2. Equal means touch, less means apart, more means a triangle.

Answer: (a) extend BC beyond B and drop the perpendicular from A to this extended line

Explanation:
1. The angle at B is 120°, an obtuse angle.
2. So the foot of the perpendicular from A does not fall between B and C; it falls beyond B.
3. Hence we must extend BC beyond B and drop the perpendicular from A to that extended line.

Teacher's Note:
1. Extend the base on the side of the obtuse angle.
2. Joining A to the midpoint gives a median, not an altitude.

Answer: (d) 115°

Explanation:
1. By the angle sum property, ∠ACB = 180° − (55° + 60°) = 180° − 115° = 65°.
2. ∠ACB and x together form a straight angle, so x = 180° − 65° = 115°.
3. Check: 115° = 55° + 60°.

Teacher's Note:
1. The exterior angle equals the sum of the two opposite interior angles.
2. 65° is the interior angle at C, not x.

Answer: (a) every triangle always has at least two acute angles

Explanation:
1. The three angles add up to 180°, so at most one of them can be 90° or more.
2. So every triangle already has at least two acute angles, including right-angled and obtuse-angled triangles.
3. 'One acute angle' would fit every triangle, so the definition must ask for all three. The best reason is every triangle has at least two acute angles.

Teacher's Note:
1. A good definition must leave out the other types.
2. Options (c) and (d) are true facts but do not answer the 'why'.

Answer: (d) A is false, but R is true.

Explanation:
1. In an equilateral triangle all three angles are equal, so each is 180° ÷ 3 = 60°. R is true.
2. So an equilateral triangle can never contain a 90° angle; A is false.
3. Hence A is false but R is true.

Teacher's Note:
1. Equal sides go with equal angles.
2. 60°, 60°, 60° leaves no room for 90°.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. An included angle of 180° makes the two sides lie along one straight line.
2. Then the three points are collinear and no closed figure is formed, so A is true.
3. R states exactly this, so R is true and correctly explains A.

Teacher's Note:
1. Three points on one line cannot make a triangle.
2. The included angle must be less than 180°.

Answer: (b) Only Statement II is true

Explanation:
1. 9 + 12 = 21 > 20, so the triangle inequality holds and a triangle does exist. Statement I is false.
2. If the lengths are in increasing order, each smaller length is already less than the longest length alone, so it is surely less than the sum of the other two. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. This is the shortcut: test only the longest length.
2. 21 beats 20, even if only by 1.

Answer: (c) 139°

Explanation:
1. Two angles of a triangle must add up to less than 180°.
2. So the second angle must be less than 180° − 40° = 140°.
3. The greatest whole-number value below 140° is 139°.

Teacher's Note:
1. 140° itself would leave 0° for the third angle.
2. 'Less than' means the boundary value is not allowed.

Answer: (a) (i)-(q), (ii)-(r), (iii)-(s), (iv)-(p)

Explanation:
1. (i) Third angle = 180° − 135° = 45°, and one angle is 90°: right-angled, so (i)-(q).
2. (ii) 120° + 70° = 190° > 180°: no triangle, so (ii)-(r).
3. (iii) Third angle = 180° − 90° = 90°: right-angled, so (iii)-(s).
4. (iv) Third angle = 180° − 150° = 30°, and the 100° angle is obtuse, so (iv)-(p). The answer is (i)-(q), (ii)-(r), (iii)-(s), (iv)-(p).

Teacher's Note:
1. First check whether the two angles add up to less than 180°.
2. Then find the third angle and look for 90° or an obtuse angle.