CBSE Class 7 Mathematics A Tale of Three Intersecting Lines Worksheet Set 01

Very Short Answer Questions

Question 1. What name is given to the three corner points of a triangle?
Answer:
They are called the vertices of the triangle; each pair of vertices is joined by a side.

Teacher's Note:
1. One corner is a vertex; many are vertices.
2. A triangle has 3 vertices and 3 sides.

Question 2. Write the three angles of △PQR in their short form.
Answer:
The angles ∠QPR, ∠PQR and ∠QRP are written in short as ∠P, ∠Q and ∠R.

Teacher's Note:
1. The short form uses only the vertex letter.
2. The vertex is the middle letter of the long form.

Question 3. Define an equilateral triangle.
Answer:
An equilateral triangle is a triangle in which all three sides are of equal length.

Teacher's Note:
1. All its angles are also equal, 60° each.
2. It is the most regular triangle.

Question 4. Name the two instruments you would use to construct a triangle with sides 4 cm, 4 cm and 4 cm.
Answer:
A marked ruler (to draw and measure the base) and a compass (to draw the two arcs of radius 4 cm).

Teacher's Note:
1. No protractor is needed when all sides are given.
2. The compass fixes the third vertex exactly.

Question 5. State the triangle inequality in one sentence.
Answer:
Three lengths satisfy the triangle inequality when each length is smaller than the sum of the other two.

Teacher's Note:
1. The direct path is shorter than the roundabout path.
2. Only such lengths can form a triangle.

Question 6. What is the sum of the three angles of a triangle?
Answer:
The three angles of any triangle add up to 180°.

Teacher's Note:
1. This is the angle sum property.
2. It holds for every triangle, big or small.

Question 7. Define an altitude of a triangle.
Answer:
An altitude is the perpendicular line segment drawn from a vertex to the opposite side (or to that side extended).

Teacher's Note:
1. Its length is the height of that vertex.
2. It always meets the base at 90°.

Question 8. What is the name of a triangle that has exactly two sides of equal length?
Answer:
It is an isosceles triangle.

Teacher's Note:
1. Two equal sides: isosceles.
2. Three equal sides: equilateral.

Question 9. A triangle has sides of 3 cm, 5 cm and 7 cm. Name this type of triangle according to its sides.
Answer:
All three sides are different, so it is a scalene triangle.

Teacher's Note:
1. Check for equal sides first.
2. No equal sides means scalene.

Question 10. What is a right-angled triangle?
Answer:
A right-angled triangle is a triangle in which one angle measures exactly 90°.

Teacher's Note:
1. A triangle can have only one right angle.
2. The other two angles then add up to 90°.

Question 11. What is an obtuse-angled triangle?
Answer:
An obtuse-angled triangle is a triangle in which one angle is obtuse, that is, greater than 90°.

Teacher's Note:
1. Only one angle can be obtuse.
2. The other two angles are then acute.

Question 12. How many altitudes does a triangle have?
Answer:
A triangle has three altitudes, one from each vertex to the opposite side.

Teacher's Note:
1. One altitude for each vertex.
2. Each side can be taken as a base.

Question 13. Can a triangle be constructed with sides 2 cm, 3 cm and 6 cm? Give a reason.
Answer:
No. 2 + 3 = 5, which is less than 6, so the triangle inequality fails and the two arcs would never meet.

Teacher's Note:
1. Compare the longest side with the sum of the other two.
2. 5 < 6, so no triangle.

Question 14. Two angles of a triangle are 36° and 72°. Find the third angle.
Answer:
Third angle = 180° − (36° + 72°) = 180° − 108° = 72°.

Teacher's Note:
1. Add the given angles first.
2. Two angles are 72°, so the triangle is isosceles.

Question 15. In △ABC, ∠B = ∠C and ∠A = 50°. Find ∠B.
Answer:
∠B + ∠C = 180° − 50° = 130°. As ∠B = ∠C, ∠B = 65°.

Teacher's Note:
1. Share the remaining angle equally.
2. Check: 50° + 65° + 65° = 180°.

Question 16. While constructing a triangle with sides 4 cm, 5 cm and 6 cm, why is the third vertex taken at the point where the two arcs cross?
Answer:
That point lies on both arcs, so it is 5 cm from A and 6 cm from B at the same time, exactly as required.

Teacher's Note:
1. Each arc gives one length condition.
2. The crossing point meets both conditions.

Question 17. When two sides of a triangle are given, which angle is called the included angle?
Answer:
The angle between the two given sides, at the vertex where they meet, is the included angle.

Teacher's Note:
1. Included means 'lying in between'.
2. For sides AB and AC, the included angle is ∠A.

Question 18. Two angles of a triangle are 90° and 85°. Can a triangle be drawn with these two angles? Give a reason.
Answer:
Yes. 90° + 85° = 175°, which is less than 180°, so a triangle is possible; its third angle is 5°.

Teacher's Note:
1. The test is: sum less than 180°.
2. A third angle of 5° is small but allowed.

Question 19. In △ABC, the side BC is extended to D. If ∠ACB = 70°, find the exterior angle ∠ACD.
Answer:
∠ACB and ∠ACD form a straight angle, so ∠ACD = 180° − 70° = 110°.

Teacher's Note:
1. Angles on a straight line add up to 180°.
2. The exterior and interior angles at C are a linear pair.

Question 20. In a right-angled triangle, which two sides of the triangle are themselves altitudes?
Answer:
The two sides that form the right angle; each is perpendicular to the other and runs from a vertex to the opposite side.

Teacher's Note:
1. No extra line is needed for these two altitudes.
2. Only the third altitude has to be drawn.

Question 21. Give one set of three lengths for which the two circles drawn during the construction only touch each other.
Answer:
One such set is 3 cm, 4 cm and 7 cm: 3 + 4 = 7, so the circles meet at just one point and no triangle is formed.

Teacher's Note:
1. The two smaller lengths must add up exactly to the longest.
2. Another example: 2, 5, 7.

Question 22. Can a triangle have all three of its angles equal to 70°? Give a reason.
Answer:
No. 70° + 70° + 70° = 210°, but the angles of a triangle must add up to 180°.

Teacher's Note:
1. Equal angles in a triangle must each be 60°.
2. Always check the total.

Question 23. Two angles of a triangle are 15° and 150°. Find the third angle.
Answer:
Third angle = 180° − (15° + 150°) = 180° − 165° = 15°.

Teacher's Note:
1. The triangle is obtuse-angled and isosceles.
2. Subtract the sum from 180°.

Question 24. Write the three angles of a triangle that is both right-angled and isosceles.
Answer:
The angles are 90°, 45° and 45°; the two equal angles share the remaining 90° equally.

Teacher's Note:
1. The right angle cannot be one of the equal pair.
2. 90° ÷ 2 = 45°.

Question 25. One angle of a triangle is 54°. Write the values that a second angle can take so that a triangle is still possible.
Answer:
The two angles must add up to less than 180°, so the second angle can be any measure greater than 0° and less than 126°.

Teacher's Note:
1. 180° − 54° = 126°.
2. 126° itself is not allowed.

Question 26. Two sides of a triangle are 1 cm and 100 cm. Describe all possible lengths of the third side.
Answer:
The third side must be longer than 99 cm and shorter than 101 cm; any length strictly between them, such as 99.5 cm or 100.7 cm, works.

Teacher's Note:
1. Third side > difference and < sum of the two sides.
2. 100 − 1 = 99 and 100 + 1 = 101.

Question 27. In △ABC, ∠B is obtuse. Explain in one line why the altitude from A to BC falls outside the triangle.
Answer:
Since ∠B is obtuse, the foot of the perpendicular from A falls beyond B, so BC must be extended and the altitude lies outside the triangle.

Teacher's Note:
1. The obtuse angle opens the sides wide.
2. Extend the base on the side of B.

Question 28. Is it possible to construct an equilateral triangle that is obtuse-angled? Give a reason.
Answer:
No. All three angles of an equilateral triangle are 60°, which is acute, so it can never have an obtuse angle.

Teacher's Note:
1. Every equilateral triangle is acute-angled.
2. 180° ÷ 3 = 60°.

Question 29. Why is it enough to compare only the longest of three given lengths with the sum of the other two to decide whether a triangle exists?
Answer:
Each of the two smaller lengths is already less than the longest length alone, so it is surely less than the sum of the other two. Only the longest needs checking.

Teacher's Note:
1. Put the lengths in increasing order first.
2. This saves two comparisons.

Question 30. Two sides of a triangle are 5 cm and 5 cm. Write all the whole-number values that the third side can take.
Answer:
The third side must be less than 5 + 5 = 10 cm and greater than 0 cm, so it can be 1, 2, 3, 4, 5, 6, 7, 8 or 9 cm.

Teacher's Note:
1. 10 cm would make the circles only touch.
2. The difference 5 − 5 = 0 gives the lower limit.