A Tale of Three Intersecting Lines — Mastery

Q1. In ΔABC, the symbol ∠A is a short form for which of the following?

• A. The side BC opposite to vertex A
• B. The exterior angle at vertex A
• C. The altitude from A to side BC
• D. The angle ∠CAB (or equivalently ∠BAC) at vertex A

Q2. A teacher writes "ΔPQR" on the board. What does the symbol Δ in front of the three letters stand for?

• A. It marks a right angle in the figure named PQR
• B. It denotes the area of the triangle PQR
• C. It tells students that PQR are collinear points
• D. It is a short form for the word "triangle", read as "triangle PQR"

Q3. Why does the chapter prefer the compass over a ruler alone to construct an equilateral triangle of a given side?

• A. A ruler can mark length, but matching the third vertex with both ends needs trial and error; the compass guarantees equal sides at once
• B. A ruler cannot draw a straight base of the required length
• C. A compass is faster than a ruler for marking any length
• D. A ruler always draws curved lines, while a compass draws straight ones

Q4. When constructing a triangle with three given sides 5 cm, 6 cm and 7 cm using a compass, the suggested first step is to

• A. Draw any one of the three given sides as the base — for example, draw the 7 cm side as base AB
• B. Draw a circle of radius 5 cm and mark three points on it
• C. Mark three random points and join them in pairs
• D. Bisect a 7 cm segment and use the two halves as the other sides

Q5. To check whether 4 cm, 5 cm and 8 cm can be the sides of a triangle using the two-circles method, which choice of base and radii is most convenient?

• A. AB = 4 cm, with circles of radius 5 cm and 8 cm at A and B
• B. AB = 5 cm, with circles of radius 4 cm and 8 cm at A and B
• C. AB = 8 cm, with circles of radius 4 cm and 5 cm at A and B
• D. Any base works, since the answer does not depend on the choice

Q6. From a fixed point O on paper, two points P and Q are marked using a compass set to the same opening. The triangle OPQ formed is always

• A. Equilateral, because all three sides are equal
• B. Right-angled, because the compass keeps the angle at O fixed at 90°
• C. Scalene, because PQ is in general different from OP and OQ
• D. Isosceles, because OP = OQ (both are the same compass radius)

Q7. To construct ΔPQR given PQ = 6 cm, ∠P = 60° and PR = 4 cm, the very first step is to

• A. Draw a 60° angle at any point and place P at its vertex
• B. Mark a point R and draw a 4 cm arc from it
• C. Draw the given base PQ = 6 cm and mark its endpoints
• D. Draw two parallel lines 6 cm apart

Q8. When a triangle is to be constructed using two angles and the included side, the "included side" is the side that

• A. Lies between the two given angles, with one given angle at each of its endpoints
• B. Faces the larger of the two given angles
• C. Is half the perimeter of the triangle
• D. Is perpendicular to both arms of the larger angle

Q9. A teacher cuts out a paper triangle and folds its three corners so that the three vertices meet at one point on the opposite side. The three angles together form a straight line. This activity is meant to verify that

• A. The sum of two sides of a triangle is greater than the third
• B. All triangles are isosceles
• C. The exterior angle of a triangle equals the sum of its sides
• D. The three angles of a triangle add up to 180°

Q10. Why is the relation "exterior angle of a triangle = sum of the two remote interior angles" essentially a consequence of the angle sum property?

• A. Because the exterior angle equals 180° minus the adjacent interior angle, and the angle sum gives 180° minus that same adjacent angle as the sum of the remote two
• B. Because every triangle is isosceles
• C. Because the exterior angle is always 90°
• D. Because the three altitudes of a triangle meet at a point

Q11. In ΔABC, the altitude from vertex A meets side BC at point D. Which of the following must be true about AD?

• A. AD bisects side BC into two equal parts
• B. AD bisects ∠A into two equal angles
• C. AD is perpendicular to BC, so ∠ADB = ∠ADC = 90°
• D. AD is parallel to BC

Q12. Can a single triangle be both right-angled and obtuse-angled at the same time? Choose the best justification.

• A. Yes, because every right triangle has at least one obtuse angle
• B. No, because the sum of a right angle (90°) and an obtuse angle (>90°) already exceeds 180°, leaving nothing positive for the third
• C. Yes, if the triangle is also equilateral
• D. No, because right and obtuse triangles must have equal sides

Q13. A Class 7 student claims, "Any three positive lengths can form a triangle." Which of the following is the most effective single counter-example a teacher can give to challenge this claim?

• A. Show three equal lengths 5, 5, 5 and form an equilateral triangle
• B. Show 3, 4, 5 and form a right-angled triangle
• C. Show 2, 3, 6 — since 2 + 3 = 5 < 6, no triangle can be drawn from these
• D. Show 6, 8, 10 and form a triangle

Q14. A Class 7 teacher wants to introduce the angle sum property of a triangle. Which approach is most aligned with the chapter's spirit?

• A. Directly state the rule "three angles add to 180°" and ask students to memorise it
• B. Let students cut paper triangles, tear off the three corners and arrange them along a straight line, then state the rule
• C. Give a list of 20 numerical problems on missing angles and check answers
• D. Show only the algebraic equation A + B + C = 180° without any figure

Q15. To draw the altitude from vertex A of a triangle ΔABC to side BC, the most practical instrument to ensure a true perpendicular at the foot is

• A. A set square aligned along BC so that its right-angle edge passes through A
• B. A simple ruler held by eye, parallel to BC
• C. A protractor placed with its base on AB
• D. A compass set to a 5 cm opening

Q16. While constructing an equilateral triangle ABC of side 5 cm, two arcs of radius 5 cm are drawn from A and B. The third vertex C is fixed as

• A. The midpoint of AB
• B. Any point on the arc from A
• C. Either point where the two arcs intersect
• D. The point where the two arcs cross AB

Q17. Among all triangles, the equilateral triangle is described as the "most symmetric". Which statement best captures this idea?

• A. It has the largest area for a given perimeter
• B. All three sides are equal and all three angles are equal (each 60°)
• C. Only one of its angles is 60°
• D. Its three sides are always whole numbers

Q18. To construct ΔABC with AB = 7 cm, BC = 5 cm and CA = 6 cm, the base AB has already been drawn. The next step is to draw an arc

• A. From A with radius 6 cm (which is CA)
• B. From A with radius 5 cm (which is BC)
• C. From the midpoint of AB with radius 5 cm
• D. From any point on AB with radius 7 cm

Q19. Continuing the construction of ΔABC with AB = 7 cm, BC = 5 cm and CA = 6 cm — the arc from A of radius 6 cm has been drawn. The third step is to draw an arc

• A. From B of radius 7 cm
• B. From B of radius 5 cm
• C. From B of radius 6 cm
• D. From the foot of the first arc, of radius 5 cm

Q20. In the three-sides construction of ΔABC, after the arcs from A and B meet at C, the final step is to

• A. Erase the two arcs and the base AB
• B. Draw a circle through A, B and C
• C. Join AC and BC with a ruler to complete the triangle
• D. Measure ∠C with a protractor and write its value

Q21. AB = 6 cm. A circle of radius 4 cm is drawn at A and a circle of radius 5 cm at B. How many points of intersection do the two circles have, and what does this mean for the triangle?

• A. 0 points; no triangle can be formed
• B. Exactly 1 point; only one triangle is possible
• C. Exactly 2 points; a triangle with sides 6, 4, 5 cm exists (two mirror positions of C)
• D. Infinite points; the circles overlap everywhere

Q22. Which triangle from the following side lists is equilateral?

• A. 3 cm, 4 cm, 5 cm
• B. 6 cm, 6 cm, 9 cm
• C. 7 cm, 7 cm, 7 cm
• D. 5 cm, 8 cm, 10 cm

Q23. To construct ΔABC with AB = 5 cm, ∠A = 60° and AC = 4 cm, the base AB has been drawn. The second step is to

• A. Construct an angle of 60° at vertex A, with AB as one of its arms
• B. Construct an angle of 60° at vertex B, with BA as one of its arms
• C. Mark a point 4 cm above the midpoint of AB
• D. Draw a 4 cm circle centred at B

Q24. After AB, ∠A and AC have been marked in the SAS construction of ΔABC, vertex C is located on the second arm of ∠A. The final step is

• A. Join BC with a ruler
• B. Erase AB and redraw it
• C. Measure ∠B and adjust
• D. Draw a parallel to AB through C

Q25. To construct ΔABC with AB = 6 cm, ∠A = 50° and ∠B = 60° (ASA), the very first step is to

• A. Construct a 50° angle in mid-air, then drop a vertical
• B. Draw the given included side AB = 6 cm using a ruler
• C. Mark a point C first and work backwards
• D. Set the compass to 6 cm and draw two arcs

Q26. In the ASA construction with given side AB and given angles ∠A and ∠B, why must the two angles be drawn at the correct endpoints (∠A at A, ∠B at B), not interchanged?

• A. It does not matter; the same triangle results either way
• B. Swapping them gives the mirror image, which is a different triangle in general — sides like CA and CB get interchanged
• C. Swapping them gives a quadrilateral instead
• D. Only the larger angle can ever be at A

Q27. In the ASA construction, after drawing AB and the angles ∠A and ∠B at the two ends, the third vertex C is obtained as

• A. The midpoint of the second arms of the two angles
• B. The point where the second arms of ∠A and ∠B (extended if needed) meet on one side of AB
• C. The foot of perpendicular from any point on AB
• D. The point of intersection of AB with itself

Q28. In ΔABC, side BC is extended to D. Why do ∠ACB and ∠ACD always add up to 180°?

• A. Because they are vertically opposite angles
• B. Because the triangle is isosceles
• C. Because BCD is a straight line and the two angles share the arm CA, forming a linear pair
• D. Because the angle sum of the triangle is 180°

Q29. A triangle has angles 110°, 40° and 30°. Which classification is correct?

• A. Acute-angled, because two angles are below 90°
• B. Right-angled, because 40° + 30° + 20° = 90°
• C. Obtuse-angled, because exactly one angle (110°) is greater than 90°
• D. Reflex-angled, because 110° is more than 90°

Q30. A Class 7 student knows two angles of a triangle (say 80° and 55°) and has to find the third. Which method is the most reliable and aligned with the chapter?

• A. Measure the third angle with a protractor on a rough sketch
• B. Subtract the sum of the two known angles from 180° using the angle sum property
• C. Guess any value below 90° and check later
• D. Assume the triangle is isosceles and copy one of the given angles