Sec 1 G2 Maths Bukit Timah | Triangle Inequality

Did you know? Three sticks measuring 3, 4 and 7 centimetres cannot form a nondegenerate triangle. The shorter two fit exactly along the longest, producing a straight line. To enclose a triangle, those shorter lengths must add to strictly more than the longest.

For families seeking a Secondary 1 G2 Mathematics tutor in Bukit Timah, the key ideas are strict triangle inequality, the permitted range of a missing side, and the connection between the longest side and largest opposite angle. These properties help students explain what can or cannot be true, rather than guess from a diagram.

The MOE G2 Mathematics syllabus includes triangle properties at Secondary 1. The unchanged eduKateSG Secondary 1 small-group reference describes weekly 90-minute classes of three near Sixth Avenue MRT. The exercises do not assume Pythagoras, generally introduced later in G2.

Triangle reasoning in one minute

All three lengths must be positive.
The shorter two sides must sum to more than the longest.
Two known lengths a and b require |a−b|
Equal endpoints produce a flattened configuration, not a triangle.
The longest side lies opposite the largest angle.
Equal sides face equal angles.
Triangle interior angles total 180 degrees.
Draw and label opposite pairs: angle A faces BC, B faces AC, C faces AB.

The central example

Take lengths 5,7,10. Their shorter sum is twelve, which exceeds ten, so a triangle is possible. Replace them with 3,4,7 and the shorter sum equals the longest. That would produce only a line. The question’s strict inequality is therefore the central fact, not a memorised appearance of a triangle.

In triangle ABC, if BC is its longest edge, angle A is the largest. BC is opposite A, not next to it. The student can deduce this ordering without measuring the picture or knowing any exact angle value.

26 explanations with worked meaning

1. Why three lengths might not form a triangle

A genuine triangle needs the two shorter sides to be long enough to meet beyond the longest. Otherwise the segments form a flattened or disconnected arrangement, not a three-sided region. Check side existence before classifying the triangle or calculating anything else.

Tutor’s check: ask the learner to state the relevant side or angle condition and illustrate it with a fresh three-length set or relabelled triangle. The first incorrect decision is more useful to diagnose than the final guess.

2. Strict inequality, not equality

For positive lengths, a+b=c produces a line without enclosed area. The correct criterion uses greater-than. The smaller two must sum to strictly greater than the longest. The equality case is the classic mistake on multiple-choice existence questions.

3. Sort first, then compare

Sorting the lengths from shortest to longest makes the decisive test clear. When the sum of the two smaller lengths exceeds the longest, the remaining triangle inequalities follow automatically for positive side lengths.

4. Every physical side must be positive

An equation may give a negative numerical value for x, but that cannot directly represent a triangle side of length x. Check the actual side expressions for positivity after solving, especially when one side is written x+5.

5. Two known sides constrain a third

For known lengths a and b, the third side x must satisfy |a−b|

6. An interval is not the same as one length

Two known sides usually do not determine one unique third side. An open range may be the complete answer. Do not guess a third length from the proportions of an unscaled picture.

7. Integer side lengths are an extra restriction

If a question asks for whole-number centimetre lengths, enumerate only integers strictly between the bounds. If no integer restriction is supplied, fractional or decimal lengths can be geometrically valid.

8. Isosceles legs do not uniquely determine the base

Two equal legs constrain but do not fix their third side. If each equal leg is s, the base must be positive and less than 2s. A pupil should not assume the base equals s simply because the other two lengths match.

9. Perimeter supplies a separate condition

Given total perimeter and two lengths, subtract to find the candidate third. A total length is additional information that can resolve ambiguity. Verify the candidate remains positive and obeys the triangle inequality.

10. Larger sides face larger opposite angles

The longest side lies opposite the largest interior angle. The rule concerns ordering rather than an equality between centimetres and degrees. It applies to acute, right and obtuse triangles.

11. Smaller sides face smaller opposite angles

The shortest edge faces the smallest interior angle. In a triangle with three different side lengths, all three opposite angles differ as well. The diagram may be rotated without changing the relationship.

12. Vertices and opposite edges need careful labels

For triangle ABC, angle A faces side BC, angle B faces side AC, and angle C faces side AB. Many incorrect answers arise from naming an angle at an endpoint rather than the vertex opposite the named edge.

13. Equal sides imply equal opposite angles

An isosceles triangle has two equal sides and their opposite angles are equal. This is why the two base angles agree when the legs meet at one apex. Side markings, rather than visual symmetry alone, establish the fact.

14. Equal angles imply equal opposite sides

The converse is valid. If angles B and C match, the opposite sides AC and AB match. The pupil should identify which side faces each angle before declaring any lengths equal.

15. Equilateral triangles have three sixty-degree angles

All sides and all angles match. Since triangle interior angles total 180 degrees, the three equal angles each measure 60 degrees. This property allows classification from either valid side or angle information.

16. A scalene triangle must first exist

Three unequal positive numbers are not automatically a scalene triangle. If they do form a genuine triangle, it is scalene when all sides differ. Invalid triples such as 3,4,7 should not be classified as triangles at all.

17. Right triangles have one 90-degree angle

The other two interior angles are acute and together form ninety degrees. The side opposite ninety is longest. This ordering needs no Pythagoras calculations and is appropriate for introductory G2 triangle properties.

18. Obtuse triangles have one angle greater than ninety

Two obtuse angles would exceed the full 180-degree total. A triangle therefore has at most one obtuse angle, and the edge opposite that angle is longest.

19. Acute triangles have three acute angles

Every interior angle measures less than ninety degrees, while the three together still total 180. An acute triangle can be scalene or isosceles depending on its side lengths and angle equalities.

20. The third angle follows from the 180-degree total

Subtract two known interior angles from 180 degrees. Once the third has been obtained, order the angles and identify which opposite side is longest or shortest.

21. Exterior and interior angles are related

Extend one triangle side. The adjacent interior angle and exterior angle sum to 180 degrees. The exterior also equals the sum of the two remote interior angles, providing a second checking method.

22. A drawing may not represent measured scale

Line lengths can look misleading in a schematic geometry question. Given labels and proved properties take priority over estimating with a ruler. Never infer equality or exact angles from appearance alone.

23. Angle values alone do not set perimeter

Triangles can have identical interior angles but different side lengths and perimeters. A numerical size or scale must be supplied to determine actual boundary lengths.

24. Uniform positive scaling preserves a valid triangle

Multiplying each side by the same positive factor preserves its strict inequalities. By contrast, changing only the longest edge can cause an otherwise valid triangle to fail the existence condition.

25. Geometry remains a constraint after algebra

An unknown side may be found by solving an equation, but a negative or excessive resulting length is not a physically valid triangle. Substitute into the actual figure and perform the strict check.

26. The first incorrect decision guides the correction

One student may include an equality boundary, another may check the wrong side pair, and a third may mix up angle labels. Targeted feedback should identify the earliest error, then test a fresh triangle without prompts.

78 worked triangle questions

Cover each answer before reading the explanation. Some examples ask whether a shape exists, some ask for a whole interval of valid lengths, and others ask which edge faces a given angle. State the exact question type first.

Triangle clinic 1: Do lengths 3, 4, 5 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 3 and 4, with sum 7. The longest is 5. The strict triangle inequality holds.

Independent check: Ask the learner to show the required strict comparison, not merely add all three lengths. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 2: Do lengths 3, 4, 7 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 3 and 4, with sum 7. The longest is 7. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 3: Do lengths 3, 4, 8 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 3 and 4, with sum 7. The longest is 8. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 4: Do lengths 4, 5, 6 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 4 and 5, with sum 9. The longest is 6. The strict triangle inequality holds.

Triangle clinic 5: Do lengths 4, 5, 9 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 4 and 5, with sum 9. The longest is 9. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 6: Do lengths 5, 5, 9 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 5 and 5, with sum 10. The longest is 9. The strict triangle inequality holds.

Triangle clinic 7: Do lengths 5, 5, 10 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 5 and 5, with sum 10. The longest is 10. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 8: Do lengths 5, 6, 10 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 5 and 6, with sum 11. The longest is 10. The strict triangle inequality holds.

Triangle clinic 9: Do lengths 5, 6, 12 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 5 and 6, with sum 11. The longest is 12. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 10: Do lengths 6, 8, 10 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 6 and 8, with sum 14. The longest is 10. The strict triangle inequality holds.

Triangle clinic 11: Do lengths 2, 7, 8 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 2 and 7, with sum 9. The longest is 8. The strict triangle inequality holds.

Triangle clinic 12: Do lengths 2, 7, 9 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 2 and 7, with sum 9. The longest is 9. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 13: Do lengths 7, 10, 16 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 7 and 10, with sum 17. The longest is 16. The strict triangle inequality holds.

Triangle clinic 14: Do lengths 7, 10, 17 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 7 and 10, with sum 17. The longest is 17. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 15: Do lengths 7, 10, 18 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 7 and 10, with sum 17. The longest is 18. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 16: Do lengths 9, 9, 17 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 9 and 9, with sum 18. The longest is 17. The strict triangle inequality holds.

Triangle clinic 17: Do lengths 9, 9, 18 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 9 and 9, with sum 18. The longest is 18. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 18: Do lengths 10, 12, 21 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 10 and 12, with sum 22. The longest is 21. The strict triangle inequality holds.

Triangle clinic 19: Do lengths 10, 12, 22 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 10 and 12, with sum 22. The longest is 22. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 20: Do lengths 10, 12, 23 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 10 and 12, with sum 22. The longest is 23. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 21: Do lengths 6, 7, 8 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 6 and 7, with sum 13. The longest is 8. The strict triangle inequality holds.

Triangle clinic 22: Do lengths 8, 15, 17 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 8 and 15, with sum 23. The longest is 17. The strict triangle inequality holds.

Triangle clinic 23: Do lengths 8, 15, 23 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 8 and 15, with sum 23. The longest is 23. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 24: Do lengths 8, 15, 24 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 8 and 15, with sum 23. The longest is 24. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 25: Do lengths 2, 3, 4 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 2 and 3, with sum 5. The longest is 4. The strict triangle inequality holds.

Triangle clinic 26: Do lengths 2, 3, 5 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 2 and 3, with sum 5. The longest is 5. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 27: Do lengths 2, 3, 6 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 2 and 3, with sum 5. The longest is 6. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 28: Do lengths 11, 13, 18 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 11 and 13, with sum 24. The longest is 18. The strict triangle inequality holds.

Triangle clinic 29: Do lengths 11, 13, 24 cm form a genuine triangle?

Answer: No. Reason: After sorting, the two smaller sides are 11 and 13, with sum 24. The longest is 24. The sum is not greater than the longest, so a triangle cannot close.

Triangle clinic 30: Do lengths 12, 12, 20 cm form a genuine triangle?

Answer: Yes. Reason: After sorting, the two smaller sides are 12 and 12, with sum 24. The longest is 20. The strict triangle inequality holds.

Triangle clinic 31: Sides 3 and 5 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 2Reason: The third side must be strictly larger than their difference 2 and strictly smaller than their sum 8. Integer values range from 3 to 7.

Independent check: Both endpoints are excluded. Decimal side lengths may also work unless the question explicitly requires integers. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 32: Sides 4 and 9 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 5Reason: The third side must be strictly larger than their difference 5 and strictly smaller than their sum 13. Integer values range from 6 to 12.

Triangle clinic 33: Sides 5 and 8 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 3Reason: The third side must be strictly larger than their difference 3 and strictly smaller than their sum 13. Integer values range from 4 to 12.

Triangle clinic 34: Sides 6 and 10 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 4Reason: The third side must be strictly larger than their difference 4 and strictly smaller than their sum 16. Integer values range from 5 to 15.

Triangle clinic 35: Sides 7 and 12 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 5Reason: The third side must be strictly larger than their difference 5 and strictly smaller than their sum 19. Integer values range from 6 to 18.

Triangle clinic 36: Sides 8 and 13 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 5Reason: The third side must be strictly larger than their difference 5 and strictly smaller than their sum 21. Integer values range from 6 to 20.

Triangle clinic 37: Sides 9 and 14 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 5Reason: The third side must be strictly larger than their difference 5 and strictly smaller than their sum 23. Integer values range from 6 to 22.

Triangle clinic 38: Sides 10 and 17 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 7Reason: The third side must be strictly larger than their difference 7 and strictly smaller than their sum 27. Integer values range from 8 to 26.

Triangle clinic 39: Sides 11 and 18 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 7Reason: The third side must be strictly larger than their difference 7 and strictly smaller than their sum 29. Integer values range from 8 to 28.

Triangle clinic 40: Sides 12 and 19 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 7Reason: The third side must be strictly larger than their difference 7 and strictly smaller than their sum 31. Integer values range from 8 to 30.

Triangle clinic 41: Sides 5 and 5 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 10. Integer values range from 1 to 9.

Triangle clinic 42: Sides 6 and 6 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 12. Integer values range from 1 to 11.

Triangle clinic 43: Sides 7 and 7 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 14. Integer values range from 1 to 13.

Triangle clinic 44: Sides 8 and 8 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 16. Integer values range from 1 to 15.

Triangle clinic 45: Sides 9 and 9 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 18. Integer values range from 1 to 17.

Triangle clinic 46: Sides 10 and 10 cm are known. Give the possible third-side interval and number of positive integer choices.

Answer: 0Reason: The third side must be strictly larger than their difference 0 and strictly smaller than their sum 20. Integer values range from 1 to 19.

Triangle clinic 47: Triangle ABC has angles A=50°, B=60°. Find C and the longest side(s).

Answer: C=70°; longest AB. Reason: Triangle sum gives C=180−50−60=70. The largest angle is 70°, and it faces AB.

Independent check: The smallest angle is 50°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 48: Triangle ABC has angles A=35°, B=75°. Find C and the longest side(s).

Answer: C=70°; longest AC. Reason: Triangle sum gives C=180−35−75=70. The largest angle is 75°, and it faces AC.

Independent check: The smallest angle is 35°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 49: Triangle ABC has angles A=40°, B=80°. Find C and the longest side(s).

Answer: C=60°; longest AC. Reason: Triangle sum gives C=180−40−80=60. The largest angle is 80°, and it faces AC.

Independent check: The smallest angle is 40°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 50: Triangle ABC has angles A=65°, B=45°. Find C and the longest side(s).

Answer: C=70°; longest AB. Reason: Triangle sum gives C=180−65−45=70. The largest angle is 70°, and it faces AB.

Independent check: The smallest angle is 45°, opposite AC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 51: Triangle ABC has angles A=70°, B=50°. Find C and the longest side(s).

Answer: C=60°; longest BC. Reason: Triangle sum gives C=180−70−50=60. The largest angle is 70°, and it faces BC.

Independent check: The smallest angle is 50°, opposite AC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 52: Triangle ABC has angles A=30°, B=100°. Find C and the longest side(s).

Answer: C=50°; longest AC. Reason: Triangle sum gives C=180−30−100=50. The largest angle is 100°, and it faces AC.

Independent check: The smallest angle is 30°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 53: Triangle ABC has angles A=45°, B=45°. Find C and the longest side(s).

Answer: C=90°; longest AB. Reason: Triangle sum gives C=180−45−45=90. The largest angle is 90°, and it faces AB.

Independent check: The smallest angle is 45°, opposite BC and AC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 54: Triangle ABC has angles A=20°, B=110°. Find C and the longest side(s).

Answer: C=50°; longest AC. Reason: Triangle sum gives C=180−20−110=50. The largest angle is 110°, and it faces AC.

Independent check: The smallest angle is 20°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 55: Triangle ABC has angles A=80°, B=60°. Find C and the longest side(s).

Answer: C=40°; longest BC. Reason: Triangle sum gives C=180−80−60=40. The largest angle is 80°, and it faces BC.

Independent check: The smallest angle is 40°, opposite AB. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 56: Triangle ABC has angles A=55°, B=75°. Find C and the longest side(s).

Answer: C=50°; longest AC. Reason: Triangle sum gives C=180−55−75=50. The largest angle is 75°, and it faces AC.

Independent check: The smallest angle is 50°, opposite AB. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 57: Triangle ABC has angles A=60°, B=60°. Find C and the longest side(s).

Answer: C=60°; longest BC and AC and AB. Reason: Triangle sum gives C=180−60−60=60. The largest angle is 60°, and it faces BC and AC and AB.

Independent check: The smallest angle is 60°, opposite BC and AC and AB. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 58: Triangle ABC has angles A=50°, B=80°. Find C and the longest side(s).

Answer: C=50°; longest AC. Reason: Triangle sum gives C=180−50−80=50. The largest angle is 80°, and it faces AC.

Independent check: The smallest angle is 50°, opposite BC and AB. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 59: Triangle ABC has angles A=35°, B=35°. Find C and the longest side(s).

Answer: C=110°; longest AB. Reason: Triangle sum gives C=180−35−35=110. The largest angle is 110°, and it faces AB.

Independent check: The smallest angle is 35°, opposite BC and AC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 60: Triangle ABC has angles A=70°, B=70°. Find C and the longest side(s).

Answer: C=40°; longest BC and AC. Reason: Triangle sum gives C=180−70−70=40. The largest angle is 70°, and it faces BC and AC.

Triangle clinic 61: Triangle ABC has angles A=25°, B=85°. Find C and the longest side(s).

Answer: C=70°; longest AC. Reason: Triangle sum gives C=180−25−85=70. The largest angle is 85°, and it faces AC.

Independent check: The smallest angle is 25°, opposite BC. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 62: Triangle ABC has angles A=90°, B=45°. Find C and the longest side(s).

Answer: C=45°; longest BC. Reason: Triangle sum gives C=180−90−45=45. The largest angle is 90°, and it faces BC.

Independent check: The smallest angle is 45°, opposite AC and AB. Name opposite vertices rather than adjacent endpoints. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 63: Equal legs each 6 cm: permitted positive integer base lengths?

Answer: 1 through 11 cm. Reason: The base x must lie in 0

Independent check: The upper endpoint twelve creates a straight line. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 64: Sides 5,8,13: triangle?

Answer: No. Reason: Five plus eight equals thirteen.

Independent check: Strict inequality is required. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 65: Sides 5,8,12.9: triangle?

Answer: Yes. Reason: Five plus eight exceeds 12.9.

Independent check: A decimal side is permissible. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 66: Perimeter 24, known sides 7,9: third side?

Answer: 8 cm. Reason: 24−7−9=8.

Independent check: Check 7+8>9. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 67: If AB=AC, which angles equal?

Answer: B and C. Reason: Equal sides face equal angles.

Independent check: AB faces C; AC faces B. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 68: If angle B equals C, which sides equal?

Answer: AB and AC. Reason: Equal angles face equal opposite sides.

Independent check: Check the vertex pairs. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 69: BC longest in triangle ABC: largest angle?

Answer: A. Reason: Vertex A is opposite edge BC.

Independent check: Avoid selecting an adjacent vertex. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 70: AC shortest in ABC: smallest angle?

Answer: B. Reason: Vertex B faces edge AC.

Independent check: The rule is about opposite angles. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 71: Angles 90°,35°,55°: type?

Answer: Right-angled. Reason: Exactly one right angle.

Independent check: The opposite side is longest. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 72: Angles 110°,40°,30°: type?

Answer: Obtuse. Reason: One angle exceeds ninety.

Independent check: The other two are acute. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 73: Angles 70°,60°,50°: type?

Answer: Acute. Reason: Every angle is under ninety.

Independent check: The largest side faces seventy. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 74: Isosceles vertex angle 40°: equal bases?

Answer: 70° each. Reason: Subtract forty from 180 then halve.

Independent check: Both base angles are equal. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 75: Isosceles base angles 40° each: vertex?

Answer: 100°. Reason: Subtract eighty from 180.

Independent check: The base/vertex distinction matters. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 76: Remote triangle angles 50° and 60°: exterior angle?

Answer: 110°. Reason: Exterior equals sum of remote angles.

Independent check: Adjacent interior equals seventy. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 77: Angles 40°,60°,80° determine perimeter?

Answer: No. Reason: These angles permit different similar-sized triangles.

Independent check: A scale or length is required. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Triangle clinic 78: Lengths 2,3,6: which condition fails?

Answer: Triangle inequality. Reason: Two plus three is less than six.

Independent check: No genuine triangle is possible. Change one number or vertex label, then justify the new outcome without copying this answer. A successful delayed attempt shows the rule has transferred.

Four physical and visual investigations

Build a triangle from strips

Use three strips with lengths 3,4,7. Join them at their endpoints and notice they cannot enclose a triangular interior. Replace the seven with six. Now the two smaller lengths add to seven, greater than six, so the arrangement can close. Write both mathematical inequalities.

Discover all third-side choices

Fix two sides of five and eight. The third side must be larger than three and smaller than thirteen. If the question requests integers, the valid choices are four through twelve. Ask why decimal lengths between those strict bounds can also work when the problem does not demand whole numbers.

Mark the opposite angle for every side

Draw triangle ABC and use arrows A↔BC, B↔AC, C↔AB. A student can now identify the angle opposite the longest side without being misled by drawing orientation. Rotate the diagram and practise again with renamed vertices.

Compare isosceles vertex and base angles

In an isosceles triangle with a forty-degree vertex angle, the base angles are seventy each. If the base angles are forty each instead, the vertex angle is one hundred. Ask what changed in the wording and why that changes the answer.

Four-week small-group teaching plan

Week 1: valid and invalid triples

Sort the three printed lengths, check strict inequality and distinguish a flattened line from a real triangle.

Keep practice compatible with schoolwork, ask for a one-sentence rule and test one unfamiliar problem without the example visible.

Week 2: missing third sides

Use the difference and sum bounds, then list allowed integer values where requested.

Keep practice compatible with schoolwork, ask for a one-sentence rule and test one unfamiliar problem without the example visible.

Week 3: opposite sides and angles

Match each angle with the opposite edge and use the triangle sum and equal-side/equal-angle properties.

Keep practice compatible with schoolwork, ask for a one-sentence rule and test one unfamiliar problem without the example visible.

Week 4: mixed independent explanations

Combine existence, intervals, angle ordering and isosceles tasks without topic headings and review mistakes several days later.

Keep practice compatible with schoolwork, ask for a one-sentence rule and test one unfamiliar problem without the example visible.

How a three-student tutor can diagnose the error

A pupil might accept equal-boundary sides, another might compare the wrong two lengths, and a third might name an angle at an endpoint rather than the angle opposite the longest side. Those errors need different explanations. A tutor should repair the first incorrect decision and then give a new problem.

The eduKateSG Secondary 1 tutoring reference describes three students per weekly 90-minute session near Sixth Avenue MRT. Parents should check suitability and availability directly; no specific assessment score can be guaranteed.

Frequently asked questions

What is the triangle inequality?

The sum of any two triangle side lengths must be strictly greater than the third.

Can 3,4,7 form a triangle?

No. The shorter two sum exactly to the longest.

How do I find the possible third side?

For known lengths a and b, use |a−b|

Which side faces the largest angle?

The longest side.

Do equal sides imply equal angles?

Yes. The angles opposite equal sides are equal.

Can a triangle have two obtuse angles?

No. Their sum would exceed 180 degrees.

Do I need Pythagoras for this?

No. Triangle inequality and opposite-side ordering can be explained without Pythagorean calculations.

Can an unknown side be decimal?

Yes, unless the question restricts lengths to integers.

What if the picture is not to scale?

Use labels and given properties, not visual estimates.

How do parents help with this?

Ask the child to identify the shorter two lengths or the vertex opposite the longest edge.

Does the tutor guarantee outcomes?

No. The purpose is to identify and improve the child’s reasoning through practice and independent checking.

Official syllabus and related learning guides

Read the MOE G2 Mathematics syllabus for Secondary 1 triangle properties and the unchanged eduKateSG tutorial reference for the small-group teaching model.

The habit to remember is to identify the constraint before calculating. Can the sides close? Which angle lies opposite the longest edge? Once those questions are answered, the rest of the reasoning becomes clear.

Secondary 1 G2 Mathematics Tutor Bukit Timah | Triangle Inequality, Longest Sides and Largest Angles

Triangle reasoning in one minute

The central example

26 explanations with worked meaning

1. Why three lengths might not form a triangle

2. Strict inequality, not equality

3. Sort first, then compare

4. Every physical side must be positive

5. Two known sides constrain a third

6. An interval is not the same as one length

7. Integer side lengths are an extra restriction

8. Isosceles legs do not uniquely determine the base

9. Perimeter supplies a separate condition

10. Larger sides face larger opposite angles

11. Smaller sides face smaller opposite angles

12. Vertices and opposite edges need careful labels

13. Equal sides imply equal opposite angles

14. Equal angles imply equal opposite sides

15. Equilateral triangles have three sixty-degree angles

16. A scalene triangle must first exist

17. Right triangles have one 90-degree angle

18. Obtuse triangles have one angle greater than ninety

19. Acute triangles have three acute angles

20. The third angle follows from the 180-degree total

21. Exterior and interior angles are related

22. A drawing may not represent measured scale

23. Angle values alone do not set perimeter

24. Uniform positive scaling preserves a valid triangle

25. Geometry remains a constraint after algebra

26. The first incorrect decision guides the correction

78 worked triangle questions

Triangle clinic 1: Do lengths 3, 4, 5 cm form a genuine triangle?

Triangle clinic 2: Do lengths 3, 4, 7 cm form a genuine triangle?

Triangle clinic 3: Do lengths 3, 4, 8 cm form a genuine triangle?

Triangle clinic 4: Do lengths 4, 5, 6 cm form a genuine triangle?

Triangle clinic 5: Do lengths 4, 5, 9 cm form a genuine triangle?

Triangle clinic 6: Do lengths 5, 5, 9 cm form a genuine triangle?

Triangle clinic 7: Do lengths 5, 5, 10 cm form a genuine triangle?

Triangle clinic 8: Do lengths 5, 6, 10 cm form a genuine triangle?

Triangle clinic 9: Do lengths 5, 6, 12 cm form a genuine triangle?

Triangle clinic 10: Do lengths 6, 8, 10 cm form a genuine triangle?

Triangle clinic 11: Do lengths 2, 7, 8 cm form a genuine triangle?

Triangle clinic 12: Do lengths 2, 7, 9 cm form a genuine triangle?

Triangle clinic 13: Do lengths 7, 10, 16 cm form a genuine triangle?

Triangle clinic 14: Do lengths 7, 10, 17 cm form a genuine triangle?

Triangle clinic 15: Do lengths 7, 10, 18 cm form a genuine triangle?

Triangle clinic 16: Do lengths 9, 9, 17 cm form a genuine triangle?

Triangle clinic 17: Do lengths 9, 9, 18 cm form a genuine triangle?

Triangle clinic 18: Do lengths 10, 12, 21 cm form a genuine triangle?

Triangle clinic 19: Do lengths 10, 12, 22 cm form a genuine triangle?

Triangle clinic 20: Do lengths 10, 12, 23 cm form a genuine triangle?

Triangle clinic 21: Do lengths 6, 7, 8 cm form a genuine triangle?

Triangle clinic 22: Do lengths 8, 15, 17 cm form a genuine triangle?

Triangle clinic 23: Do lengths 8, 15, 23 cm form a genuine triangle?

Triangle clinic 24: Do lengths 8, 15, 24 cm form a genuine triangle?

Triangle clinic 25: Do lengths 2, 3, 4 cm form a genuine triangle?

Triangle clinic 26: Do lengths 2, 3, 5 cm form a genuine triangle?

Triangle clinic 27: Do lengths 2, 3, 6 cm form a genuine triangle?

Triangle clinic 28: Do lengths 11, 13, 18 cm form a genuine triangle?

Triangle clinic 29: Do lengths 11, 13, 24 cm form a genuine triangle?

Triangle clinic 30: Do lengths 12, 12, 20 cm form a genuine triangle?

Triangle clinic 31: Sides 3 and 5 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 32: Sides 4 and 9 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 33: Sides 5 and 8 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 34: Sides 6 and 10 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 35: Sides 7 and 12 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 36: Sides 8 and 13 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 37: Sides 9 and 14 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 38: Sides 10 and 17 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 39: Sides 11 and 18 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 40: Sides 12 and 19 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 41: Sides 5 and 5 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 42: Sides 6 and 6 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 43: Sides 7 and 7 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 44: Sides 8 and 8 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 45: Sides 9 and 9 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 46: Sides 10 and 10 cm are known. Give the possible third-side interval and number of positive integer choices.

Triangle clinic 47: Triangle ABC has angles A=50°, B=60°. Find C and the longest side(s).

Triangle clinic 48: Triangle ABC has angles A=35°, B=75°. Find C and the longest side(s).

Triangle clinic 49: Triangle ABC has angles A=40°, B=80°. Find C and the longest side(s).