A Tale of Three Intersecting Lines | FIO

Question 11

For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category:

(a) 30∘30^\circ

(b) 70∘70^\circ

(c) 54∘54^\circ

(d) 144∘144^\circ

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Solution
Understand the Question
  • The sum of all three interior angles in a triangle is always 180∘180^\circ: A+B+C=180∘A + B + C = 180^\circ
  • Since every angle must be positive (C>0∘C > 0^\circ): C=180∘−(A+B)>0∘  ⟹  A+B<180∘C = 180^\circ - (A + B) > 0^\circ \implies A + B < 180^\circ
  • Therefore, given an angle AA:
    • Triangle is possible: The second angle BB must satisfy B<180∘−AB < 180^\circ - A (with B>0∘B > 0^\circ).
    • Triangle is not possible: The second angle BB satisfies B≥180∘−AB \ge 180^\circ - A.

(a) 30∘30^\circ

Step 1 · Find Angles for a Possible Triangle

For a triangle to be possible, the sum of two angles must be strictly less than 180∘180^\circ.Diagram 1

30∘+B<180∘30^\circ + B < 180^\circ

B<180∘−30∘B<150∘\begin{aligned} B &< 180^\circ - 30^\circ \\ B &< 150^\circ \end{aligned}

Thus, any angle B<150∘B < 150^\circ forms a valid triangle.

  • Example 1: Choose B=60∘B = 60^\circ 30∘+60∘=90∘<180∘30^\circ + 60^\circ = 90^\circ < 180^\circ C=180∘−90∘=90∘>0∘(Valid)C = 180^\circ - 90^\circ = 90^\circ > 0^\circ \quad (\text{Valid})
  • Example 2: Choose B=90∘B = 90^\circ 30∘+90∘=120∘<180∘30^\circ + 90^\circ = 120^\circ < 180^\circ C=180∘−120∘=60∘>0∘(Valid)C = 180^\circ - 120^\circ = 60^\circ > 0^\circ \quad (\text{Valid})

Step 2 · Find Angles for an Impossible Triangle

For a triangle to not be possible, the sum of the two angles must be 180∘180^\circ or more:

30∘+B≥180∘30^\circ + B \ge 180^\circ

B≥180∘−30∘B≥150∘\begin{aligned} B &\ge 180^\circ - 30^\circ \\ B &\ge 150^\circ \end{aligned}

Thus, any angle B≥150∘B \ge 150^\circ will not form a triangle.

  • Example 1: Choose B=170∘B = 170^\circ 30∘+170∘=200∘>180∘(Not possible)30^\circ + 170^\circ = 200^\circ > 180^\circ \quad (\text{Not possible})
  • Example 2: Choose B=160∘B = 160^\circ 30∘+160∘=190∘>180∘(Not possible)30^\circ + 160^\circ = 190^\circ > 180^\circ \quad (\text{Not possible})
Answer

(a) Possible: Any angle <150∘< 150^\circ (e.g., 60∘,90∘60^\circ, 90^\circ)

Not possible: Any angle ≥150∘\ge 150^\circ (e.g., 160∘,170∘160^\circ, 170^\circ)

(b) 70∘70^\circ

Step 1 · Find Angles for a Possible Triangle

For a possible triangle:

70∘+B<180∘70^\circ + B < 180^\circ

B<180∘−70∘B<110∘\begin{aligned} B &< 180^\circ - 70^\circ \\ B &< 110^\circ \end{aligned}
  • Example 1: Choose B=70∘B = 70^\circ 70∘+70∘=140∘<180∘70^\circ + 70^\circ = 140^\circ < 180^\circ C=180∘−140∘=40∘>0∘(Valid)C = 180^\circ - 140^\circ = 40^\circ > 0^\circ \quad (\text{Valid})
  • Example 2: Choose B=40∘B = 40^\circ 70∘+40∘=110∘<180∘70^\circ + 40^\circ = 110^\circ < 180^\circ C=180∘−110∘=70∘>0∘(Valid)C = 180^\circ - 110^\circ = 70^\circ > 0^\circ \quad (\text{Valid})

Step 2 · Find Angles for an Impossible Triangle

For an impossible triangle:

70∘+B≥180∘70^\circ + B \ge 180^\circ

B≥180∘−70∘B≥110∘\begin{aligned} B &\ge 180^\circ - 70^\circ \\ B &\ge 110^\circ \end{aligned}
  • Example 1: Choose B=120∘B = 120^\circ 70∘+120∘=190∘>180∘(Not possible)70^\circ + 120^\circ = 190^\circ > 180^\circ \quad (\text{Not possible})
  • Example 2: Choose B=150∘B = 150^\circ 70∘+150∘=220∘>180∘(Not possible)70^\circ + 150^\circ = 220^\circ > 180^\circ \quad (\text{Not possible})
Answer

(b) Possible: Any angle <110∘< 110^\circ (e.g., 70∘,40∘70^\circ, 40^\circ)

Not possible: Any angle ≥110∘\ge 110^\circ (e.g., 120∘,150∘120^\circ, 150^\circ)

(c) 54∘54^\circ

Step 1 · Find Angles for a Possible Triangle

For a possible triangle:

54∘+B<180∘54^\circ + B < 180^\circ

B<180∘−54∘B<126∘\begin{aligned} B &< 180^\circ - 54^\circ \\ B &< 126^\circ \end{aligned}
  • Example 1: Choose B=72∘B = 72^\circ 54∘+72∘=126∘<180∘54^\circ + 72^\circ = 126^\circ < 180^\circ C=180∘−126∘=54∘>0∘(Valid)C = 180^\circ - 126^\circ = 54^\circ > 0^\circ \quad (\text{Valid})
  • Example 2: Choose B=54∘B = 54^\circ 54∘+54∘=108∘<180∘54^\circ + 54^\circ = 108^\circ < 180^\circ C=180∘−108∘=72∘>0∘(Valid)C = 180^\circ - 108^\circ = 72^\circ > 0^\circ \quad (\text{Valid})

Step 2 · Find Angles for an Impossible Triangle

For an impossible triangle:

54∘+B≥180∘54^\circ + B \ge 180^\circ

B≥180∘−54∘B≥126∘\begin{aligned} B &\ge 180^\circ - 54^\circ \\ B &\ge 126^\circ \end{aligned}
  • Example 1: Choose B=140∘B = 140^\circ 54∘+140∘=194∘>180∘(Not possible)54^\circ + 140^\circ = 194^\circ > 180^\circ \quad (\text{Not possible})
  • Example 2: Choose B=130∘B = 130^\circ 54∘+130∘=184∘>180∘(Not possible)54^\circ + 130^\circ = 184^\circ > 180^\circ \quad (\text{Not possible})
Answer

(c) Possible: Any angle <126∘< 126^\circ (e.g., 72∘,54∘72^\circ, 54^\circ)

Not possible: Any angle ≥126∘\ge 126^\circ (e.g., 140∘,130∘140^\circ, 130^\circ)

(d) 144∘144^\circ

Step 1 · Find Angles for a Possible Triangle

For a possible triangle:

144∘+B<180∘144^\circ + B < 180^\circ

B<180∘−144∘B<36∘\begin{aligned} B &< 180^\circ - 144^\circ \\ B &< 36^\circ \end{aligned}
  • Example 1: Choose B=10∘B = 10^\circ 144∘+10∘=154∘<180∘144^\circ + 10^\circ = 154^\circ < 180^\circ C=180∘−154∘=26∘>0∘(Valid)C = 180^\circ - 154^\circ = 26^\circ > 0^\circ \quad (\text{Valid})
  • Example 2: Choose B=26∘B = 26^\circ 144∘+26∘=170∘<180∘144^\circ + 26^\circ = 170^\circ < 180^\circ C=180∘−170∘=10∘>0∘(Valid)C = 180^\circ - 170^\circ = 10^\circ > 0^\circ \quad (\text{Valid})

Step 2 · Find Angles for an Impossible Triangle

For an impossible triangle:

144∘+B≥180∘144^\circ + B \ge 180^\circ

B≥180∘−144∘B≥36∘\begin{aligned} B &\ge 180^\circ - 144^\circ \\ B &\ge 36^\circ \end{aligned}
  • Example 1: Choose B=40∘B = 40^\circ 144∘+40∘=184∘>180∘(Not possible)144^\circ + 40^\circ = 184^\circ > 180^\circ \quad (\text{Not possible})
  • Example 2: Choose B=50∘B = 50^\circ 144∘+50∘=194∘>180∘(Not possible)144^\circ + 50^\circ = 194^\circ > 180^\circ \quad (\text{Not possible})
Answer

(d) Possible: Any angle <36∘< 36^\circ (e.g., 10∘,26∘10^\circ, 26^\circ)

Not possible: Any angle ≥36∘\ge 36^\circ (e.g., 40∘,50∘40^\circ, 50^\circ)

Common Mistakes
  • Boundary Angle: Forgetting that if A+B=180∘A + B = 180^\circ, the third angle is 0∘0^\circ, which means a triangle cannot be formed. Thus, B=180∘−AB = 180^\circ - A falls into the "not possible" category.
  • Strict Inequality: Writing A+B≤180∘A + B \le 180^\circ for a possible triangle instead of the strict inequality A+B<180∘A + B < 180^\circ.

More questions in FIO

Q1

Use the points on the circle and/or the centre to form isosceles triangles.

Q2

Use the points on the circles and/or their centres to form isosceles and equilateral triangles. The circles are of the same size.

Q3

We checked by construction that there are no triangles having sidelengths 3 cm3\text{ cm}, 4 cm4\text{ cm} and 8 cm8\text{ cm}; and 2 cm2\text{ cm}, 3 cm3\text{ cm} and 6 cm6\text{ cm}. Check if you could have found this without trying to construct the triangle.

Q4

Can we say anything about the existence of a triangle for each of the following sets of lengths?

(a) 10 km10\text{ km}, 10 km10\text{ km} and 25 km25\text{ km}

(b) 5 mm5\text{ mm}, 10 mm10\text{ mm} and 20 mm20\text{ mm}

(c) 12 cm12\text{ cm}, 20 cm20\text{ cm} and 40 cm40\text{ cm}

Q5

Which of the following lengths can be the sidelengths of a triangle? Explain your answers. Note that for each set, the three lengths have the same unit of measure.

(a) 2,2,52, 2, 5 (b) 3,4,63, 4, 6 (c) 2,4,82, 4, 8 (d) 5,5,85, 5, 8 (e) 10,20,2510, 20, 25 (f) 10,20,3510, 20, 35 (g) 24,26,2824, 26, 28

Q6

Check if a triangle exists for each of the following set of lengths:

(a) 1, 100, 100

(b) 3, 6, 9

(c) 1, 1, 5

(d) 5, 10, 12

Q7

Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any sidelength? Justify your answer.

Q8

For each of the following, give at least 5 possible values for the third length so there exists a triangle having these as sidelengths (decimal values could also be chosen):

(a) 11, 100100

(b) 55, 55

(c) 33, 77

See if you can describe all possible lengths of the third side in each case, so that a triangle exists with those sidelengths. For example, in case (a), all numbers strictly between 9999 and 101101 would be possible.

Q9

Construct triangles for the following measurements where the angle is included between the sides:

(a) 3 cm3\text{ cm}, 75∘75^\circ, 7 cm7\text{ cm}

(b) 6 cm6\text{ cm}, 25∘25^\circ, 3 cm3\text{ cm}

(c) 3 cm3\text{ cm}, 120∘120^\circ, 8 cm8\text{ cm}

Q10

Construct triangles for the following measurements:

(a) 75∘75^\circ, 5 cm5\text{ cm}, 75∘75^\circ

(b) 25∘25^\circ, 3 cm3\text{ cm}, 60∘60^\circ

(c) 120∘120^\circ, 6 cm6\text{ cm}, 30∘30^\circ

Q11

For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category:

(a) 30∘30^\circ

(b) 70∘70^\circ

(c) 54∘54^\circ

(d) 144∘144^\circ

Q12

Determine which of the following pairs can be the angles of a triangle and which cannot:

(a) 35∘35^\circ, 150∘150^\circ

(b) 70∘70^\circ, 30∘30^\circ

(c) 90∘90^\circ, 85∘85^\circ

(d) 50∘50^\circ, 150∘150^\circ

Q13

Find the third angle of a triangle (using a parallel line) when two of the angles are:

(a) 36∘,72∘36^\circ, 72^\circ

(b) 150∘,15∘150^\circ, 15^\circ

(c) 90∘,30∘90^\circ, 30^\circ

(d) 75∘,45∘75^\circ, 45^\circ

Q14

Can you construct a triangle all of whose angles are equal to 70∘70^\circ? If two of the angles are 70∘70^\circ what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

Q15

Here is a triangle in which we know ∠B=∠C\angle B = \angle C and ∠A=50∘\angle A = 50^\circ. Can you find ∠B\angle B and ∠C\angle C?

Q16

Construct a triangle ABCABC with BC=5 cmBC = 5\text{ cm}, AB=6 cmAB = 6\text{ cm}, CA=5 cmCA = 5\text{ cm}. Construct an altitude from AA to BCBC.

Q17

Construct a triangle TRYTRY with RY=4 cmRY = 4\text{ cm}, TR=7 cmTR = 7\text{ cm}, ∠R=140∘\angle R = 140^\circ. Construct an altitude from TT to RYRY.

Q18

Construct a right-angled triangle ΔABC\Delta \text{ABC} with ∠B=90∘\angle B = 90^\circ, AC=5 cm\text{AC} = 5 \text{ cm}. How many different triangles exist with these measurements?

[Hint: Note that the other measurements can take any values. Take AC\text{AC} as the base. What values can ∠A\angle A and ∠C\angle C take so that the other angle is 90∘90^\circ?]

Q19

Through construction, explore if it is possible to construct an equilateral triangle that is: (i) right-angled (ii) obtuse-angled.

Also construct an isosceles triangle that is: (i) right-angled (ii) obtuse-angled.

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