A Tale of Three Intersecting Lines | FIO

Question 12

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

(a) $35^\circ$ , $150^\circ$

(b) $70^\circ$ , $30^\circ$

(d) $50^\circ$ , $150^\circ$

Solution

Understand the Question

The sum of all three interior angles of a triangle is always $180^\circ$ (Angle Sum Property).
For any two angles to be part of a triangle, their sum must be strictly less than $180^\circ$ so that the third angle is greater than $0^\circ$ .
If the sum of two angles is greater than or equal to $180^\circ$ , they cannot form a triangle.

(a) $35^\circ$ , $150^\circ$

Step 1 · Check the Sum of the Angles

Sum of the two angles

\begin{aligned} 35^\circ + 150^\circ &= 185^\circ \end{aligned}

Since $185^\circ > 180^\circ$ , the sum exceeds the total angle sum of a triangle.

Answer

(a) Cannot be angles of a triangle

(b) $70^\circ$ , $30^\circ$

Step 1 · Check the Sum of the Angles

Sum of the two angles

\begin{aligned} 70^\circ + 30^\circ &= 100^\circ \end{aligned}

Since $100^\circ < 180^\circ$ , these can form a triangle.

Third angle

\begin{aligned} 180^\circ - 100^\circ &= 80^\circ \end{aligned}

Answer

(b) Can be angles of a triangle (Third angle $= 80^\circ$ )

(c) $90^\circ$ , $85^\circ$

Step 1 · Check the Sum of the Angles

Sum of the two angles

\begin{aligned} 90^\circ + 85^\circ &= 175^\circ \end{aligned}

Since $175^\circ < 180^\circ$ , these can form a triangle.

Third angle

\begin{aligned} 180^\circ - 175^\circ &= 5^\circ \end{aligned}

Answer

(c) Can be angles of a triangle (Third angle $= 5^\circ$ )

(d) $50^\circ$ , $150^\circ$

Step 1 · Check the Sum of the Angles

Sum of the two angles

\begin{aligned} 50^\circ + 150^\circ &= 200^\circ \end{aligned}

Since $200^\circ > 180^\circ$ , the sum exceeds the total angle sum of a triangle.

Answer

(d) Cannot be angles of a triangle

Common Mistakes

Summing to $180^\circ$ : Confusing the condition for two angles with the sum of all three angles. The sum of two angles must be strictly less than $180^\circ$ , not equal to $180^\circ$ .
Missing the Third Angle: Forgetting that every triangle must have three positive angles ( $> 0^\circ$ ).

More questions in FIO

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

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

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

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

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

(b) $5\text{ mm}$ , $10\text{ mm}$ and $20\text{ mm}$

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) $2, 2, 5$ (b) $3, 4, 6$ (c) $2, 4, 8$ (d) $5, 5, 8$ (e) $10, 20, 25$ (f) $10, 20, 35$ (g) $24, 26, 28$

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

(a) 1, 100, 100

(b) 3, 6, 9

(d) 5, 10, 12

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.

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) $1$ , $100$

(b) $5$ , $5$

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 $99$ and $101$ would be possible.

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

(a) $3\text{ cm}$ , $75^\circ$ , $7\text{ cm}$

(b) $6\text{ cm}$ , $25^\circ$ , $3\text{ cm}$

Q10

Construct triangles for the following measurements:

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

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

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) $30^\circ$

(b) $70^\circ$

(d) $144^\circ$

Q12

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

(a) $35^\circ$ , $150^\circ$

(b) $70^\circ$ , $30^\circ$

(d) $50^\circ$ , $150^\circ$

Q13

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

(a) $36^\circ, 72^\circ$

(b) $150^\circ, 15^\circ$

(d) $75^\circ, 45^\circ$

Q14

Can you construct a triangle all of whose angles are equal to $70^\circ$ ? If two of the angles are $70^\circ$ 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 $\angle B = \angle C$ and $\angle A = 50^\circ$ . Can you find $\angle B$ and $\angle C$ ?

Q16

Construct a triangle $ABC$ with $BC = 5\text{ cm}$ , $AB = 6\text{ cm}$ , $CA = 5\text{ cm}$ . Construct an altitude from $A$ to $BC$ .

Q17

Construct a triangle $TRY$ with $RY = 4\text{ cm}$ , $TR = 7\text{ cm}$ , $\angle R = 140^\circ$ . Construct an altitude from $T$ to $RY$ .

Q18

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

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

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.