A Tale of Three Intersecting Lines | IT

Question 34

Try experimenting with different triangles to see if there is a relation between any two angles and the third one. To find this relation, what data will you keep track of and how will you organise the data you collect?

Solution

Understand the Question

To investigate the relationship between the angles of different triangles (acute, right-angled, and obtuse), we measure the three interior angles $\angle A$ , $\angle B$ , and $\angle C$ using a protractor.
We record the measurements of the three angles along with their sum in a structured table to observe patterns.
The sum of the interior angles of any triangle is always $180^\circ$ , meaning the sum of any two angles is always $180^\circ$ minus the third angle ( $A + B = 180^\circ - C$ ).

Step 1 · Draw and Measure Different Triangles

Draw various types of triangles: an acute-angled triangle, a right-angled triangle, and an obtuse-angled triangle. Label the vertices and angles as $A$ , $B$ , and $C$ , then measure each angle accurately using a protractor. Diagram 1

Step 2 · Organise the Collected Data

Organise the collected data into a tabular format recording the triangle type, individual angle measurements, and their sum.

For example, for a sample triangle with $\angle A = 60^\circ$ , $\angle B = 70^\circ$ , and $\angle C = 50^\circ$ :

\begin{aligned} 60^\circ + 70^\circ + 50^\circ &= 130^\circ + 50^\circ \\ &= 180^\circ \end{aligned}

\begin{array}{|c|c|c|c|c|} \hline \text{Triangle} & \angle A & \angle B & \angle C & \text{Sum } (\angle A + \angle B + \angle C) \\ \hline \text{Acute} & 60^\circ & 70^\circ & 50^\circ & 180^\circ \\ \hline \text{Right-angled} & 90^\circ & 45^\circ & 45^\circ & 180^\circ \\ \hline \text{Obtuse} & 110^\circ & 40^\circ & 30^\circ & 180^\circ \\ \hline \end{array}

Step 3 · Deduce the Relation

From the data, the sum of all three angles is always constant: $A + B + C = 180^\circ$

Rearranging to relate any two angles to the third: $A + B = 180^\circ - C$

Answer

Relation: The sum of the three angles of any triangle is $180^\circ$ ( $A + B + C = 180^\circ$ ), so the sum of any two angles is equal to $180^\circ$ minus the third angle ( $A + B = 180^\circ - C$ ).

Data tracked: The measures of each of the three interior angles and their sum.
Organisation: A table with columns for Triangle Type/Name, $\angle A$ , $\angle B$ , $\angle C$ , and Sum of Angles.

Common Mistakes

Measurement Errors: Inaccurate readings with a protractor may yield sums like $179^\circ$ or $181^\circ$ ; ensuring the protractor's baseline and centre point align precisely with the vertex is essential.
Generalisation Hesitation: Thinking the angle sum property varies depending on the size or shape of the triangle — the sum of interior angles in Euclidean geometry is always strictly $180^\circ$ for all triangles.

More questions in IT

What happens when the three vertices lie on a straight line?

Construct a triangle in which all the sides are of length 4 cm.

Context: Construct a triangle in which all the sides are of length 4 cm.

Q. How did you construct this triangle and what tools did you use? Can this construction be done only using a marked ruler (and a pencil)?

Q. How do we make this construction more efficient?

Context: Step 2: Construct another arc of radius $4\text{ cm}$ from $B$ . Let $C$ be the point of intersection of the arcs.

Q. The construction ensures that both $AC$ and $BC$ are of length $4\text{ cm}$ . Can you see why?

How do we construct triangles that are not equilateral?
Construct a triangle of sidelength $4\text{ cm}$ , $5\text{ cm}$ and $6\text{ cm}$ .

Context: Construct a triangle of sidelength $4\text{ cm}$ , $5\text{ cm}$ and $6\text{ cm}$ .

Q. How do we construct this triangle more efficiently?

Construct

Construct triangles having the following sidelengths (all the units are in cm):

(a) $4$ , $4$ , $6$

(b) $3$ , $4$ , $5$

(d) $4$ , $6$ , $8$

(e) $3.5$ , $3.5$ , $3.5$

Construct a triangle with sidelengths $3\text{ cm}$ , $4\text{ cm}$ , and $8\text{ cm}$ .

What is happening? Are you able to construct the triangle?

Q10

Here is another set of lengths: $2\text{ cm}$ , $3\text{ cm}$ , and $6\text{ cm}$ . Check if a triangle is possible for these sidelengths.

Q11

Try to find more sets of lengths for which a triangle construction is impossible. See if you can find any pattern in them.

Q12

Context: Clearly, the direct straight-line path from the tent to the tree is shorter than the roundabout path via the pole. In fact, the direct straight-line path is the shortest possible path to the tree from the tent. Will the direct path between any two points be shorter than the roundabout path via a third point? Clearly, the answer is yes.

Q. Can this understanding be used to tell something about the existence of a triangle having sidelengths $10\text{ cm}$ , $15\text{ cm}$ and $30\text{ cm}$ ?

Q13

Can we say anything about the existence of a triangle having sidelengths $3\text{ cm}$ , $3\text{ cm}$ and $7\text{ cm}$ ? Verify your answer by construction.

Q14

In the rough diagram in Fig. 7.4, is it possible to assign lengths in a different order such such that the direct paths are always coming out to be shorter than the roundabout paths? If this is possible, then a triangle might exist.

Q15

Context: "In the rough diagram in Fig. 7.4, is it possible to assign lengths in a different order such that the direct paths are always coming out to be shorter than the roundabout paths? If this is possible, then a triangle might exist."

Q. Is such rearrangement of lengths possible in the triangle?

Q16

Will this always happen? That is, for any set of lengths, will there be at least two comparisons where the direct length is less than the sum of the other two? Explore for different sets of lengths.

Q17

Further, for a given set of lengths, is it possible to identify which lengths will immediately be less than the sum of the other two, without calculations?

[Hint: Consider the direct lengths in the increasing order.]

Q18

Given three sidelengths, what do we need to compare to check for the existence of a triangle?

Q19

Context: Does a triangle exist with sidelengths $4\text{ cm}$ , $5\text{ cm}$ and $8\text{ cm}$ ? This satisfies the triangle inequality:

$8 < 4 + 5 = 9$

Q. Why do we not need to check the other two sides?

Q20

Now, suppose that a circle of radius $5 \text{ cm}$ is constructed, centred at $B$ . Can you draw a rough diagram of the resulting figure?

Q21

Will triangles always exist when a set of lengths satisfies the triangle inequality? How can we be sure?

Q22

Q. Let us study each of these cases by finding the relation between the radii (the smaller two lengths) and $AB$ (longest length).

Q23

Context: Case 2: Circles do not intersect internally

Q. For this case to happen, what should be the relation between the radii and $AB$ ?

Q24

Can we use this analysis to tell if a triangle exists when the lengths satisfy the triangle inequality?

Q25

How will the two circles turn out for a set of lengths that do not satisfy the triangle inequality? Find 3 examples of sets of lengths for which the circles:

(a) touch each other at a point,

(b) do not intersect.

Q26

Frame a complete procedure that can be used to check the existence of a triangle.

Q27

We have seen that triangles do not exist for all sets of sidelengths. Is there a combination of measurements in the case of two sides and the included angle where a triangle is not possible? Justify your answer using what you observe during construction.

Q28

Do triangles exist for every combination of two angles and their included side? Explore.

Q29

Find examples of measurements of two angles with the included side where a triangle is not possible.

Q30

It is clear that if the line from $B$ is "inclined" sufficiently to the right, then it will not meet the line $l$ .

(a) Try to find a possible $\angle B$ (marked in the figure) for this to happen.

(b) What could be smallest value of $\angle B$ for the lines to not meet?

Q31

Like the triangle inequality, can you form a rule that describes the two angles for which a triangle is possible?

Can the sum of the two angles be used for framing this rule?

Q32

Context: Let us take two angles, say $60^\circ$ and $70^\circ$ , whose sum is less than $180^\circ$ . Let the included side be $5\text{ cm}$ .

Q. What could the measure of the third angle be? Does this measure change if the base length is changed to some other value, say $7\text{ cm}$ ? Construct and find out.

Q33

In general, once the two angles are fixed, does the third angle depend on the included sidelength? Try with different pairs of angles and lengths.

Q34

Q35

Context: Consider a triangle ABC with $\angle B = 50^\circ$ and $\angle C = 70^\circ$ . Suppose we construct a line XY parallel to BC through vertex A.

Q. We can see new angles being formed here: $\angle XAB$ and $\angle YAC$ . What are their values?

Q36

Angle Sum Property

What can we say about the sum of the angles of any triangle?

Q37

There is a convenient way of verifying the angle sum property by folding a triangular cut-out of a paper. Do you see how this shows that the sum of the angles in this triangle is $180^\circ$ ?

Q38

Find the exterior angle for different measures of $\angle A$ and $\angle B$ . Do you see any relation between the exterior angle and these two angles?

[Hint: From angle sum property, we have $\angle A + \angle B + \angle ACB = 180^\circ$ .]

We also have $\angle ACD + \angle ACB = 180^\circ$ , since they form a straight angle.

What does this show?

Q39

What would the altitude from $A$ to $\text{BC}$ be in this triangle?

Q40

Construct an arbitrary triangle. Label the vertices $A, B, C$ taking $BC$ to be the base.

Construct the altitude from $A$ to $BC$ ,

Q41

Context: Construction of the Altitudes of a Triangle Construct an arbitrary triangle. Label the vertices $A$ , $B$ , $C$ taking $\text{BC}$ to be the base. Construct the altitude from $A$ to $\text{BC}$ .

Q. Constructing the altitude using just a ruler is not accurate. To get a more precise angle of $90^\circ$ , we use a set square along with a ruler.

Can you see how to do this?

Q42

Does there exist a triangle in which a side is also an altitude?

Visualise such a triangle and draw a rough diagram.

Q43

Context: In our study of triangles, we have encountered the following types of triangles; equilateral, isosceles, scalene and right-angled triangles.

Q. Did you spot any other type of triangle?

Q44

What are the other types of triangles based on angle measures?

Q45

What could an acute-angled triangle be? Can we define it as a triangle with one acute angle? Why not?