Triangles

Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about Chapter 7, which is all about Triangles. Now, triangles are something you have been studying since your earlier classes, but in this chapter, we are going to go much deeper and understand some really important properties and rules about triangles. So let's begin, shall we?

First, let's recall what we mean by a triangle. You all know that a triangle is a closed figure formed by three intersecting lines. The word 'tri' actually means 'three', so it makes perfect sense. Now, a triangle has three sides, three angles, and three vertices. For example, if I take a triangle and name it as triangle ABC, which we write as ΔABC, then AB, BC, and CA are the three sides. The three angles are angle A, angle B, and angle C. And the three vertices are the points A, B, and C. So that's the basic structure of any triangle.

Now, in Chapter 6, you have already studied some properties of triangles. In this chapter, we are going to study in detail about the congruence of triangles, which means when two triangles are exactly alike in shape and size. We will learn the rules or criteria for congruence, some more properties of triangles, and also inequalities in a triangle. Many of these properties you have already verified in your earlier classes through activities, and now we will prove some of them mathematically. So get ready to learn something really interesting!

Let's start with the concept of congruence. Have you ever noticed that two copies of your photographs of the same size are identical? Or have you seen two bangles of the same size? They look exactly the same, don't they? Similarly, if you take two one rupee coins minted in the same year and place one on top of the other, they cover each other completely. What do we call such figures? We call them congruent figures. The word 'congruent' means equal in all respects. So congruent figures are figures that have the same shape and the same size.

Now, let me ask you to draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely, and we call them congruent circles. Similarly, if you take two squares with sides of the same measure and place one on the other, they are congruent. The same thing happens with two equilateral triangles of equal sides. You will see that they cover each other perfectly.

Now, why are we studying congruence? Think about the ice tray in your refrigerator. The moulds for making ice are all congruent, aren't they? The cast used for making these moulds has congruent depressions. So whenever identical objects have to be produced, the concept of congruence is used. Another example is when you need to replace the refill in your pen. If the new refill is not of the same size as the one you want to remove, it won't fit. Only if the two refills are congruent, the new one fits properly. So congruence is a very useful concept in our daily life.

Now, let's understand what congruence means for triangles. You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. But we need to be careful about the correspondence. When we say triangle PQR is congruent to triangle ABC, we write it as ΔPQR ≅ ΔABC. This means that when we place these triangles on each other, the sides of triangle PQR fall on the corresponding equal sides of triangle ABC, and the same happens for the angles. So PQ covers AB, QR covers BC, and RP covers CA. Also, angle P covers angle A, angle Q covers angle B, and angle R covers angle C. There is a one-to-one correspondence between the vertices, which we write as P ↔ A, Q ↔ B, R ↔ C.

Now, it is very important to write the correspondence correctly. For example, if we have triangle FDE and triangle ABC, and the correspondence is F ↔ A, D ↔ B, E ↔ C, then we write ΔFDE ≅ ΔABC. But it would not be correct to write ΔDEF ≅ ΔABC because the order of vertices matters. So always remember that in congruent triangles, corresponding parts are equal, and we write in short 'CPCT' which stands for 'corresponding parts of congruent triangles'.

Now let's move on to the criteria for congruence of triangles. How do we know if two triangles are congruent? We need some rules or criteria. In earlier classes, you have learned four criteria for congruence. Let's recall them one by one.

The first criterion is the SAS congruence rule. SAS stands for Side-Angle-Side. This means that if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. Now, what does 'included angle' mean? It means the angle between the two equal sides. So if we have triangle ABC and triangle PQR, and if AB = PQ, angle A = angle P, and AC = PR, then triangle ABC is congruent to triangle PQR. This is accepted as an axiom, which means it cannot be proved with the help of previously known results. We accept it as true.

Let me give you an example. Suppose we have a point O, and we have OA = OB and OD = OC. We need to show that triangle AOD is congruent to triangle BOC. Notice that in triangles AOD and BOC, we have OA = OB, OD = OC, and angle AOD equals angle BOC because they are vertically opposite angles. So by the SAS rule, triangle AOD is congruent to triangle BOC. Now, since they are congruent, the corresponding parts are equal. So angle OAD equals angle OBC. And these form a pair of alternate angles for line segments AD and BC. Therefore, AD is parallel to BC. That's a very useful result!

Here's another example. Suppose we have a line segment AB, and line l is its perpendicular bisector. That means line l is perpendicular to AB and passes through the midpoint of AB. Let's call the midpoint C. Now, if a point P lies on line l, we need to show that P is equidistant from A and B, which means PA = PB. Consider triangles PCA and PCB. We have AC = BC because C is the midpoint of AB. Angle PCA and angle PCB are both 90 degrees because line l is perpendicular to AB. And PC is common to both triangles. So by the SAS rule, triangle PCA is congruent to triangle PCB. Therefore, PA = PB because they are corresponding sides of congruent triangles. So we have proved that any point on the perpendicular bisector of a line segment is equidistant from the two endpoints.

Now, very important point: the equal angles must be included between the pairs of equal sides. If the angle is not included, then the triangles may not be congruent. For example, if we construct two triangles with sides 4 cm and 5 cm, and one angle of 50 degrees, but this angle is not between the equal sides, then the two triangles are not necessarily congruent. So the SAS rule works, but not the ASS or SSA rule, because the angle must be the included angle.

Now, let's look at the second criterion, which is the ASA congruence rule. ASA stands for Angle-Side-Angle. This means that if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent. This is actually a theorem, which means it can be proved. Let me explain the proof.

Suppose we have two triangles ABC and DEF. We are given that angle B equals angle E, angle C equals angle F, and side BC equals side EF. We need to prove that triangle ABC is congruent to triangle DEF. Now, there are three cases. In the first case, let us assume that AB equals DE. Then we have AB = DE, angle B = angle E, and BC = EF. So by the SAS rule, triangle ABC is congruent to triangle DEF.

In the second case, suppose AB is greater than DE. Then we can take a point P on AB such that PB equals DE. Now consider triangles PBC and DEF. We have PB = DE, angle B = angle E, and BC = EF. So by the SAS rule, triangle PBC is congruent to triangle DEF. Since the triangles are congruent, their corresponding parts are equal. So angle PCB equals angle DFE. But we are given that angle ACB equals angle DFE. So angle ACB equals angle PCB. This is possible only if point P coincides with point A. So AB equals DE, and then by the SAS rule, triangle ABC is congruent to triangle DEF.

The third case is when AB is less than DE. We can choose a point M on DE such that ME equals AB, and by repeating the same arguments, we can prove that triangle ABC is congruent to triangle DEF. So the ASA rule is proved.

Now, what if two pairs of angles are equal, but the side is not included between them? You know that the sum of the three angles of a triangle is 180 degrees. So if two pairs of angles are equal, the third pair is also equal automatically. So we can say that two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. This is called the AAS congruence rule, which stands for Angle-Angle-Side. This is very useful because we don't need the side to be included between the equal angles.

Now, let me make something very clear. Equality of three angles is not sufficient for congruence of triangles. For example, if you draw triangles with angles 40 degrees, 50 degrees, and 90 degrees, you can draw as many triangles as you want with different lengths of sides. These triangles may or may not be congruent to each other. So for congruence, we need at least one side to be equal along with the angles.

Now, let's look at some more criteria for congruence. The third criterion is the SSS congruence rule. SSS stands for Side-Side-Side. This means that if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent. You have already verified this in earlier classes. For example, if you construct two triangles with sides 4 cm, 3.5 cm, and 4.5 cm, and place them on each other, they cover each other completely. So they are congruent.

The fourth criterion is the RHS congruence rule, which is specifically for right triangles. RHS stands for Right angle-Hypotenuse-Side. This means that if in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. Note that the right angle is not the included angle in this case. You have verified this in earlier classes. For example, if you construct two right-angled triangles with hypotenuse equal to 5 cm and one side equal to 4 cm, they are congruent.

Now, let's study some properties of triangles. Specifically, let's look at isosceles triangles. A triangle in which two sides are equal is called an isosceles triangle. For example, if in triangle ABC, AB equals AC, then it is an isosceles triangle. Now, what do you think is special about isosceles triangles? Let's do an activity. Construct a triangle in which two sides are equal, say each equal to 3.5 cm, and the third side equal to 5 cm. Now measure the angles opposite to the equal sides. What do you observe? You will notice that the angles opposite to the equal sides are equal. This is a very important result.

So, we have Theorem 7.2: Angles opposite to equal sides of an isosceles triangle are equal. Let me prove this for you. We are given an isosceles triangle ABC in which AB equals AC. We need to prove that angle B equals angle C. Let's draw the bisector of angle A and let it intersect BC at point D. Now, in triangles BAD and CAD, we have AB equals AC, angle BAD equals angle CAD because AD is the bisector of angle A, and AD is common. So by the SAS rule, triangle BAD is congruent to triangle CAD. Therefore, angle ABD equals angle ACD, because they are corresponding angles of congruent triangles. So angle B equals angle C. That's the proof!

Now, is the converse true? That is, if two angles of a triangle are equal, then are the sides opposite to them equal? Let's perform an activity. Construct a triangle ABC with BC of any length and angle B equals angle C equals 50 degrees. Draw the bisector of angle A and let it intersect BC at D. Now, cut out the triangle and fold it along AD so that vertex C falls on vertex B. What do you observe? You will see that side AC covers side AB completely. So AC equals AB. Repeat this activity with some more triangles. Each time you will observe that the sides opposite to equal angles are equal. So we have Theorem 7.3: The sides opposite to equal angles of a triangle are equal. This is the converse of Theorem 7.2. You can prove this theorem by using the ASA congruence rule.

Now, let's take some examples to apply these results. Example 4: In triangle ABC, the bisector AD of angle A is perpendicular to side BC. We need to show that AB equals AC and triangle ABC is isosceles. In triangles ABD and ACD, we have angle BAD equals angle CAD because AD is the bisector of angle A. AD is common. And angle ADB equals angle ADC because both are 90 degrees. So by the ASA rule, triangle ABD is congruent to triangle ACD. Therefore, AB equals AC, or triangle ABC is an isosceles triangle.

Example 5: E and F are respectively the mid-points of equal sides AB and AC of triangle ABC. We need to show that BF equals CE. In triangles ABF and ACE, we have AB equals AC, angle A is common, and AF equals AE because they are halves of equal sides. So by the SAS rule, triangle ABF is congruent to triangle ACE. Therefore, BF equals CE.

Example 6: In an isosceles triangle ABC with AB equals AC, D and E are points on BC such that BE equals CD. We need to show that AD equals AE. In triangles ABD and ACE, we have AB equals AC, angle B equals angle C because angles opposite to equal sides are equal, and BD equals CE because BE equals CD. So by the SAS rule, triangle ABD is congruent to triangle ACE. Therefore, AD equals AE.

Now, let's summarize what we have learned so far. We have learned about congruence of triangles and the different criteria for congruence. We have learned that two figures are congruent if they are of the same shape and same size. We have learned the SAS rule, the ASA rule, the AAS rule, the SSS rule, and the RHS rule. We have also learned that in an isosceles triangle, the angles opposite to equal sides are equal, and conversely, the sides opposite to equal angles are equal.

Now, let's look at some more examples. Example 7: AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B. We need to show that the line PQ is the perpendicular bisector of AB. We are given that PA equals PB and QA equals QB. We need to show that PQ is perpendicular to AB and PQ bisects AB. Let PQ intersect AB at C. First, consider triangles PAQ and PBQ. We have PA equals PB, QA equals QB, and PQ is common. So by the SSS rule, triangle PAQ is congruent to triangle PBQ. Therefore, angle APQ equals angle BPQ. Now, consider triangles PAC and PBC. We have PA equals PB, angle APC equals angle BPC because angle APQ equals angle BPQ, and PC is common. So by the SAS rule, triangle PAC is congruent to triangle PBC. Therefore, AC equals BC, and angle ACP equals angle BCP. Also, angle ACP and angle BCP form a linear pair, so their sum is 180 degrees. Therefore, 2 times angle ACP equals 180 degrees, which means angle ACP is 90 degrees. So we have proved that PQ is the perpendicular bisector of AB.

Example 8: P is a point equidistant from two lines l and m intersecting at point A. We need to show that the line AP bisects the angle between them. We are given that lines l and m intersect at A. Let PB be perpendicular to l, and PC be perpendicular to m. It is given that PB equals PC. We need to show that angle PAB equals angle PAC. Consider triangles PAB and PAC. We have PB equals PC, angle PBA equals angle PCA because both are 90 degrees, and PA is common. So by the RHS rule, triangle PAB is congruent to triangle PAC. Therefore, angle PAB equals angle PAC. So the line AP bisects the angle between the two lines.

Now, let's also recall some important facts. Each angle of an equilateral triangle is 60 degrees. This is because the sum of the angles is 180 degrees, and all three angles are equal, so each angle is 180 divided by 3, which is 60 degrees.

Now, let's go through the entire chapter one more time to make sure we have covered everything.

In this chapter, we started with the introduction of triangles, understanding what a triangle is, its sides, angles, and vertices. Then we learned about congruence of figures, and specifically congruence of triangles. We understood that two triangles are congruent if their corresponding sides and angles are equal. We learned about CPCT, which stands for corresponding parts of congruent triangles.

Then we learned about the criteria for congruence. The first criterion is the SAS rule, which says that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. The second criterion is the ASA rule, which says that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. The third criterion is the AAS rule, which says that if two angles and one side of one triangle are equal to two angles and the corresponding side of another triangle, then the triangles are congruent. The fourth criterion is the SSS rule, which says that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. The fifth criterion is the RHS rule, which is specifically for right triangles and says that if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, then the triangles are congruent.

We also learned some very important properties of triangles. We learned that in an isosceles triangle, the angles opposite to the equal sides are equal. This is Theorem 7.2. And we learned the converse of this theorem, which is Theorem 7.3: the sides opposite to equal angles of a triangle are equal. We applied these results to solve various examples.

We also learned that in an equilateral triangle, each angle is 60 degrees.

So, to summarize the entire chapter:

1. Two figures are congruent if they are of the same shape and of the same size. 2. Two circles of the same radii are congruent. 3. Two squares of the same sides are congruent. 4. If two triangles ABC and PQR are congruent under the correspondence A ↔ P, B ↔ Q, and C ↔ R, then we write ΔABC ≅ ΔPQR. 5. If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. This is the SAS Congruence Rule. 6. If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent. This is the ASA Congruence Rule. 7. If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent. This is the AAS Congruence Rule. 8. Angles opposite to equal sides of a triangle are equal. 9. Sides opposite to equal angles of a triangle are equal. 10. Each angle of an equilateral triangle is 60 degrees. 11. If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent. This is the SSS Congruence Rule. 12. If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. This is the RHS Congruence Rule.

That, students, is the complete lesson on Chapter 7: Triangles. I hope you have understood all the concepts clearly. Remember, practice is very important in mathematics. So, make sure you solve problems based on these concepts to strengthen your understanding. Thank you for listening, and I will see you in the next lesson. Good luck with your studies!