ICSE Class 7 Maths Chapter 29 Introduction of Theorem

Chapter 29: Introduction of Theorem

Basic Treatment

Some of the properties of geometrical figures are self-evident and are accepted without any proof. Such self-evident truths are called axioms and the statements, stating these self-evident truths, are called postulates.

For example, if ∠a = ∠b and ∠b = ∠c, then ∠a = ∠c.

With the help of these axioms and postulates, some other important properties, in geometry, can easily be proved. The course of proving such properties is called a theorem.

A theorem is a statement about geometrical figures for which a proof is required.

To prove a theorem following steps are required:

1. General statement or general enunciation - This is a combined statement of the facts which are given and those which are to be proved.

(i) Figure - A figure helps in making the proof more understandable.

(ii) Particular statement or enunciation - This is the description of the theorem which is explained with the help of a labelled figure. This part is referred to as given.

2. To prove - The proposition to be proved is written in brief.

3. Construction - Sometimes a line is to be drawn or extended or some points are to be joined in the given figure, to make the proof possible. This step is mentioned here but is not always necessary.

4. (i) Proof - This gives the step by step statements with reasons, so that the proof appears to be logical. It is preferable to write the statements and reasons separately.

(ii) Q.E.D. - The letters Q.E.D. may be written at the end of a theorem and stand for "Quod Erat Demonstrandum", that means - "which was to be proved". Usually we write "Proved or Hence Proved" in place of Q.E.D.

Theorem 1

The sum of the angles of a triangle is equal to two right angles.

Given - Triangle ABC

To Prove - ∠ABC + ∠ACB + ∠BAC = 2 right angles, i.e., 180°.

Construction - At C, draw CE parallel to BA. Produce BC to any point D.

Proof

Alternative Method

Construction - Through vertex A draw PQ parallel to BC.

Proof:

1. Since, PQ // BC and AB is transversal, ∠ABC = ∠BAP (Alternate angles)

2. Since, PQ // BC and AC is transversal, ∠ACB = ∠CAQ (Alternate angles)

∠ABC + ∠ACB = ∠BAP + ∠CAQ (Adding results of 1 and 2)

Therefore ∠ABC + ∠ACB + ∠BAC = ∠BAP + ∠CAQ + ∠BAC (Adding ∠BAC on both the sides)

But, ∠BAP + ∠CAQ + ∠BAC = Straight line angle PAQ = 180°

Therefore ∠ABC + ∠ACB + ∠BAC = 180° (Hence Proved.)

Teacher's Note

Understanding how to prove geometric theorems helps students develop logical reasoning skills that apply to everyday problem-solving, from building construction to computer programming.

Theorem 2

If any side of a triangle is produced, then the exterior angle so obtained is equal to the sum of the two interior opposite angles.

Given - A triangle ABC whose side BC is produced to the point D to form an exterior angle ACD.

To Prove - ∠ACD = ∠ABC + ∠BAC

Construction - At the point C, draw CE parallel to BA.

Proof

Teacher's Note

This theorem explains why exterior angles in polygons are important - they appear in navigation, engineering, and architectural design when extending walls or pathways.

Exercise 29

1. In the given figure, prove that: ∠a = ∠b + ∠c.

2. Given: ∠b = ∠d. Prove: ∠a + ∠c = 180°.

3. Prove: ∠ABD + ∠ACE = 180° + ∠BAC.

4. From the given figure, prove that: (i) Triangle ABC and Triangle CDA are congruent. (ii) ∠B = ∠D.

5. Given: AB = AC and BD = DC. Prove: (i) Triangle ABD ≅ Triangle ACD. (ii) ∠BAD = ∠CAD.

6. Given: AB = AC, BE ⊥ AC and CD ⊥ AB. Prove: (i) Triangle ABE ≅ Triangle ACD. (ii) ∠B = ∠C.

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