This theorem can be proved using a suitable construction. Theorem 7.4(SSS congruence rule): If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. What do you observe? They cover each other completely, if the equal sides are placed on each other. Cut them out and place them on each other. To be sure, construct two triangles with sides 4 cm, 3.5 cm and 4.5 cm. You have already verified in earlier classes that this is indeed true. You may wonder whether equality of three sides of one triangle to three sides of another triangle is enough for congruence of the two triangles. You have seen earlier in this chapter that equality of three angles of one triangle to three angles of the other is not sufficient for the congruence of the two triangles. Some More Criteria for Congruence of Triangles So, AB = AC (CPCT) or, Δ ABC is an isosceles triangle. You can prove this theorem by ASA congruence rule.Let us take some examples to apply these results.Įxample: In Δ ABC, the bisector AD of ∠ A is perpendicular to side BC.Show that AB = AC and Δ ABC is isosceles. Theorem : The sides opposite to equal angles of a triangle are equal Each time you will observe that the sides opposite to equal angles are equal. Repeat this activity with some more triangles. Cut out the triangle from the sheet of paper and fold it along AD so that vertex C falls on vertex B.What can you say about sides AC and AB? Observe that AC covers AB completely So, AC = AB Draw the bisector of ∠ A and let it intersect BC at D. Is the converse also true?That is: If two angles of any triangle are equal, can we conclude that the sides opposite to them are also equal?Ĭonstruct a triangle ABC with BC of any length and ∠ B = ∠ C = 50°. So, ∠ABD = ∠ACD, since they are corresponding angles of congruent triangles. Let us draw the bisector of ∠ A and let D be the point of intersection of this bisector of ∠ A and BC. We are given an isosceles triangle ABC in which AB = AC. Theorem : Angles opposite to equal sides of an isosceles triangle are equal. Repeat this activity with other isosceles triangles with different sides. You may observe that in each such triangle, the angles opposite to the equal sides are equal. This is a very important result and is indeed true for any isosceles triangle. So, Δ ABC is an isosceles triangle with AB = AC. Now, measure ∠ B and ∠ C. You have done such constructions in earlier classes.Ī triangle in which two sides are equal is called an isosceles triangle. Let us apply the results of congruence of triangles to study some properties related to a triangle whose two sidesare equal.Ĭonstruct a triangle in which two sides are equal, say each equal to 3.5 cm and the third side equal to 5 cm. The sides opposite to equal angles of a triangle are equal.Angles opposite to equal sides of an isosceles triangle are equal.This topic gives an overview of the theorems
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