7.2-Properties of a Triangle

7.2-Properties of a Triangle Important Formulae

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Grade 9 → Math → Triangles → 7.2-Properties of a Triangle

After successful completion of this topic, you should be able to:

  • Understand properties of triangles.

Theorem 7.2 : Angles opposite to equal sides of an isosceles triangle are equal.
Theorem 7.3 : The sides opposite to equal angles of a triangle are equal.


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In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that:
(i) OB = OC
(ii) AO bisects ∠A

In ∆ ABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that ∆ ABC is an isosceles triangle in which AB = AC.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig. 7.32). Show that:
(i) ∆ABE ≅ ∆ACF
(ii) AB = AC, i.e., ABC is an isosceles triangle.

ABC and DBC are two isosceles triangles on the same base BC (see Fig. 7.33). Show that ∠ABD=∠ACD.

∆ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠ BCD is a right angle.

Solution:
∠BCD=∠BCA+∠DCA=∠B+∠D

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ABC is a right angled triangle in which ∠ A = 90° and AB = AC. Find ∠B and ∠C.

Solution:
Each is of 45°

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Show that the angles of an equilateral triangle are 60° each.