Isosceles Triangles

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An isosceles triangle is a triangle in which at least two sides are equal to each other. The two equal sides are called the legs, and the third side is called the base.

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An equilateral triangle is a triangle in which all three sides are equal. Since it has at least two equal sides, every equilateral triangle is also an isosceles triangle, but not every isosceles trian

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Theorem 1: If two sides of a triangle are equal, then the angles opposite to them are also equal. In other words, in triangle ABC, if AB = AC, then ∠B = ∠C.

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Theorem 2: If two angles of a triangle are equal, then the sides opposite to them are also equal. In other words, in triangle ABC, if ∠B = ∠C, then AB = AC.

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The base angles are the two angles that are opposite to the equal sides of an isosceles triangle. These angles are always equal to each other according to the properties of isosceles triangles.

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The vertex angle is the angle between the two equal sides of an isosceles triangle. It is the angle that is not a base angle and may have a different measure than the base angles.

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Since AB = AC, triangle ABC is isosceles. By Theorem 1, ∠B = ∠C. Using angle sum property: ∠A + ∠B + ∠C = 180°. So 40° + ∠B + ∠B = 180°, which gives 2∠B = 140°, therefore ∠B = ∠C = 70°.

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Given: Triangle ABC with AB = AC, AD bisects ∠A. To prove: AD ⊥ BC and BD = DC. Proof: In triangles ABD and ACD, AB = AC (given), ∠BAD = ∠CAD (AD bisects ∠A), AD = AD (common). So △ABD ≅ △ACD by SAS.