Proving Congruence of Isosceles Triangles: Understanding the Conditions and Method

If the congruent sides in one isosceles triangle have the same measure as the congruent sides inanother isosceles triangle, then the triangles are congruent

To prove that two isosceles triangles are congruent, we need to show that all corresponding sides and angles are equal

To prove that two isosceles triangles are congruent, we need to show that all corresponding sides and angles are equal.

In an isosceles triangle, there are two congruent sides and two congruent angles. Let’s refer to the given triangles as Triangle ABC and Triangle DEF, where AB = AC and DE = DF.

To show that the congruent sides in Triangle ABC have the same measure as the congruent sides in Triangle DEF, we need to show that AB = DE and AC = DF.

Since it is given that AB = AC, we can substitute AC for AB in Triangle DEF, so AB = DE = AC.

Now let’s look at the angles. In an isosceles triangle, the base angles opposite the congruent sides are also congruent. We can denote these angles as angle A and angle D.

Since Triangle ABC is isosceles, angle A = angle B.
Similarly, since Triangle DEF is isosceles, angle D = angle E.

Using the given information that the congruent sides have the same measure, we can state that angle A = angle D.

So far, we have AB = DE = AC and angle A = angle D.

Now, let’s consider the remaining side of Triangle DEF, which is EF. Since Triangle DEF is isosceles, angle D = angle E. By the Angle-Side-Angle Criterion of Congruence, we can state that Triangle ABC is congruent to Triangle DEF.

In conclusion, when the congruent sides in one isosceles triangle have the same measure as the congruent sides in another isosceles triangle, we can conclude that the triangles are congruent. This is because all corresponding sides and angles are equal, fulfilling the requirements for triangle congruence.

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