Understanding the Isosceles Triangle Theorem | Proof and Applications for Angle Measurements in Isosceles Triangles

Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that if a triangle has two sides that are congruent (equal in length), then the angles opposite those sides are also congruent

The Isosceles Triangle Theorem states that if a triangle has two sides that are congruent (equal in length), then the angles opposite those sides are also congruent. In other words, if you have an isosceles triangle with two sides of equal length, then the angles opposite those sides will have the same measure.

Let’s consider an isosceles triangle ABC where AB = AC. The theorem states that angle BAC is congruent to angle BCA. This can be written as ∠BAC ≅ ∠BCA.

To understand why this theorem holds, let’s look at a proof:

Proof:
1. Given: Triangle ABC with AB = AC
2. Draw the altitude from point A to side BC, and label the point of intersection as D.
3. Since the altitude is drawn, AD is perpendicular to BC, which means angle BAD and angle DAC are right angles.
4. In an isosceles triangle, such as ABC in this case, the altitude is also a median and an angle bisector.
5. Therefore, AD is a median, which means it divides side BC into two congruent segments, BD = CD.
6. Now we have two congruent triangles, triangle ABD and triangle ACD, because they have AB = AC, AD is common, and BD = CD.
7. By triangle congruence criteria (such as Side-Angle-Side, or SAS), we can conclude that triangle ABD ≅ triangle ACD.
8. Therefore, angle BAD is congruent to angle CAD.
9. But we know that angle BAD is congruent to angle BAC, so BAC is also congruent to CAD.
10. So, the Isosceles Triangle Theorem is proved.

The Isosceles Triangle Theorem is useful in geometry to identify and relate angles within isosceles triangles. It allows us to determine the measures of angles in an isosceles triangle by using the congruency of sides.

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