Fundamental Ideas
Triangles
Isosceles triangles are special and because of that there are unique relationships that involve their internal line segments. Consider isosceles triangle
ABC in Figure
1 .
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With a median drawn from the vertex to the base, BC , it can be proven that Δ BAX ≅ Δ CAX, which leads to several important theorems.
Theorem 32: If two sides of a triangle are equal, then the angles opposite those sides are also equal.
Theorem 33: If a triangle is equilateral, then it is also equiangular.
Theorem 34: If two angles of a triangle are equal, then the sides opposite these angles are also equal.
Theorem 35: If a triangle is equiangular, then it is also equilateral.
Example 1: Figure
2 has Δ
QRS with
QR =
QS. If
m ∠
Q = 50°, find
m ∠
R and
m ∠
S.
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Because m ∠ Q + m ∠ R + m ∠ S = 180°, and because QR = QS implies that m ∠ R = m ∠ S,
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Example 2: Figure
3 has Δ
ABC with
m ∠
A =
m ∠
B =
m ∠
C, and
AB = 6. Find
BC and
AC.
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Because the triangle is equiangular, it is also equilateral. Therefore, BC = AC = 6.
