(a) Define what it means for two triangles AABC and ADEF to be congruent. (b) State the SAS Postulate. (c) State and prove the Isosceles Triangle Theorem.
(a) Define what it means for two triangles AABC and ADEF to be congruent. (b) State the SAS Postulate. (c) State and prove the Isosceles Triangle Theorem.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#### Geometric Concepts
**(a) Define Congruence of Triangles:**
For two triangles \( \triangle ABC \) and \( \triangle DEF \) to be congruent, they must have exactly the same shape and size. This occurs when there is a one-to-one correspondence between their vertices such that their corresponding sides and angles are equal. In simpler terms, triangle \( \triangle ABC \) can be mapped onto triangle \( \triangle DEF \) through a series of rigid transformations (translations, rotations, and reflections).
**(b) State the SAS Postulate:**
The Side-Angle-Side (SAS) Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
**(c) State and Prove the Isosceles Triangle Theorem:**
The Isosceles Triangle Theorem asserts that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent.
*Proof:*
1. Consider \( \triangle ABC \) where \( AB = AC \).
2. By definition, the angles opposite these sides, \( \angle B \) and \( \angle C \), must be congruent.
3. To prove this, draw the altitude from vertex \( A \) to the midpoint \( D \) of base \( BC \).
4. This altitude creates two congruent right triangles, \( \triangle ABD \) and \( \triangle ACD \), by the Hypotenuse-Leg (HL) theorem.
5. Hence, \( \angle B = \angle C \).
This theorem confirms that an isosceles triangle has two equal angles opposite its equal sides, reinforcing the properties of symmetry and congruence in geometry.
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