Problem 1 - A History of Mathematics: An Introduction

Question

Prove Proposition \(\mathrm{I}-5\), that the base angles of an isosceles triangle are equal to one another.

Step-by-Step Solution

Answer

Question: Prove Proposition I-5 which states that the base angles of an isosceles triangle are equal to one another. Solution: First, draw the isosceles triangle and an auxiliary line. Then, show that the two right triangles formed are congruent using the Hypotenuse-Leg Congruence. Finally, use the congruent triangles to show that the base angles are equal using the Corresponding Parts of Congruent Triangles theorem.

Step 1: Draw the isosceles triangle and an auxiliary line

Draw isosceles triangle \(ABC\) with \(AB = AC\) (equal sides). Let \(D\) be a point on side \(BC\) such that \(AD\) is perpendicular to \(BC\). Now, we have four right triangles: \(\triangle ABD\), \(\triangle ADC\), \(\triangle BAD\), and \(\triangle DAC\).

Step 2: Show that \(\triangle ABD\) is congruent to \(\triangle ADC\)

Since \(AB = AC\), \(AD\) is common, and \(\angle BAD \cong \angle DAC\) (both are right angles), we can use the Hypotenuse-Leg Congruence to show \(\triangle ABD \cong \triangle ADC\). That is, if two right triangles have a congruent hypotenuse and a corresponding congruent leg, the triangles are congruent.

Step 3: Use congruent triangles to show that base angles are equal

Since \(\triangle ABD \cong \triangle ADC\), we can use the Corresponding Parts of Congruent Triangles theorem: \(\angle ABC \cong \angle ACB\). Thus, the base angles of isosceles triangle \(ABC\) are equal to one another. Therefore, we have proved Proposition I-5: The base angles of an isosceles triangle are equal to one another.

Key Concepts

Congruent Triangles

When you hear the term "congruent triangles," it refers to triangles that are identical in shape and size. This means all their corresponding sides and angles are equal.

In the context of isosceles triangles, identifying congruent triangles helps us prove properties like equal base angles.

In the original exercise, we identify that \( \triangle ABD \cong \triangle ADC \). By showing these two triangles are congruent, we establish that the base angles of the isosceles triangle are equal. This concept is crucial because, through congruence, any property true for one triangle is true for the other.

• All sides match
• All angles match

Understanding this can make many geometry problems easier to solve.

Hypotenuse-Leg Congruence

The Hypotenuse-Leg (HL) Congruence Theorem is a valuable tool when working with right triangles. It states that if two right triangles have a hypotenuse and one corresponding leg that are congruent, then the triangles are congruent.

In the exercise, the HL theorem is used to show that \( \triangle ABD \cong \triangle ADC \).

Here's why:

• These triangles share a right angle.
• The hypotenuse, \( AB \) and \( AC \), are equal as given in the isosceles triangle.
• They share the leg \( AD \).

Having congruent right triangles allows us to use their properties to understand more about the isosceles triangle, like its equal base angles.

Base Angles

Base angles in an isosceles triangle are the angles opposite the equal sides. In \( \triangle ABC \), these are \( \angle ABC \) and \( \angle ACB \). Proving that these angles are equal is a common problem and relies on understanding congruence.

When we've shown that \( \triangle ABD \cong \triangle ADC \) using the HL congruence, it automatically tells us that \( \angle ABC \cong \angle ACB \).

• Corresponding Parts of Congruent Triangles (CPCTC) help us here.
• The equal base angles theorem simplifies isosceles triangle problems.

This fundamental property is what allows their heights and other line segments to draw fascinating conclusions about symmetry and balance in these shapes.