Problem 21 - Algebra 2 and Trigonometry

Question

In \(\triangle A B C, \mathrm{m} \angle B=30\) and in \(\triangle D E F, \mathrm{m} \angle E=150 .\) Show that if \(A B=D E\) and \(B C=E F,\) the areas of the two triangles are equal.

Step-by-Step Solution

Answer

The triangles are congruent by ASA, so their areas are equal.

Step 1: Understand Triangle Properties

Given two triangles, \( \triangle ABC \) and \( \triangle DEF \), we have \( m \angle B = 30^\circ \) and \( m \angle E = 150^\circ \). Note that these angles are supplementary since \( 30^\circ + 150^\circ = 180^\circ \). This is important because it suggests the roles of angles in determining congruent triangles.

Step 2: Examine Side Conditions

According to the problem, \( AB = DE \) and \( BC = EF \). These conditions imply that two sides of \( \triangle ABC \) are equal in length to the corresponding two sides of \( \triangle DEF \).

Step 3: Check Angle-Side-Angle Congruency

Since \( m \angle B = 30^\circ \) and \( m \angle E = 150^\circ \), angle \( \angle B \) is equal to the supplementary angle of \( \angle E \). When two sides and the included angle in one triangle are equal to two sides and the included angle in another triangle, the triangles are congruent by the Angle-Side-Angle (ASA) postulate. Thus, \( \triangle ABC \cong \triangle DEF \).

Step 4: Conclude Based on Congruency

Since \( \triangle ABC \cong \triangle DEF \), corresponding parts of congruent triangles are congruent. Consequently, their areas are equal as well.

Key Concepts

Angle-Side-Angle Postulate

The Angle-Side-Angle (ASA) Postulate is a critical tool for determining triangle congruence. A simple way to remember this is that if two triangles have two angles and the included side between them equal, the triangles are congruent. This means not only are the triangles the same size, but they also have the same shape.

In our exercise, consider triangles \( \triangle ABC \) and \( \triangle DEF \). If \( \angle B = 30^{\circ} \) in \( \triangle ABC \) and that's the corresponding angle to \( \angle E = 150^{\circ} \) in \( \triangle DEF \), and if both have equal included sides like \( AB = DE \) and \( BC = EF \), then these conditions meet the ASA Postulate requirements.

Therefore, \( \triangle ABC \) and \( \triangle DEF \) are congruent. This indicates that every corresponding angle and side has its match in the other triangle, which is key to understanding triangle congruence through ASA.

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. When you think about them, imagine that together they form a straight line. In the context of triangles, supplementary angles help understand relationships between angles in different triangles.

In the given problem, \( \angle B = 30^{\circ} \) from triangle \( \triangle ABC \) and \( \angle E = 150^{\circ} \) from triangle \( \triangle DEF \) together form 180 degrees. This makes these angles supplementary, suggesting a unique relationship between the two triangles.

Understanding that the two angles are supplementary helps reinforce the application of ASA postulate here. Knowing that one angle complements the other fills the gap needed when comparing angle measures and lengths between corresponding triangles.

Triangle Area Comparison

Comparing the areas of two triangles often involves checking the congruence first. When two triangles are congruent by the Angle-Side-Angle postulate, their areas are simply equal. This means you don't need exhaustive calculations to compare areas.

• If \( \triangle ABC \) and \( \triangle DEF \) are congruent, then every aspect of one triangle, including the area, matches those of the other.
• We determined the congruence through ASA in our step-by-step solution: both triangles share two angles and the included side between those angles.

This congruence directly impacts the area comparison because it implies identical dimensions: each side and angle pair role mirrors in both triangles. Hence, the equality of their areas is guaranteed by the congruence itself, confirming that the two triangles cover the same space without needing more complex calculations.