Given that ABCD is a square, if we consider triangles …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

We are given that ABCD is a square. If we consider triangle AEB and triangle AED, we see that side is congruent to side AD because sides of a square are congruent. We know that side AE is congruent to side AE by using the . Finally, we know that side DE is congruent to side because the diagonals of a square bisect each other. Therefore, triangle AEB is congruent to triangle AED by congruency. We see that angle AED and angle AEB are a linear pair, and congruent by CPCTC. Thus, the measure of these angles will be 90°, and diagonal AC is perpendicular to diagonal BD by the .

Short Answer

The proof of congruence between Triangle AEB and Triangle AED involves establishing congruent sides and applying the Reflexive Property. By demonstrating that sides AB, AD, AE, DE, and BE are congruent, we conclude the triangles are congruent via the SSS postulate, leading to the identification of angles AED and AEB as a linear pair measuring 90 degrees.

Step-by-Step Solution

Step 1: Understand the Congruent Sides

To begin solving the problem, we are given two triangles, Triangle AEB and Triangle AED. We notice that side AB is congruent to side AD. This establishes a fundamental basis for proving the triangles are congruent, as having at least one pair of congruent sides is essential.

Step 2: Apply the Reflexive Property

Next, we utilize the Reflexive Property, which states that any segment is congruent to itself. Therefore, side AE is congruent to itself (AE ‚ÄöaO AE). This helps reinforce the structure of the triangles as it provides another side that is congruent and necessary for our congruence proof.

Step 3: Prove Triangle Congruence and Define Angles

Now, we look at side DE, which is congruent to side BE due to the property that the diagonals of a square bisect each other. With the sides AB ‚ÄöaO AD, AE ‚ÄöaO AE, and DE ‚ÄöaO BE, we can conclude that Triangle AEB is congruent to Triangle AED by the SSS (Side-Side-Side) congruency postulate. Consequently, using CPCT (Corresponding Parts of Congruent Triangles), we find that angles AED and AEB are a linear pair, measuring 90 degrees because diagonal AC is perpendicular to diagonal BD, as stated in the definition of perpendicularity.

Related Concepts

Congruent Sides

Sides of two or more geometric figures that are equal in length, which is crucial in proving the figures’ similarity or congruence

Reflexive Property

A property in geometry stating that any geometric figure is congruent to itself, thereby establishing relationships between angles or sides in proofs

Sss Congruency Postulate

A theorem stating that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Table Of Contents
  1. We are given that ABCD is a square. If we consider triangle AEB and triangle AED, we see that side is congruent to side AD because sides of a square are congruent. We know that side AE is congruent to side AE by using the . Finally, we know that side DE is congruent to side because the diagonals of a square bisect each other. Therefore, triangle AEB is congruent to triangle AED by congruency. We see that angle AED and angle AEB are a linear pair, and congruent by CPCTC. Thus, the measure of these angles will be 90°, and diagonal AC is perpendicular to diagonal BD by the .

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