Question

given abcd is a square. prove: ac ⊥ bd. 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 definition of right angles definition of perpendicular definition of diagonal

Explanation:

Step1: Identify congruent sides in square

In square ABCD, AB is congruent to AD as sides of a square are congruent.

Step2: Recognize reflexive property

AE is congruent to AE by the reflexive property.

Step3: Use diagonal - bisection property

In a square, diagonals bisect each other, so DE is congruent to BE.

Step4: Determine triangle - congruence

By SSS (Side - Side - Side) congruency, triangle AEB is congruent to triangle AED.

Step5: Analyze angle relationship

Since angle AED and angle AEB are a linear pair and congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), each angle measures 90°.

Step6: Prove perpendicularity

By the definition of perpendicular (two lines are perpendicular if they intersect at a 90 - degree angle), AC is perpendicular to BD.

Answer:

We are given that ABCD is a square. If we consider triangle AEB and triangle AED, we see that side AB 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 reflexive property. Finally, we know that side DE is congruent to side BE because the diagonals of a square bisect each other. Therefore, triangle AEB is congruent to triangle AED by SSS 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 definition of perpendicular.