Given: ABCD is a square.

Prove: [tex]AC \perp BD[/tex].

Square ABCD is shown. Diagonals are drawn from point A to point C and from point B to point D, intersecting at point E.

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 BE is congruent to side DE because the diagonals of a square bisect each other. Therefore, triangle AEB is congruent to triangle AED by the Side-Side-Side (SSS) congruence theorem. We see that angle AED and angle AEB are a linear pair and congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Thus, the measure of these angles will be 90°, and diagonal AC is perpendicular to diagonal BD by definition.

Question

26-06-2024
Mathematics

High School

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Given: ABCD is a square.

Prove: [tex]AC \perp BD[/tex].

Square ABCD is shown. Diagonals are drawn from point A to point C and from point B to point D, intersecting at point E.

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 BE is congruent to side DE because the diagonals of a square bisect each other. Therefore, triangle AEB is congruent to triangle AED by the Side-Side-Side (SSS) congruence theorem. We see that angle AED and angle AEB are a linear pair and congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Thus, the measure of these angles will be 90°, and diagonal AC is perpendicular to diagonal BD by definition.

Answer :

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abidemiokin abidemiokin · Answer 1 · 2025-01-30T18:48:48-05:00

The measure of these angles will be 90°, and diagonal AC is perpendicular to diagonal BD by the definition of perpendicular lines.

Given: ABCD is a square.

Prove: AC ⊥ BD.

Proof:

Square ABCD is shown. Diagonals are drawn from point A to point C and from point B to point D and intersect at point E.

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 of Congruence. 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. Therefore, AC ⊥ BD.