Chapter 4.2, Problem 26WE

To determine

To prove: that $△ABF△BCG$ are congruent.

Answer to Problem 26WE

  $\text{Δ}ABF\cong \text{Δ}BCG$ are congruent

Explanation of Solution

Given information:

Cube whose faces are congruent squares

Calculation:

SSS Postulate: If three sides of a triangle are equal (congruent) to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of a triangle are equal (congruent) to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of a triangle are equal (congruent) to the two corresponding angles and the included side of another triangle, then the triangles are congruent.

In the above three-dimensional figure the triangles listed are coloured red.

  

Proof:

1. Cube with congruent square faces

Reason: Given

2. $AB=FB=BC=GC$

Reason: Definition of square

3. Square $ABFE\text{\&}BCGF$

  $\angle B={90}^{\circ }\text{\&}\angle C={90}^{\circ }$

Reason: Definition of square

4. $\angle B\cong \angle C$

Reason: Definition of congruent angles.

5. $\text{Δ}ABF\cong \text{Δ}BCG$

Reason: SAS postulate

Hence, $\text{Δ}ABF\cong \text{Δ}BCG$ are congruent