Question

Which postulate or theorem proves that these two triangles are congruent?

Answer 1

From the figure,

JHGF is a parallelogram.

From the properties of parallelogram,

∠ F = ∠ H (opposite angles are equal)

FG = JH (opposite sides are equal)

Also, note that FG || JH and JG is a transversal.

Therefore,

∠ FGJ = ∠ GJH (Alternate interior angles are equal)

Now, in Δ FGJ and Δ GJH,

FG = JH

∠ F = ∠ H and

∠ FGJ = ∠ GJH

Hence, the triangles are congruent by AAS Congruence Theorem.

Answer 2

The triangles $\triangle \text{FJG}$ and $\triangle \text{JHG}$ are congruent by $\fbox{\begin\\\ \bf {AAS congruence theorem}\\\end{minispace}}$.

Further explanation:

From the given figure it is observed that there are two triangles $\triangle \text{FJG}$ and $\triangle \text{JHG}$ inside the quadrilateral.

Two triangles are said to be congruent if both the triangles have same angles and sides or it can be said that two triangles are congruent if they completely overlap with each other.

The theorems to check the congruency for the two triangles are as follows:

1) SSS congruence theorem:

As per the SSS congruence theorem if all the three sides of the first triangle are equal to all the three sides of the second triangle then the two triangles are called congruent triangles.

2) SAS congruence theorem:

As per the SAS congruence theorem if the two sides of first triangle are equal to the two sides of the second triangle and the including angles are equal then the two triangles are called congruent triangles.

3) AAS congruence rule:

As per the AAS congruence theorem if the two angles of first triangle are equal to the two angles of the second triangle and the similarly located sides are equal then the two triangles are called congruent triangles.

4) HL congruence rule:

As per the HL congruence theorem if the hypotenuse and the one other side of a triangle is equal to the hypotenuse and the corresponding other side of the second triangle then the two triangles are called congruent triangles.

From the given figure it is observed that $\angle \text{GJH}$ is equal to the $\angle \text{JGF}$, $\angle \text{GHJ}$ is equal to the $\angle \text{GFJ}$ and the side JG is the common side for $\triangle \text{FJG}$ and $\triangle \text{JHG}$.

From figure 1 (attached in the end) it is observed that the $\angle \text{GJH}$ and the $\angle \text{JGF}$ lies on same side, $\angle \text{GHJ}$ and the $\angle \text{GFJ}$ lies on the same side and the side JG is common for the $\triangle \text{FJG}$ and $\triangle \text{JHG}$.  

For triangles $\triangle \text{FJG}$ and $\triangle \text{JHG}$.  

$\fbox{\begin\\\ \begin{aligned}\angle\text{GJH}&=\angle\text{JGF}\\ \angle\text{GHJ}&=\angle\text{GFJ}\\ \text{JG}&=\text{JG}\end{aligned}\\\end{minispace}}$

So, as per the concept of AAS congruence theorem the two triangles $\triangle\text{FJG}$ and $\triangle\text{JHG}$ are the congruent triangles.

Therefore, the theorem according to which the triangles $\triangle\text{FJG}$ and $\triangle\text{JHG}$ are congruent is $\fbox{\begin\\\ \bf {AAS congruence theorem}\\\end{minispace}}$.

Learn more:  

1. A problem to complete the square of quadratic function brainly.com/question/12992613

2. A problem to determine the slope intercept form of a line brainly.com/question/1473992

3. Inverse function brainly.com/question/1632445

Answer details  

Grade: Middle school  

Subject: Mathematics

Chapter: Congruency

Keywords: Congruency, congruent, congruent triangles, AAS, SSS, SAS, HL, angles, sides, quadrilateral, theorem, postulate, geometry, mathematics, hypotenuse.