Geometric Twins | FIO

Question 9

In the figure below, are ΔDFE and ΔGED congruent to each other? It is given that DF = DG and FE = GE.

Question diagram 1
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Solution

We need to check if triangle DFE and triangle GED are congruent.

Step 1 — Identify common and equal sides

Let us look at the diagram and the given information. We are told that side DF is equal to side DG. This is shown by the single tick marks on them. We are also told that side FE is equal to side GE. This is shown by the double tick marks on them. The side DE is a common side for both triangle DFE and triangle DGE. So, side DE in triangle DFE is equal to side DE in triangle DGE.

Step 2 — Check congruence of △DFE\triangle DFE and △DGE\triangle DGE

Let us compare triangle DFE and triangle DGE. We have found three pairs of equal sides: Side DF = Side DG (This was given to us) Side FE = Side GE (This was given to us) Side DE = Side DE (This is a common side) So, by the SSS (Side-Side-Side) congruence rule, triangle DFE is congruent to triangle DGE. This means that triangle DFE and triangle DGE have the exact same size and shape.

△DFE≅△DGE\triangle DFE \cong \triangle DGE

Diagram 1

Step 3 — Check the specific naming △DFE\triangle DFE and △GED\triangle GED

The question asks if triangle DFE and triangle GED are congruent. When we write a congruence statement, the order of the letters (vertices) is very important. It tells us which parts of one triangle match up with the parts of the other. Let us check if the statement △DFE≅△GED\triangle DFE \cong \triangle GED is correct. For this statement to be true, the matching sides must be equal. Side DF from △DFE\triangle DFE must be equal to side GE from △GED\triangle GED. Side FE from △DFE\triangle DFE must be equal to side ED from △GED\triangle GED. Side ED from △DFE\triangle DFE must be equal to side DG from △GED\triangle GED.

Let us check if these conditions are met by our given information:

  1. Is DF = GE? We know DF = DG and FE = GE. So, this would mean DG = FE. This is not given to us.
  2. Is FE = ED? This is not given to us.
  3. Is ED = DG? This is not given to us.

Since these matching sides are not necessarily equal, the given information does not support the congruence statement △DFE≅△GED\triangle DFE \cong \triangle GED. This means that if we try to match the triangles in the order DFE and GED, they are not congruent.

Answer

(i) The triangles DFE and DGE are congruent by the SSS rule. (ii) The given statements do not support the congruence of triangle DFE and triangle GED because the corresponding sides are not equal when matched in that specific order.

More questions in FIO

Q1

Check if the two figures are congruent.

Q2

Circle the pairs that appear congruent.

Q3

What measurements would you take to create a figure congruent to a given:

(a) Circle

(b) Rectangle

Q4

Using this, state how would you check if two —

(a) Circles are congruent?

(b) Rectangles are congruent?

Q5

How would we check if two figures like the one below are congruent?

Use this to identify whether each of the following pairs are congruent.

Q6

Suppose ΔHEN\Delta\text{HEN} is congruent to ΔBIG\Delta\text{BIG}. List all the other correct ways of expressing this congruence.

Q7

Determine whether the triangles are congruent. If yes, express the congruence.

Q8

In the figure below, AB = AD, CB = CD.

Can you identify any pair of congruent triangles? If yes, explain why they are congruent.

Does AC divide ∠BAD and ∠BCD into two equal parts? Give reasons.

Q9

In the figure below, are ΔDFE and ΔGED congruent to each other? It is given that DF = DG and FE = GE.

Q10

Identify whether the triangles below are congruent. What conditions did you use to establish their congruence? Express the congruence.

Q11

Given that CD and AB are parallel, and AB = CD, what are the other equal parts in this figure? (Hint: When the lines are parallel, the alternate angles are equal. Are the two resulting triangles congruent? If so, express the congruence.)

Q12

Given that ∠ABC=∠DBC\angle ABC = \angle DBC and ∠ACB=∠DCB\angle ACB = \angle DCB, show that ∠BAC=∠BDC\angle BAC = \angle BDC. Are the two triangles congruent?

Q13

Identify the equal parts in the following figure, given that ∠ABD=∠DCA\angle ABD = \angle DCA and ∠ACB=∠DBC\angle ACB = \angle DBC.

Q14

ΔAIR≅ΔFLY\Delta\text{AIR} \cong \Delta\text{FLY}. Identify the corresponding vertices, sides and angles.

Q15

Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent.

(a) AB = DE, BC = EF, CA = DF

(b) AB = EF, ∠\angleA = ∠\angleE, AC = ED

(c) AB = DF, ∠\angleB = ∠\angleD = 90°, AC = FE

(d) ∠\angleA = ∠\angleD, ∠\angleB = ∠\angleE, AC = DF

(e) AB = DF, ∠\angleB = ∠\angleF, AC = DE

Q16

It is given that OB = OC, and OA = OD. Show that AB is parallel to CD.
[Hint: AD is a transversal for these two lines. Are there any equal alternate angles?]

Q17

ABCD is a square. Show that ΔABC≅ΔADC\Delta\text{ABC} \cong \Delta\text{ADC}. Is ΔABC\Delta\text{ABC} also congruent to ΔCDA\Delta\text{CDA}?

Give more examples of two triangles where one triangle is congruent to the other in two different ways, as in the case above. Can you give an example of two triangles where one is congruent to the other in six different ways?

Q18

Find ∠B\angle\text{B} and ∠C\angle\text{C}, if A is the centre of the circle.

Q19

Find the missing angles. As per the convention that we have been following, all line segments marked with a single '|' are equal to each other and those marked with a double '|' are equal to each other, etc.

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