Geometric Twins | FIO

Question 15

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

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Solution

We will check each pair of triangles to see if they are congruent. We will use our known rules for congruence.

Step 1 — Case (a): SSS Congruence

We are given three pairs of equal sides. The side AB in the first triangle is equal to the side DE in the second triangle. The side BC in the first triangle is equal to the side EF in the second triangle. The side CA in the first triangle is equal to the side DF in the second triangle. All three corresponding sides are equal. This matches the Side-Side-Side (SSS) congruence rule. So, the triangles are congruent.

triangle ABC ≅ triangle DEF\boxed{\text{triangle ABC } \cong \text{ triangle DEF}}

Diagram 1

Step 2 — Case (b): SAS Congruence

We are given two pairs of equal sides and one pair of equal angles. The side AB is equal to the side EF. The angle A is equal to the angle E. The side AC is equal to the side ED. The angle A is between sides AB and AC. The angle E is between sides EF and ED. So, the equal angle is the included angle for both triangles. This matches the Side-Angle-Side (SAS) congruence rule. So, the triangles are congruent.

triangle ABC ≅ triangle EFD\boxed{\text{triangle ABC } \cong \text{ triangle EFD}}

Diagram 2

Step 3 — Case (c): RHS Congruence

We are given an angle, a side, and another side. The angle B is 90 degrees. The angle D is also 90 degrees. So, both triangles are right-angled triangles. The side AC is the hypotenuse of triangle ABC. The side FE is the hypotenuse of triangle FDE. We are given that AC is equal to FE. The side AB is equal to the side FD. This matches the Right-angle-Hypotenuse-Side (RHS) congruence rule. So, the triangles are congruent.

triangle ABC ≅ triangle FDE\boxed{\text{triangle ABC } \cong \text{ triangle FDE}}

Diagram 3

Step 4 — Case (d): AAS Congruence

We are given two pairs of equal angles and one pair of equal sides. The angle A is equal to the angle D. The angle B is equal to the angle E. The side AC is equal to the side DF. The side AC is not between angle A and angle B. The side DF is not between angle D and angle E. This means the equal side is a non-included side. This matches the Angle-Angle-Side (AAS) congruence rule. So, the triangles are congruent.

triangle ABC ≅ triangle DEF\boxed{\text{triangle ABC } \cong \text{ triangle DEF}}

Diagram 4

Step 5 — Case (e): SSA Condition

We are given two pairs of equal sides and one pair of equal angles. The side AB is equal to the side DF. The angle B is equal to the angle F. The side AC is equal to the side DE. The angle B is not between sides AB and AC. The angle F is not between sides DF and DE. This means the equal angle is a non-included angle. This is called the Side-Side-Angle (SSA) condition. The SSA condition is not a valid rule for congruence. So, the triangles are not necessarily congruent.

triangle ABC need not be congruent to triangle DFE\boxed{\text{triangle ABC need not be congruent to triangle DFE}}

Diagram 5

Answer

(a) Here, AB = DE, BC = EF, CA = FD. All three corresponding sides are equal. Thus, triangles are congruent by the side-side-side (SSS) congruence criterion. Hence, triangle ABC ≅ triangle DEF. (b) Given AB = EF, ∠A = ∠E, AC = ED. Two corresponding sides and the included angle are equal. Thus, triangles satisfy the SAS condition. Hence, triangle ABC ≅ triangle EFD. (c) Here, AB = FD, ∠B = ∠D = 90°, AC = FE. The triangles have equal right angles, equal hypotenuses, and one equal corresponding side. Thus, triangles satisfy the RHS conditions. Hence, triangle ABC ≅ triangle FDE. (d) Here, ∠A = ∠D, ∠B = ∠E, AC = DF. Clearly, two corresponding angles and one corresponding side are equal. Thus, triangles satisfy the AAS conditions. Hence, triangle ABC ≅ triangle DEF. (e) Here, AB = DF, ∠B = ∠F, AC = DE. Here, two corresponding sides and a non-included angle are equal. Thus, the triangles satisfy the SSA condition, which is not a valid congruence rule. Hence, triangle ABC need not be congruent to triangle DFE.

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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