Geometric Twins | FIO

Question 17

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

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?

Solution

Congruent shapes are identical in both size and form.

Step 1 — Proving $\Delta\text{ABC} \cong \Delta\text{ADC}$

We have a square named ABCD. All sides of a square are equal. So, side AB is equal to side AD. Also, side BC is equal to side DC. Side AC is a common side for both triangles. It is present in $\Delta\text{ABC}$ and $\Delta\text{ADC}$ . We use the SSS congruence rule. This rule states three sides must be equal. One triangle's sides must match the other's. So, $\Delta\text{ABC}$ is congruent to $\Delta\text{ADC}$ .

Diagram 1

Step 2 — Proving $\Delta\text{ABC} \cong \Delta\text{CDA}$

We now compare $\Delta\text{ABC}$ with $\Delta\text{CDA}$ . Side AB of $\Delta\text{ABC}$ matches side CD. In a square, AB is equal to CD. Side BC of $\Delta\text{ABC}$ matches side DA. In a square, BC is equal to DA. Side AC of $\Delta\text{ABC}$ matches side CA. This is the common diagonal. We use the SSS congruence rule again. So, $\Delta\text{ABC}$ is congruent to $\Delta\text{CDA}$ .

Step 3 — Triangles congruent in six ways

We want triangles congruent in six ways. Let us consider two equilateral triangles. Let us name them $\Delta\text{HEN}$ and $\Delta\text{BIG}$ . All sides of an equilateral triangle are equal. All angles of an equilateral triangle are equal. This allows many ways to match their vertices. There are six different ways to write their congruence. We list these six ways below.

Answer

(i) $\Delta\text{ABC}$ is congruent to $\Delta\text{ADC}$ . This is because AB = AD, BC = DC, and AC = AC (common side). We use the SSS congruence rule. (ii) Yes, $\Delta\text{ABC}$ is also congruent to $\Delta\text{CDA}$ . This is because AB = CD, BC = DA, and AC = CA (common side). We use the SSS congruence rule. (iii) Let us take two congruent equilateral triangles, $\Delta\text{HEN}$ and $\Delta\text{BIG}$ . All sides and angles are equal in equilateral triangles. So, there are six ways to write their congruence: (i) $\Delta\text{HEN} \cong \Delta\text{BIG}$ (ii) $\Delta\text{HNE} \cong \Delta\text{BGI}$ (iii) $\Delta\text{EHN} \cong \Delta\text{IBG}$ (iv) $\Delta\text{ENH} \cong \Delta\text{IGB}$ (v) $\Delta\text{NHE} \cong \Delta\text{GBI}$ (vi) $\Delta\text{NEH} \cong \Delta\text{GIB}$

More questions in FIO

Check if the two figures are congruent.

Circle the pairs that appear congruent.

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

(a) Circle

(b) Rectangle

Using this, state how would you check if two —

(a) Circles are congruent?

(b) Rectangles are congruent?

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.

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

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

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.

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

Q13

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

Q14

$\Delta\text{AIR} \cong \Delta\text{FLY}$ . 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, $\angle$ A = $\angle$ E, AC = ED

(d) $\angle$ A = $\angle$ D, $\angle$ B = $\angle$ E, AC = DF

(e) AB = DF, $\angle$ B = $\angle$ F, 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 $\Delta\text{ABC} \cong \Delta\text{ADC}$ . Is $\Delta\text{ABC}$ also congruent to $\Delta\text{CDA}$ ?

Q18

Find $\angle\text{B}$ and $\angle\text{C}$ , 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.