Geometric Twins | FIO

Question 8

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.

Solution

We can show two triangles are identical if their sides match up perfectly.

Step 1 — Identify equal sides

We look at the two triangles in the figure. These are triangle ABC and triangle ADC. The problem tells us some sides are equal. Side AB is equal to side AD. Side CB is equal to side CD. The line AC is part of both triangles. So, side AC is equal to side AC.

$AB = AD$

$CB = CD$

$AC = AC$

Step 2 — Apply congruence rule

We have found three pairs of equal sides. One side from triangle ABC matches one side from triangle ADC. This is true for all three sides. This means the triangles are congruent. We use the Side-Side-Side (SSS) rule. So, triangle ABC is congruent to triangle ADC.

$\boxed{\text{Triangle ABC } \cong \text{ Triangle ADC}}$

Diagram 1

Step 3 — Check angle bisection

We know the two triangles are congruent. This means all their matching parts are equal. We call this Corresponding Parts of Congruent Triangles (CPCT). Angle BAC is a part of triangle ABC. Angle DAC is a part of triangle ADC. These angles are matching parts. So, angle BAC is equal to angle DAC. This means AC cuts angle BAD into two equal parts. Angle BCA is a part of triangle ABC. Angle DCA is a part of triangle ADC. These angles are also matching parts. So, angle BCA is equal to angle DCA. This means AC cuts angle BCD into two equal parts.

$\angle BAC = \angle DAC$

$\angle BCA = \angle DCA$

Answer

(i) The pair of congruent triangles is triangle ABC and triangle ADC. They are congruent by the Side-Side-Side (SSS) congruence criterion because AB = AD, CB = CD, and AC = AC (common side). (ii) Yes, AC divides ∠BAD into two equal parts. (iii) Yes, AC divides ∠BCD into two equal parts.

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 $\Delta\text{CDA}$ ?

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 $\angle\text{B}$ 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.