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 $\angle BAD$ and $\angle BCD$ into two equal parts? Give reasons.

Solution

Understand the Question

We are given a quadrilateral $ABCD$ where $AB = AD$ and $CB = CD$ , with diagonal $AC$ dividing it into two triangles: $\Delta ABC$ and $\Delta ADC$ .
To check for congruence, compare the three pairs of corresponding sides of both triangles.
Once congruence is proven using the SSS criterion, use CPCT (Corresponding Parts of Congruent Triangles) to determine whether $AC$ bisects $\angle BAD$ and $\angle BCD$ .

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

Step 1 · Prove Triangle Congruence using SSS Criterion

Diagram 1

In $\Delta ABC$ and $\Delta ADC$ :

\begin{aligned} AB &= AD \quad (\text{Given}) \\ CB &= CD \quad (\text{Given}) \\ AC &= AC \quad (\text{Common side}) \end{aligned}

By SSS (Side-Side-Side) congruence criterion: $\Delta ABC \cong \Delta ADC$

Answer

(i) Yes, $\Delta ABC \cong \Delta ADC$ by the SSS congruence criterion.

(ii) Does $AC$ divide $\angle BAD$ and $\angle BCD$ into two equal parts? Give reasons.

Step 1 · Check Angle Bisection using CPCT

Since $\Delta ABC \cong \Delta ADC$ , their corresponding angles are equal by $\text{CPCT}$ (Corresponding Parts of Congruent Triangles):

\begin{aligned} \angle BAC &= \angle DAC \\[0.6em] \angle BCA &= \angle DCA \end{aligned}

Therefore, $AC$ divides both $\angle BAD$ and $\angle BCD$ into two equal parts.

Answer

(ii) Yes, $AC$ divides $\angle BAD$ and $\angle BCD$ into two equal parts because $\angle BAC = \angle DAC$ and $\angle BCA = \angle DCA$ (by $\text{CPCT}$ ).

Common Mistakes

Missing the Common Side: Forgetting to mention that $AC = AC$ is the shared common side needed to complete the three side pairs for the SSS criterion.
Incorrect Vertex Correspondence: Writing the congruence statement incorrectly as $\Delta ABC \cong \Delta CDA$ instead of $\Delta ABC \cong \Delta ADC$ , since vertex $B$ corresponds to $D$ .
Omitting the CPCT Reason: Stating that the angles are equal without citing $\text{CPCT}$ following the congruence proof.

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 $\angle BAD$ and $\angle BCD$ into two equal parts? Give reasons.

In the figure below, are $\triangle DFE$ and $\triangle GED$ 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

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