Geometric Twins | FIO

Question 14

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

Solution

Understand the Question

When two triangles are congruent (written as $\Delta \text{AIR} \cong \Delta \text{FLY}$ ), their corresponding vertices, sides, and angles match exactly in the given order.
The order of letters in the notation determines the correspondences:
- 1st vertex $\leftrightarrow$ 1st vertex
- 2nd vertex $\leftrightarrow$ 2nd vertex
- 3rd vertex $\leftrightarrow$ 3rd vertex

Step 1 · Identify Corresponding Vertices

From the congruence statement $\Delta \text{AIR} \cong \Delta \text{FLY}$ , match the vertices by their positions: Diagram 1

Vertex $A \leftrightarrow$ Vertex $F$
Vertex $I \leftrightarrow$ Vertex $L$
Vertex $R \leftrightarrow$ Vertex $Y$

Step 2 · Identify Corresponding Sides

Using the matching vertices, the corresponding sides are:

$\text{AI} \leftrightarrow \text{FL}$
$\text{IR} \leftrightarrow \text{LY}$
$\text{AR} \leftrightarrow \text{FY}$

Step 3 · Identify Corresponding Angles

Using the matching vertices, the corresponding angles are:

$\angle A \leftrightarrow \angle F$
$\angle I \leftrightarrow \angle L$
$\angle R \leftrightarrow \angle Y$

Answer

Corresponding Vertices: $A \leftrightarrow F$ , $I \leftrightarrow L$ , $R \leftrightarrow Y$

Corresponding Sides: $\text{AI} \leftrightarrow \text{FL}$ , $\text{IR} \leftrightarrow \text{LY}$ , $\text{AR} \leftrightarrow \text{FY}$

Corresponding Angles: $\angle A \leftrightarrow \angle F$ , $\angle I \leftrightarrow \angle L$ , $\angle R \leftrightarrow \angle Y$

Common Mistakes

Ignoring Vertex Order: Assuming any vertex, side, or angle can correspond arbitrarily. In congruent triangles, the correspondence is strictly defined by the order of the letters in the statement $\Delta \text{AIR} \cong \Delta \text{FLY}$ .
Incorrect Side Pairing: Naming matching sides out of vertex order (e.g., writing $\text{AI} \leftrightarrow \text{LF}$ instead of $\text{AI} \leftrightarrow \text{FL}$ ).

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.