Geometric Twins | FIO

Question 11

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.)

Solution

Understand the Question

Given that $AB \parallel CD$ and $AB = CD$ , with line segments $AC$ and $BD$ intersecting at $O$ to form two triangles: $\triangle OAB$ and $\triangle OCD$ .
When parallel lines are intersected by transversals ( $AC$ and $BD$ ), alternate interior angles are equal.
Using the ASA (Angle-Side-Angle) congruence criterion, we can prove that $\triangle OAB \cong \triangle OCD$ .
By CPCT (Corresponding Parts of Congruent Triangles), the corresponding sides and angles between the two triangles are equal.

Step 1 · Identify Equal Angles

Diagram 1

Given $AB \parallel CD$ :

With transversal $AC$ , alternate interior angles are equal: $\angle OAB = \angle OCD$
With transversal $BD$ , alternate interior angles are equal: $\angle OBA = \angle ODC$
Since lines $AC$ and $BD$ intersect at point $O$ , vertically opposite angles are equal: $\angle AOB = \angle DOC$

Step 2 · Prove Congruence of Triangles

In $\triangle OAB$ and $\triangle OCD$ :

$\angle OAB = \angle OCD$ (Alternate interior angles)
$AB = CD$ (Given)
$\angle OBA = \angle ODC$ (Alternate interior angles)

By the ASA (Angle-Side-Angle) congruence criterion: $\triangle OAB \cong \triangle OCD$

Step 3 · Find Remaining Equal Parts

Since $\triangle OAB \cong \triangle OCD$ , their corresponding parts (CPCT) are equal:

\begin{aligned} OA &= OC \\[0.6em] OB &= OD \end{aligned}

Answer

The two triangles are congruent: $\triangle OAB \cong \triangle OCD$

The other equal parts are:

Angles: $\angle OAB = \angle OCD$ , $\angle OBA = \angle ODC$ , and $\angle AOB = \angle DOC$
Sides: $OA = OC$ and $OB = OD$

Common Mistakes

Incorrect Vertex Correspondence: Writing $\triangle OAB \cong \triangle ODC$ instead of $\triangle OAB \cong \triangle OCD$ . Vertex $A$ corresponds to $C$ and vertex $B$ corresponds to $D$ due to alternate interior angles.
Assuming Extra Symmetries: Incorrectly concluding that $OA = OB$ or $AC = BD$ . The triangles are congruent to each other, but not necessarily isosceles.

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

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.