Geometric Twins | FIO

Question 16

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?]

Question diagram 1
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Solution

We will show that two triangles are congruent, then use their equal angles to prove that the lines are parallel.

Step 1 — Finding equal parts

We are given some information about the lengths of line segments. We are given that OB = OC. We are also given that OA = OD. Look at the point where the lines AD and BC cross. This point is O. When two straight lines cross each other, the angles opposite each other are called vertically opposite angles. Angle AOB and angle DOC are vertically opposite angles. Vertically opposite angles are always equal. So, ∠AOB = ∠DOC.

Diagram 1

Step 2 — Proving triangle congruence

Let us look at triangle AOB and triangle DOC. We have side OA equal to side OD. We have angle AOB equal to angle DOC. We have side OB equal to side OC. This matches the Side-Angle-Side (SAS) congruence rule. The SAS rule says if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. So, triangle AOB is congruent to triangle DOC.

△AOB≅△DOC\boxed{\triangle AOB \cong \triangle DOC}

Step 3 — Identifying equal alternate angles

Since triangle AOB is congruent to triangle DOC, their corresponding parts are equal. This is called CPCTC (Corresponding Parts of Congruent Triangles are Congruent). The angle ∠OAB in triangle AOB corresponds to angle ∠ODC in triangle DOC. So, ∠OAB = ∠ODC. These angles are on opposite sides of the transversal line AD. They are also between the lines AB and CD. This means they are alternate interior angles.

Step 4 — Concluding parallelism

We have found that the alternate interior angles ∠OAB and ∠ODC are equal. When a transversal line cuts two other lines, and the alternate interior angles are equal, then the two lines must be parallel. Here, AD is the transversal line. AB and CD are the two lines. Since ∠OAB = ∠ODC, the line AB is parallel to the line CD.

Answer

AB is parallel to CD.

More questions in FIO

Q1

Check if the two figures are congruent.

Q2

Circle the pairs that appear congruent.

Q3

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

(a) Circle

(b) Rectangle

Q4

Using this, state how would you check if two —

(a) Circles are congruent?

(b) Rectangles are congruent?

Q5

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.

Q6

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

Q7

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

Q8

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.

Q9

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

Q13

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

Q14

ΔAIR≅ΔFLY\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, ∠\angleA = ∠\angleE, AC = ED

(c) AB = DF, ∠\angleB = ∠\angleD = 90°, AC = FE

(d) ∠\angleA = ∠\angleD, ∠\angleB = ∠\angleE, AC = DF

(e) AB = DF, ∠\angleB = ∠\angleF, 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 ΔABC≅ΔADC\Delta\text{ABC} \cong \Delta\text{ADC}. Is ΔABC\Delta\text{ABC} also congruent to ΔCDA\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 ∠B\angle\text{B} and ∠C\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.

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