Given that and , show that . Are the two triangles congruent?

Geometric Twins | FIO

Question 12

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

Solution

We can show the triangles are identical by matching their parts.

Step 1 — List What We Know

Let us look at the two triangles, $\triangle ABC$ and $\triangle DBC$ . We are told that $\angle ABC$ is equal to $\angle DBC$ . Let us call this angle 'a'. We are also told that $\angle ACB$ is equal to $\angle DCB$ . Let us call this angle 'b'. The side BC is shared by both triangles. So, BC is a common side.

Step 2 — Check for Congruence

We have an angle, then a side, then another angle. This pattern is Angle-Side-Angle, or ASA. The side BC is between the two angles we know. So, $\triangle ABC$ is congruent to $\triangle DBC$ .

$\boxed{\triangle ABC \cong \triangle DBC \text{ by ASA}}$

Diagram 1

Step 3 — Find the Equal Angle

When two triangles are congruent, all their matching parts are equal. This means their corresponding angles are equal. The angle $\angle BAC$ in $\triangle ABC$ matches $\angle BDC$ in $\triangle DBC$ . So, these two angles must be equal.

Answer

(i) Yes, the two triangles are congruent. (ii) $\angle BAC = \angle BDC$ .

More questions in FIO

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(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.

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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.