Question 23

Context: The symbol on this signboard needs to be recreated on another board.

Q. How do we do it?

Q. Can we take some measurements that would allow us to exactly recreate this figure? If yes, what measurements should we take?

Context: Let us name the corner points of this symbol as shown.

Q. Are the arm lengths $AB$ and $BC$ sufficient to exactly recreate this figure?

Context: Suppose the lengths of the arms of the symbol are $\text{AB} = 4\text{ cm}$ and $\text{BC} = 8\text{ cm}$. Several such symbols can be constructed with the same lengths.

Q. To get the exact replica, would it help to take any other measurement?

Can you draw the symbol if it is known that $AB = 4\text{ cm}$, $BC = 8\text{ cm}$, and $\angle ABC = 80^\circ$?

If it is known that both symbols have the same arm lengths, can it be concluded that the two symbols are congruent?

What do you think they can do?

Context: Meera says: "The angles of the triangle are not required! With the side lengths we have measured, we can create a triangle congruent to this one."

Q. Do you agree with Meera?

Instead of the lengths being $40\text{ cm}$, $60\text{ cm}$, and $80\text{ cm}$, suppose the sidelengths had been $4\text{ cm}$, $6\text{ cm}$, $8\text{ cm}$ (this triangle can fit on our page).

Context: Sidelengths of a triangle are $4\text{ cm}$, $6\text{ cm}$, and $8\text{ cm}$.

Q. Is this information sufficient to replicate the triangle with the same size and shape? If yes, can you do so?

Examine whether $\Delta \text{ABE}$ and $\Delta \text{ABF}$ are congruent.

How can these two triangles be superimposed? Which vertices of $\Delta XYZ$ and $\Delta ABC$ should we overlap? This has to be done so that the equal sides overlap. Figure out how.

Are there other ways of overlapping the vertices so that the triangles fit exactly over each other?

Can you identify a pair of congruent triangles below? Why are they congruent?

Consider $\Delta ABD$ and $\Delta CDB$. Since $ABCD$ is a rectangle, we have

If the remaining sides of $\Delta ABD$ and $\Delta CDB$ have the same length then the SSS condition is satisfied, confirming the congruence of the two triangles. Is this the case?

Verify this by superimposing paper cutouts of the triangles obtained from the rectangle $ABCD$ (Fig. 1.1).

Identify the correct correspondence of vertices and express the congruence between the two triangles.

Suppose the angles are $30^\circ$, $70^\circ$, and $80^\circ$. Can we create an exact copy of the frame with this?

$\Delta ABC$ and $\Delta XYZ$ are two triangles such that

$AB = XY = 6\text{ cm}$, $AC = XZ = 5\text{ cm}$, and $\angle A = \angle X = 30^\circ$

Are they congruent?

Context: $\Delta ABC$ and $\Delta XYZ$ are two triangles such that $AB = XY = 6\text{ cm}$, $AC = XZ = 5\text{ cm}$, and $\angle A = \angle X = 30^\circ$.

Q. Construct a triangle having the above measurements.

Compare it with the triangles constructed by your classmates. Are the triangles all congruent? Explain why all such triangles with these measurements are congruent.

$\Delta ABC$ and $\Delta XYZ$ are two triangles such that

$AB = XY = 6\text{ cm}$, $AC = XZ = 4\text{ cm}$, and $\angle B = \angle Y = 30^\circ$

Are they congruent?

Context: $\Delta ABC$ and $\Delta XYZ$ are two triangles such that $AB = XY = 6\text{ cm}$, $AC = XZ = 4\text{ cm}$, and $\angle B = \angle Y = 30^\circ$.

Q. Can there exist non-congruent triangles having these measurements? Construct and find out.

$\Delta \text{ABC}$ and $\Delta \text{XYZ}$ are two triangles with,

$\text{BC} = \text{YZ} = 5 \text{ cm}, \quad \angle \text{B} = \angle \text{Y} = 50^\circ \quad \text{and} \quad \angle \text{C} = \angle \text{Z} = 30^\circ$

Are they congruent?

In the figure, Point $O$ is the midpoint of $AD$ and $BC$. What can one say about the lengths $AB$ and $CD$?

The following triangles $\Delta ABC$ and $\Delta XYZ$ are such that $\angle A = \angle X = 35^\circ$, $\angle C = \angle Z = 75^\circ$, and $BC = YZ = 4\text{ cm}$. Are the triangles congruent? Give a reason.

$\Delta \text{ABC}$ and $\Delta \text{XYZ}$ are right-angled triangles such that $\text{BC} = \text{YZ} = 4\text{ cm}$, $\angle \text{B} = \angle \text{Y} = 90^\circ$ and $\text{AC} = \text{XZ} = 5\text{ cm}$. Are they congruent?

Context: $\Delta ABC$ and $\Delta XYZ$ are right-angled triangles such that $BC = YZ = 4\text{ cm}$, $\angle B = \angle Y = 90^\circ$, and $AC = XZ = 5\text{ cm}$.

Q. Can there exist non-congruent triangles having these measurements? Construct and find out.

Consider the downward extension of line $l$ below $\text{QR}$. Would the arc from $\text{R}$ meet this line downwards as well (as in the case of triangle construction when the sidelengths are given)? If so, would this lead to a triangle whose size and shape are different from $\Delta \text{PQR}$, and yet has the given measurements?

$\Delta ABC$ is isosceles with $AB = AC$, and $\angle A = 80^\circ$. What can we say about $\angle B$ and $\angle C$?

Construct the altitude from $A$ to $BC$.

Context: In $\Delta ABC$, $AB = AC$ and $\angle A = 80^\circ$.

Q. Can you use this fact to find $\angle B$ and $\angle C$?

Context: All the three angles of an equilateral triangle are equal.

Q. What could be their measures?

Context: In an equilateral triangle, each angle is $60^\circ$.

Q. Verify this by construction.

Describe the congruent triangles you see in each picture.