Geometric Twins | IT

Question 23

Context: $\Delta\text{ABC}$ and $\Delta\text{XYZ}$ are two triangles with $\text{BC} = \text{YZ} = 5\text{ cm}, \angle\text{B} = \angle\text{Y} = 50^\circ \text{ and } \angle\text{C} = \angle\text{Z} = 30^\circ$ .

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

Solution

We need to check if the given measurements satisfy a congruence rule.

Step 1 — Identify Given Information

We are given two triangles.

They are $\Delta\text{ABC}$ and $\Delta\text{XYZ}$ .

Let us list the measurements for $\Delta\text{ABC}$ .

Side $\text{BC}$ is $\mathbf{5\text{ cm}}$ .

Angle $\angle\text{B}$ is $\mathbf{50^\circ}$ .

Angle $\angle\text{C}$ is $\mathbf{30^\circ}$ .

Let us list the measurements for $\Delta\text{XYZ}$ .

Side $\text{YZ}$ is $\mathbf{5\text{ cm}}$ .

Angle $\angle\text{Y}$ is $\mathbf{50^\circ}$ .

Angle $\angle\text{Z}$ is $\mathbf{30^\circ}$ .

Diagram 1

Step 2 — Compare Corresponding Parts

We compare the parts of $\Delta\text{ABC}$ and $\Delta\text{XYZ}$ .

Side $\text{BC}$ in $\Delta\text{ABC}$ matches side $\text{YZ}$ in $\Delta\text{XYZ}$ .

$\text{BC} = \text{YZ}$ .

Both measure $\mathbf{5\text{ cm}}$ .

Angle $\angle\text{B}$ in $\Delta\text{ABC}$ matches angle $\angle\text{Y}$ in $\Delta\text{XYZ}$ .

$\angle\text{B} = \angle\text{Y}$ .

Both measure $\mathbf{50^\circ}$ .

Angle $\angle\text{C}$ in $\Delta\text{ABC}$ matches angle $\angle\text{Z}$ in $\Delta\text{XYZ}$ .

$\angle\text{C} = \angle\text{Z}$ .

Both measure $\mathbf{30^\circ}$ .

Step 3 — Apply Congruence Rule

We have two angles and the side between them.

This is called the included side.

The side $\text{BC}$ is included between $\angle\text{B}$ and $\angle\text{C}$ .

The side $\text{YZ}$ is included between $\angle\text{Y}$ and $\angle\text{Z}$ .

This situation perfectly fits the ASA congruence rule.

ASA stands for Angle-Side-Angle.

The ASA rule says if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

Step 4 — Formulate Conclusion

Our triangles $\Delta\text{ABC}$ and $\Delta\text{XYZ}$ satisfy the ASA rule.

This means they are congruent.

Congruent triangles are identical in shape and size.

They are exact copies of each other.

So, it is not possible to have non-congruent triangles.

All triangles made with these measurements will be the same.

Answer

(i) No, non-congruent triangles cannot exist with these measurements. (ii) The triangles are congruent by the ASA (Angle-Side-Angle) rule.

More questions in IT

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Q10

Context: Sidelengths of a triangle are 4 cm, 6 cm, and 8 cm.

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Q11

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Q12

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Q13

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

Q14

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

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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?

Q15

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Q16

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Q17

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Q18

ΔABC and ΔXYZ are two triangles such that

AB = XY = 6 cm, AC = XZ = 5 cm, and ∠A = ∠X = 30°

Are they congruent?

Q19

Context: ΔABC and ΔXYZ are two triangles such that AB = XY = 6 cm, AC = XZ = 5 cm, and ∠A = ∠X = 30°.

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.

Q20

ΔABC and ΔXYZ are two triangles such that

AB = XY = 6 cm, AC = XZ = 4 cm, and ∠B = ∠Y = 30°

Are they congruent?

Q21

Context: ΔABC and ΔXYZ are two triangles such that AB = XY = 6 cm, AC = XZ = 4 cm, and ∠B = ∠Y = 30°.

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

Q22

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

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

Are they congruent?

Q23

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

Q24

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

Q25

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.

Q26

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

Q27

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

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Q28

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Q29

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

Construct the altitude from A to BC.

Q30

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Q31

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

Q. What could be their measures?

Q32

Context: In an equilateral triangle, each angle is 60°.

Q. Verify this by construction.

Q33

Describe the congruent triangles you see in each picture.