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

We will check if the triangles meet the conditions for the RHS congruence rule. Step 1 — Identify equal parts Let us look at the given information. We have two triangles. They are ABC and XYZ. Both triangles are right-angled. This means they each have a 90-degree angle. In ABC, B is 90°. In XYZ, Y is 90°. The side opposite the right angle is called the hypotenuse. In ABC, the hypotenuse AC is 5 cm. In XYZ, the hypotenuse XZ is 5 cm. We are also given another pair of sides. In ABC, side BC is 4 cm. In XYZ, side YZ is 4 cm. Step 2 — Apply RHS congruence rule Let us check the conditions for RHS congruence. RHS stands for Right angle, Hypotenuse, Side. This rule applies to right-angled triangles. First, we need one right angle in each triangle. We have B = Y. Second, we need the hypotenuses to be equal. We have AC = XZ. Third, we need one other corresponding side to be equal. We have BC = YZ. All three conditions are met. So, the triangles are congruent by the RHS rule. Answer > (i) Yes, ABC and XYZ are congruent by the RHS congruence rule.

Geometric Twins | IT

Question 26

$\Delta$ 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?

Solution

We will check if the triangles meet the conditions for the RHS congruence rule.

Step 1 — Identify equal parts

Let us look at the given information. We have two triangles. They are $\Delta$ ABC and $\Delta$ XYZ. Both triangles are right-angled. This means they each have a 90-degree angle. In $\Delta$ ABC, $\angle$ B is 90°. In $\Delta$ XYZ, $\angle$ Y is 90°. The side opposite the right angle is called the hypotenuse. In $\Delta$ ABC, the hypotenuse AC is 5 cm. In $\Delta$ XYZ, the hypotenuse XZ is 5 cm. We are also given another pair of sides. In $\Delta$ ABC, side BC is 4 cm. In $\Delta$ XYZ, side YZ is 4 cm.

Diagram 1

Step 2 — Apply RHS congruence rule

Let us check the conditions for RHS congruence. RHS stands for Right angle, Hypotenuse, Side. This rule applies to right-angled triangles. First, we need one right angle in each triangle. We have $\angle$ B = $\angle$ Y. $\angle\text{B} = \angle\text{Y} = \mathbf{90°}$ Second, we need the hypotenuses to be equal. We have AC = XZ. $\text{AC} = \text{XZ} = \mathbf{5 \text{ cm}}$ Third, we need one other corresponding side to be equal. We have BC = YZ. $\text{BC} = \text{YZ} = \mathbf{4 \text{ cm}}$ All three conditions are met. So, the triangles are congruent by the RHS rule.

Answer

(i) Yes, $\Delta$ ABC and $\Delta$ XYZ are congruent by the RHS congruence rule.

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