Question

8
the triangles shown below must be congruent.

a. true
b. false

Explanation:

Step1: Identify triangle properties

Both are right triangles (one right angle), have a \(40^\circ\) angle, and a side of length 12. For the first triangle, the side of length 12 is opposite the \(40^\circ\) angle? Wait, no—wait, in a right triangle, angles sum to \(180^\circ\), so right angle (\(90^\circ\)), \(40^\circ\), so the third angle is \(50^\circ\). Now, check the side: in the first triangle, the side labeled 12 is adjacent to the \(40^\circ\) angle? Wait, no, looking at the diagrams: first triangle, right angle at top, \(40^\circ\) at bottom right, base 12 (opposite the right angle? No, base is the side between the two acute angles? Wait, no, in a right triangle, the hypotenuse is opposite the right angle. Wait, maybe using AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle). Let's see: both triangles have a right angle (\(90^\circ\)), a \(40^\circ\) angle, and a side of length 12. Let's check the side: in the first triangle, the side of length 12 is the side opposite the \(40^\circ\) angle? Wait, no—wait, in the first triangle, the \(40^\circ\) angle is at the bottom right, so the side adjacent to \(40^\circ\) is the base? Wait, maybe better: both triangles have two angles (right angle and \(40^\circ\)) and a side. Let's confirm the side: in the first triangle, the side of length 12 is between the right angle and the \(40^\circ\) angle? No, wait, the first triangle: right angle at top, \(40^\circ\) at bottom right, base 12 (so the side opposite the right angle? No, the right angle is at top, so the base is the hypotenuse? Wait, no, hypotenuse is opposite right angle. Wait, maybe I'm overcomplicating. Let's use ASA: angle, side, angle. Both triangles have a right angle (\(90^\circ\)), a \(40^\circ\) angle, and the side of length 12 is between these two angles (since in the first triangle, the side 12 is adjacent to the \(40^\circ\) angle and the right angle? Wait, no, the first triangle: right angle at top, \(40^\circ\) at bottom right, so the side between the right angle and \(40^\circ\) angle is the leg, and the base is 12. Wait, maybe the side of length 12 is the side opposite the \(40^\circ\) angle? No, let's calculate the sides. In a right triangle, \(\sin(40^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos(40^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\). But maybe using AAS: two angles and a non-included side. Wait, both triangles have angles \(90^\circ\), \(40^\circ\), and a side of length 12. Let's check the side: in the first triangle, the side of length 12 is the side opposite the \(40^\circ\) angle? Wait, no—wait, the first triangle: \(40^\circ\) at bottom right, so the side opposite \(40^\circ\) is the left leg (from top to bottom left). The base is 12, which is the side adjacent to \(40^\circ\). In the second triangle, \(40^\circ\) at bottom left, base 12 (adjacent to \(40^\circ\)). Wait, but both triangles have a right angle, a \(40^\circ\) angle, and a side of length 12 (which is the side adjacent to \(40^\circ\) and the right angle? No, maybe the side of length 12 is the hypotenuse? Wait, no, hypotenuse is opposite right angle. Wait, maybe the side of length 12 is the leg. Let's use ASA: angle (right angle), side (12), angle (40°). Wait, in the first triangle, the side of length 12 is between the right angle and the 40° angle? Let's see the diagram: first triangle, right angle at top, 40° at bottom right, base 12 (so the side from bottom left to bottom right is 12, which is adjacent to the 40° angle and the right angle? No, the right angle is at top, so the sides are: top to bottom left, top to bottom right (legs), and bottom left to bottom right (hypotenuse). Wait, so hypotenuse is 12, with a 40° angle at bottom right. Then the second triangle: right angle at top, 40° at bottom left, hypotenuse 12. Wait, but in that case, the triangles are congruent by AAS (two angles and a side). Wait, no—wait, in the first triangle, the 40° angle is at bottom right, hypotenuse 12. In the second triangle, 40° angle at bottom left, hypotenuse 12. But since it's a right triangle, the other angle is 50°, so both triangles have angles 90°, 40°, 50°, and hypotenuse 12. So by AAS (angle-angle-side: 90°, 40°, hypotenuse 12), they are congruent. Wait, but maybe the side is a leg. Wait, maybe I made a mistake. Wait, let's check the side: in the first triangle, the side of length 12 is the side opposite the 40° angle? No, if the 40° angle is at bottom right, then the side opposite is the leg from top to bottom left. But in the second triangle, the 40° angle is at bottom left, so the side opposite is the leg from top to bottom right. But the hypotenuse is 12 in both? Wait, maybe the side of length 12 is the leg. Wait, no, in a right triangle, the hypotenuse is the longest side. If the side is 12, and it's a leg, then the hypotenuse would be longer. But maybe the diagram shows that the side of length 12 is the leg. Wait, maybe the correct approach is: both triangles are right-angled, have a 40° angle, and a side of length 12. Let's use ASA: angle (right angle), side (12), angle (40°). Wait, in the first triangle, the side of length 12 is between the right angle and the 40° angle? Let's see the first triangle: right angle at top, 40° at bottom right, so the side between right angle and 40° angle is the leg (from top to bottom right), and the base is 12 (from bottom left to bottom right). Wait, no, the base is 12, so that's the side between the two acute angles (bottom left and bottom right). So in the first triangle, the side of length 12 is adjacent to the 40° angle (bottom right) and the other acute angle (bottom left, which is 50°). In the second triangle, the side of length 12 is adjacent to the 40° angle (bottom left) and the other acute angle (bottom right, 50°). So both triangles have: angle (40°), side (12), angle (50°) (since 180 - 90 - 40 = 50). So by ASA (angle-side-angle: 40°, 50°, side 12), they are congruent. Therefore, the statement is True.

Step2: Confirm congruence

Since both triangles have two angles (90° and 40°) and the included side (12) equal (or AAS, as angles are 90°, 40°, and side 12), they must be congruent. So the answer is True.

Answer:

A. True