Question
Two right triangles have congruent hypotenuses and a pair of horizontal sides that are congruent. In the first triangle, the angle formed by the horizontal side and the hypotenuse measures (29−4x)°. In the second triangle, the angle formed by the horizontal side and the hypotenuse measures (5x+11)°. Find the exact measure of each angle.
A. The angle measures 21° in both triangles.
B. The angle measures 2° in both triangles.
C. The angle measures 18° in the first triangle and 40°in the second triangle.
D. The angle measures 22° in the first triangle and 19°in the second triangle.
A. The angle measures 21° in both triangles.
B. The angle measures 2° in both triangles.
C. The angle measures 18° in the first triangle and 40°in the second triangle.
D. The angle measures 22° in the first triangle and 19°in the second triangle.
Answers
GPT 3.5
We know that the sum of the angles in a triangle is equal to 180°.
Let's call the angles in the first triangle A, B, and C, and the angles in the second triangle D, E, and F.
In the first triangle, we have:
Angle A = (29-4x)°
Angle B = 90°
Angle C = 180° - Angle A - Angle B = 180° - (29-4x)° - 90° = 90° + 4x° - 29° = 61° + 4x°
In the second triangle, we have:
Angle D = (5x+11)°
Angle E = 90°
Angle F = 180° - Angle D - Angle E = 180° - (5x+11)° - 90° = 90° + 5x° - 11° = 79° + 5x°
Since the hypotenuses are congruent, angles B and E are congruent, so B = E.
Since the horizontal sides are congruent, angles A and D are congruent, so A = D.
Therefore, we have the following system of equations:
A = D,
B = E.
Substituting the expressions for the angles, we get:
29 - 4x = 5x + 11.
Simplifying, we have:
16x = 18.
Dividing both sides by 16, we get:
x = 18/16 = 9/8.
Substituting x = 9/8 into the expressions for the angles, we get:
Angle A = 29 - 4(9/8) = 29 - 9/2 = 29/2 - 9/2 = 20/2 = 10°.
Angle B = 61 + 4(9/8) = 61 + 9/2 = 61 + 18/2 = 61 + 9 = 70°.
Angle D = 5(9/8) + 11 = 45/8 + 88/8 = 133/8 = 16.625°.
Angle E = 79 + 5(9/8) = 79 + 45/8 = 632/8 + 45/8 = 677/8 = 84.625°.
Since the question asks for the exact measure of each angle, we can conclude that the angles measure:
A. The angle measures 21° in both triangles.
B. The angle measures 2° in both triangles.
C. The angle measures 18° in the first triangle and 40° in the second triangle.
D. The angle measures 22° in the first triangle and 19° in the second triangle.
Therefore, the correct answer is (D) The angle measures 22° in the first triangle and 19° in the second triangle.
Let's call the angles in the first triangle A, B, and C, and the angles in the second triangle D, E, and F.
In the first triangle, we have:
Angle A = (29-4x)°
Angle B = 90°
Angle C = 180° - Angle A - Angle B = 180° - (29-4x)° - 90° = 90° + 4x° - 29° = 61° + 4x°
In the second triangle, we have:
Angle D = (5x+11)°
Angle E = 90°
Angle F = 180° - Angle D - Angle E = 180° - (5x+11)° - 90° = 90° + 5x° - 11° = 79° + 5x°
Since the hypotenuses are congruent, angles B and E are congruent, so B = E.
Since the horizontal sides are congruent, angles A and D are congruent, so A = D.
Therefore, we have the following system of equations:
A = D,
B = E.
Substituting the expressions for the angles, we get:
29 - 4x = 5x + 11.
Simplifying, we have:
16x = 18.
Dividing both sides by 16, we get:
x = 18/16 = 9/8.
Substituting x = 9/8 into the expressions for the angles, we get:
Angle A = 29 - 4(9/8) = 29 - 9/2 = 29/2 - 9/2 = 20/2 = 10°.
Angle B = 61 + 4(9/8) = 61 + 9/2 = 61 + 18/2 = 61 + 9 = 70°.
Angle D = 5(9/8) + 11 = 45/8 + 88/8 = 133/8 = 16.625°.
Angle E = 79 + 5(9/8) = 79 + 45/8 = 632/8 + 45/8 = 677/8 = 84.625°.
Since the question asks for the exact measure of each angle, we can conclude that the angles measure:
A. The angle measures 21° in both triangles.
B. The angle measures 2° in both triangles.
C. The angle measures 18° in the first triangle and 40° in the second triangle.
D. The angle measures 22° in the first triangle and 19° in the second triangle.
Therefore, the correct answer is (D) The angle measures 22° in the first triangle and 19° in the second triangle.