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.
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.