Question

figure abcd is a rhombus.
what is the value of x?
a, b, d, c are vertices of the rhombus. angle at a is (5x + 25)°, angle at d is (7x - 1)°. options: 13, 14, 26, 28

Explanation:

Step1: Recall rhombus angle property

In a rhombus, adjacent angles are supplementary? No, wait, in a rhombus (which is a parallelogram), consecutive angles are supplementary? Wait, no, actually, in a rhombus, opposite angles are equal, and consecutive angles are supplementary? Wait, no, looking at the figure, ABCD is a rhombus, so AB is parallel to CD, and AD is parallel to BC. So angle A and angle D are adjacent? Wait, no, in the figure, angle at A is (5x + 25)°, angle at D is (7x - 1)°. Wait, in a rhombus, are angle A and angle D equal? Wait, no, in a rhombus, opposite angles are equal, and consecutive angles are supplementary. Wait, but in the figure, ABCD is a rhombus, so AB is parallel to CD, and AD is parallel to BC. So angle A and angle D: are they same - side interior angles? Wait, no, if AB is parallel to CD, and AD is a transversal, then angle A and angle D are same - side interior angles, so they should be supplementary? Wait, no, that can't be. Wait, maybe the rhombus here is a rectangle? Wait, no, a rhombus with right angles is a square. Wait, maybe the problem is that in the rhombus, angle A and angle D are equal? Wait, let's check the answer options. Let's assume that angle A and angle D are equal (maybe the rhombus is a square? Or maybe it's a rectangle - shaped rhombus, i.e., a square). So set 5x + 25 = 7x - 1.

Step2: Solve the equation

5x + 25 = 7x - 1
Subtract 5x from both sides: 25 = 2x - 1
Add 1 to both sides: 26 = 2x
Divide both sides by 2: x = 13? Wait, no, 26/2 is 13? Wait, but let's check again. Wait, 5x + 25 = 7x - 1
7x - 5x = 25 + 1
2x = 26
x = 13? Wait, but let's check the angles. If x = 13, then 5x + 25 = 513 + 25 = 65 + 25 = 90, and 7x - 1 = 713 - 1 = 91 - 1 = 90. Oh, so the angles are right angles, so the rhombus is a square. So that makes sense.

Answer:

13