Question

ABCD and CDEF are parallelograms. ∠A = 3x, ∠E = 5x and ∠BCF = 112°. Find x.

Answer 1

Given:

• ABCD and CDEF are parallelograms.
• ∠A = 3x
• ∠E = 5x
• ∠BCF = 112°

Stepwise Calculation:

1. Since ABCD is a parallelogram, ∠A and ∠C are equal and adjacent angles sum to 180°.
2. Similarly, CDEF is also a parallelogram, so ∠E and ∠F are related similarly.
3. Based on the figure, note that ∠BCF is an exterior angle at vertex C to the parallelogram ABCD and CDEF.
4. Consider the geometry around point C:
   • ∠BCA and ∠BCF are supplementary (sum to 180°), so ∠BCA = 180° – 112° = 68°.
5. In parallelogram ABCD, ∠A + ∠B = 180°.
6. The angle ∠D in ABCD is equal to ∠B since opposite angles are equal.
7. Angles at point C in the two parallelograms help relate ∠A and ∠E.
8. Express these in terms of x and solve the equation obtained by summing angles around C or from the external angle given.

After laying out the equations and solving, x = 31°.