Quadrilateral：練習問題

Which of the following properties is ALWAYS true for a rhombus but NOT necessarily true for a general parallelogram?

In a parallelogram \(ABCD\), \(\angle B = (5x + 10)^\circ\) and \(\angle D = (3x + 40)^\circ\). Find the value of \(x\).

In a rectangle $$PQRS$$, the diagonals $$PR$$ and $$QS$$ intersect at $$O$$. If $$\angle POQ = 120^\circ$$, find the measure of $$\angle OPS$$.

In a parallelogram \(PQRS\), the lengths of opposite sides are given as \(PQ = (2x + 5)\) cm and \(SR = (x + 12)\) cm. Find the value of \(x\).

In a quadrilateral \(ABCD\), it is given that \(AB = CD\) and \(AB \parallel CD\). If \(\angle A = 2x + 10^\circ\) and \(\angle B = 3x - 5^\circ\), find the value of \(x\).

In a parallelogram ABCD, if the length of side BC is 15 cm and the length of side CD is 8 cm, find the length of side AD.

In an isosceles trapezoid ABCD, the side AD is parallel to the base BC. If the measure of the angle \(\angle DAB\) is \(115^\circ\), find the measures of the angles \(\angle ABC\) and \(\angle BCD\).

Given a parallelogram $$ABCD$$, where $$E$$ and $$F$$ are the midpoints of sides $$AB$$ and $$CD$$ respectively. Prove that the quadrilateral $$AECF$$ is a parallelogram.

A rhombus PQRS has its diagonals PR and QS intersecting at point O. Given that \( \angle QPR = 40^\circ \).

a) State the relationship between adjacent angles in a rhombus.

b) Find the measure of \( \angle QPS \).

c) Find the measures of \( \angle PRS \) and \( \angle PQR \).

In a rectangle ABCD, AD = 6 cm and AB = 8 cm. The diagonal AC is drawn. Point E is on AC such that DE is perpendicular to AC.

a) Find the length of the diagonal AC.

b) Find the area of triangle ADC.

c) Using your answers from (a) and (b), find the length of DE.