Geometry: แบบฝึกหัด

A square has all sides equal and all interior angles equal to $$90^\circ$$. If one of its diagonals is drawn, what is the measure of the angle formed between this diagonal and one of the sides of the square?

Two parallel lines, \(L_1\) and \(L_2\), are intersected by a transversal. If two consecutive interior angles measure \((2x + 10)^\circ\) and \((3x - 50)^\circ\), find the value of \(x\).

The interior angles of a convex polygon form an arithmetic progression. The smallest angle is \( 120^\circ \) and the common difference is \( 5^\circ \). Find the number of sides of this polygon.

Calculate the measure of one interior angle of a regular octagon.

In a regular polygon, the sum of the interior angles is \(1080^\circ\). How many sides does this polygon have?

If two angles are complementary and one of the angles measures $$30^\circ$$, what is the measure of the other angle?

In a regular \(n\)-sided polygon, each interior angle is \(8\) times the size of each exterior angle. If the number of sides of this polygon is increased by \(k\), the ratio of each interior angle to each exterior angle becomes \(11:1\). Find the value of \(k\).

What type of quadrilateral has exactly one pair of parallel sides?

A quadrilateral $$ABCD$$ has interior angles given by the following expressions:

$$\angle A = (2x)^\circ$$
$$\angle B = (3x+10)^\circ$$
$$\angle C = (x+40)^\circ$$
$$\angle D = (4x-10)^\circ$$

(a) Find the value of $$x$$.

(b) Calculate the measure of each interior angle of the quadrilateral.

(c) Determine if the quadrilateral $$ABCD$$ is a parallelogram. Justify your answer.

In the figure, the straight line \(AB\) is parallel to the straight line \(CD\). Point \(P\) lies between the parallel lines. Given that \(\angle ABP = 35^\circ\) and \(\angle CDP = 60^\circ\).

(a) By drawing an auxiliary straight line \(XY\) passing through \(P\) and parallel to \(AB\), find the measure of \(\angle BPD\). Show your steps clearly and state the geometric reasons.

(b) If \(BP\) is extended to a point \(Q\) such that \(B, P, Q\) are collinear, find the measure of \(\angle DPQ\).