Deductive Geometry: แบบฝึกหัด

Lines L1 and L2 are parallel. A transversal intersects L1 at point A and L2 at point B. If one of the interior angles on the same side of the transversal, formed with line L1, is \(75^\circ\), what is the measure of the other interior angle on the same side, formed with line L2?

In parallelogram ABCD, the angle bisector of \(\angle ABC\) intersects the side AD at point E. If the length of side AB is 5 cm and the perimeter of the parallelogram ABCD is 32 cm, find the length of the segment AE.

In \(\triangle ABC\), \(AD\) and \(BE\) are medians intersecting at \(G\). If \(AD = 24\) cm, what is the length of \(AG\)?

What is the measure of each interior angle of a regular octagon?

In a parallelogram ABCD, the angle bisector of \(\angle A\) intersects side DC at point E. If AD = 7 cm and \(\angle DAB = 110^\circ\), find the length of DE.

Three angles are formed at a point. If two of the angles are identified as \( 125^\circ \) and \( 145^\circ \), find the magnitude of the third angle.

In \(\triangle ABC\), a line \(DE\) is drawn parallel to \(BC\) such that \(D\) is on \(AB\) and \(E\) is on \(AC\). If \(\angle ADE = 75^\circ\) and \(\angle ACB = 40^\circ\), find the measure of \(\angle BAC\).

A regular polygon has a total of 20 diagonals. Determine the measure of each interior angle of this polygon.

In a triangle \(ABC\), \(\angle A = 60^\circ\) and \(\angle B = 70^\circ\). The side \(BC\) is extended through \(C\) to a point \(D\).

(a) Find the measure of \(\angle C\) of the triangle.

(b) Find the measure of the exterior angle \(\angle ACD\).

(c) Verify that the exterior angle \(\angle ACD\) is equal to the sum of the interior opposite angles.

Consider an isosceles triangle $$PQR$$ with $$PQ = PR$$. The exterior angle at vertex $$R$$ (formed by extending $$QR$$ to a point $$X$$) is $$110^\circ$$. A line segment $$ST$$ is drawn through $$Q$$ such that $$ST$$ is parallel to side $$PR\$>.

(a) Find the measure of the interior angle $$\angle PRQ$>.

(b) Find the measure of $$\angle PQR\$>.

(c) Find the measure of $$\angle QPR$>.

(d) Determine the measure of $$\angle TQR\$>.

(e) If side $$PQ$$ is extended to a point $$U$$, find the measure of $$\angle RQU$>.