Introduction to Geometry: แบบฝึกหัด

What is the sum of the interior angles of any convex quadrilateral?

In the figure, the straight lines \(L_1\) and \(L_2\) are parallel. A transversal intersects them such that a pair of alternate interior angles are \((2x + 15)^\circ\) and \((x + 45)^\circ\). Find the value of \(x\).

In the figure, AB is parallel to CD. A point M lies between the lines such that \(\angle BAM = 30^\circ\) and \(\angle DCM = 40^\circ\). If the angle \(\angle AMC\) is bisected by a line MN which is parallel to AB, find the measure of \(\angle NMA\).

Three angles, \(a\), \(b\), and \(c\), meet at a single point. If \(a = 110^\circ\) and \(b = 130^\circ\), calculate the measure of angle \(c\).

In the figure, three straight lines intersect at a single point \(O\). If the measures of two adjacent angles are given by \((5x + 10)^\circ\) and \((2x + 15)^\circ\), and the measure of an angle vertically opposite to the first one is \((3x + 40)^\circ\), find the value of \(x\).

Three angles meet at a point. Two of them measure \( 90^\circ \) and \( 120^\circ \). What is the measure of the third angle?

Calculate the measure of each interior angle of a regular 15-sided polygon.

In the figure, \(AOB\) is a straight line and rays \(OC\) and \(OD\) are drawn on the same side of the line such that \(OC \perp OD\). If \(\angle BOD\) is \(30^\circ\) less than \(\angle AOC\), find the measure of the reflex angle \(\angle BOC\).

An angle measures \( (2x+5)^\circ \). Its complementary angle measures \( (x+10)^\circ \).

Find the value of \( x \).

A convex hexagon has six interior angles. The measures of these angles are \(x^\circ\), \((x + 20)^\circ\), \((x + 40)^\circ\), \((x + 60)^\circ\), \((x + 80)^\circ\), and \((x + 100)^\circ\).

a) Using the formula for the sum of interior angles of a polygon, calculate the total sum of the interior angles for a hexagon.
b) Set up an equation and solve for the value of \(x\).
c) Determine the measure of the largest interior angle of this hexagon.