Exercises from the chapter 11 — Angles on a circle

Determine the value of the angle $x$ in the following cases:

Determine the value of the arc $x$ in the following cases:

In the figures, calculate the value of $x$.

In the figures, calculate the value of $\alpha$.

In the figures, calculate the value of the arc $\overset{\frown}{ABC}$.

In the figures, calculate $x$.

In the figure, with $\overset{\frown}{ABC} = 260^{\circ}$, calculate the value of $\alpha$.

Calculate the value of $x$.

The arc $\overset{\frown}{CD}$ in the figure measures $105^{\circ}$. Calculate the value of $x$.

On the circumference, the arc $\overset{\frown}{CFD}$ exceeds the arc $\overset{\frown}{AEB}$ by $50^{\circ}$. Determine its measurements, knowing that the angle $\alpha$ measures $70^{\circ}$.

In the figure, angle $A\hat{C}D$ is equal to $70^{\circ}$ and angle $A\hat{P}D$ is equal to $110^{\circ}$. Determine the measure of the angle $B\hat{A}C$.

Calculate $x$ in the figures:

Calculate $x$ in the figures:

If $y = 75^{\circ}$ and $\overset{\frown}{AB} = 100^{\circ}$, calculate $x$.

In the figure, what is the value of $\alpha$?

In the figure, the arc $\overset{\frown}{CMD}$ is equal to $100^{\circ}$ and the arc $\overset{\frown}{ANB}$ measures $30^{\circ}$. Calculate the value of $x$.

Determine the measure of the angle $\alpha$, knowing that, in the figure, $CD = R$.

Calculate $x$ in the figures:

Calculate $x$ in the figures (in b, $ABCDE$ is a regular pentagon):

In the figure, the arc $\overset{\frown}{BEC}$ measures $60^{\circ}$ and $\overline{OB}$ is perpendicular to $\overline{AC}$. Determine the arc measure $\overset{\frown}{AFB}$ and the angle measure $A\hat{D}C$.

Determine the measurements of the angles of the triangle obtained by the points of tangency of the circle inscribed with the sides of a triangle $ABC$, being $\hat{A} = 60^{\circ}$, $\hat{B} = 40^{\circ}$ and $\hat{C} = 80^{\circ}$.

Determine the ratio between the angles $\alpha$ and $\beta$ in the figure, knowing that the straight line $r$ tangents the circle at the point $A$ and that the arcs $\overset{\frown}{AB}$, $\overset{\frown}{BC}$ and $\overset{\frown}{AC}$ are proportional to the numbers 2, 9 and 7.

In the figure, determine the measure of the angle $\alpha$, knowing that the arc $\overset{\frown}{AB}$ measures $100^{\circ}$ and that the chord $\overline{CD}$ measures $R$, with $R$ being the radius of the circle.

Determine the smallest angle formed by two straight lines secant to a circle, led by an external point $P$, knowing that these secants determine two arcs on the circle whose measures are worth $30^{\circ}$ and $90^{\circ}$.

In the figure, $\overline{AB}$ and $\overline{AC}$ are tangents to the center circle $O$ and $Q$ is a point on the minor arc $\overset{\frown}{BC}$. $\overline{PQR}$ is tangent to the circle, and $\hat{A} = 28^{\circ}$. Find $P\hat{O}R$.

In the figure, $\overline{AB}$ is a diameter, the chord $\overline{AM}$ is the side of the inscribed equilateral triangle and $\overline{BN}$ is the side of the inscribed square. Calculate the angle $\alpha$, formed by the tangents $\overline{PM}$ and $\overline{PN}$.

Determine the measurements $x$ and $y$.

Let us consider a equilateral triangle $ABC$ inscribed in a circle. Determine the smallest angle formed by the tangent lines to this circle at points $A$ and $B$.

Determine the value of $x$ in the following cases:

Show that, if $\overset{\frown}{AB}$ and $\overset{\frown}{CD}$ are arcs of equal length on a circle, then the chords $\overline{AB}$ and $\overline{CD}$ are congruent.

Prove that distinct parallel lines, secant to a circle, determine on the circle, between the parallel lines, arcs of the same size.

Prove that a trapezoid inscribed in a circle is isosceles.

Let $r$ and $R$ be the radii of the circles inscribed and circumscribed to a right-angled triangle of legs $a$ and $b$. Prove that $a + b = 2(R + r)$.

Prove that the sum of the diameters of the inscribed and circumscribed circles of a right-angled triangle is equal to the sum of the legs of that triangle.

If the sides $\overline{AB}$ and $\overline{AC}$ of a triangle are diameters of two circles, prove that the other point common to the circles is at $\overline{BC}$.

Let $ABC$ be an acute triangle and $H_1$, $H_2$, $H_3$ be the feet of the heights. Prove that orthocenter $H$ of triangle $ABC$ is incenter of triangle $H_1H_2H_3$.