Geometry

A line segment $PQ$ is parallel to $RS$. Point $M$ lies between them such that $\angle PQM = 30^{\circ}$ and $\angle RSM = 45^{\circ}$. Find $\angle QMS$.

The exterior angles of a triangle are in the ratio $3:4:5$. What is the largest interior angle of the triangle?

Two parallel lines are cut by a transversal. If one of the interior angles is $72^{\circ}$, what is the measure of its co-interior angle?

Which of the following is the measure of an interior angle of a regular hexagon?

In a regular polygon, the sum of the interior angles is 10 times the measure of one of its exterior angles. Find the number of sides $n$.

Two parallel lines are intersected by a transversal. The bisectors of the interior angles on the same side of the transversal meet at a point. What is the measure of the angle formed at that point?

Each interior angle of a regular $n$-gon is $162^{\circ}$. Find the number of diagonals in this polygon.

In the figure, $AB \parallel CD$. The angle $\angle EAB = 120^\circ$ and $\angle ECD = 140^\circ$. If $EA = AC$, determine the measure of $\angle AEC$.

A polygon has $n$ sides. If the sum of its interior angles is four times the sum of its exterior angles, find the number of sides $n$ and the measure of each interior angle if the polygon is regular.

The interior angles of a hexagon are in an arithmetic progression. If the smallest angle is $100^\circ$, calculate the measure of the largest angle in the hexagon.

If $\angle APB = 40^\circ$ and $P$ is a point on the circumference of a circle with center $O$, and $AB$ is a chord. What is the reflex angle $\angle AOB$?

The angle between a tangent and a chord at the point of contact is $62^\circ$. What is the angle in the alternate segment?

In the diagram, $O$ is the center. $\angle BOC = 130^\circ$. Find $\angle BAC$.

In the diagram, $\angle ABC = 70^\circ$ where $A, B, C$ are on the circle and $AC$ is not a diameter. If $D$ is a point on the minor arc $AC$, find $\angle ADC$.

In the diagram, $\angle BAC = 40^\circ$ and $\angle ACB = 30^\circ$. Points $A, B, C, D$ are on the circle. Find $\angle BDC$.

In the diagram, $O$ is the center and $AB$ is a chord. $OC$ is perpendicular to $AB$. If $AB = 16$ cm and the radius is 10 cm, find the length of $OC$.

In a cyclic quadrilateral $ABCD$, $\angle A = 2x + 10$ and $\angle C = 3x - 20$. Find the value of $x$.

In a semi-circle with diameter $AB$, a point $C$ lies on the arc. A perpendicular is dropped from $C$ to $AB$ at point $D$. If $AD = 4$ cm and $DB = 9$ cm, calculate the length of $CD$.

A designer creates a logo with a circle. Chords $AB$ and $CD$ intersect at $E$. If $AE = x+2$, $EB = x-2$, $CE = 3$, and $ED = 4$, determine the value of $x$.

In the diagram, $O$ is the center of the circle and $PT$ is a tangent at $T$. The line $PAB$ passes through the center $O$ such that $PA = 4$ units and the radius of the circle is $6$ units. Calculate the length of the tangent $PT$.

Identify the congruence criterion for the two triangles shown, given that the segments with tick marks are equal and the right angles are indicated.

If $\triangle ABC \sim \triangle PQR$ and the ratio of their corresponding sides is $k$, what is the ratio of their altitudes?

Which condition is sufficient to prove that two triangles are similar?

Are the two triangles shown congruent? If so, by which criterion?

In the diagram, $\triangle ADE \sim \triangle ABC$. If $BC = 10$, $DE = 4$, and $AE = 6$, find the length of $EC$.

Two cylinders are similar. The ratio of their heights is $1:4$. If the surface area of the smaller cylinder is $50$ cm$^2$, find the surface area of the larger cylinder.

In the diagram, $L_1 \parallel L_2$. Find the value of $x$.

A structural engineer is designing a truss support consisting of two nested triangles, $\triangle ABC$ and $\triangle ADE$, where $DE$ is parallel to $BC$. The line segment $AE$ lies on $AC$ and $AD$ lies on $AB$. It is given that the area of the quadrilateral $BCED$ is exactly $3$ times the area of $\triangle ADE$. If the perimeter of $\triangle ABC$ is $120$ cm, and the length of $DE$ is $15$ cm, calculate the length of the base $BC$.

A frustum of a right circular cone has a base radius of $12$ cm and a top radius of $6$ cm. The height of the frustum is $8$ cm. By considering the similar triangles formed by extending the sides to form a full cone, find the slant height of the original full cone.

In $\triangle ABC$, $\angle A = 90^\circ$. A square $PQRS$ is inscribed in the triangle such that $P$ and $Q$ lie on $BC$, $R$ lies on $AC$, and $S$ lies on $AB$. If $BP = 18$ cm and $QC = 8$ cm, find the side length of the square.