Geometry

What is the sum of angles forming a linear pair?

Two angles are in the ratio $1:2$. If they are complementary, find the smaller angle.

If $\angle A$ and $\angle B$ are vertically opposite angles and $\angle A = 2x + 10^\circ$, $\angle B = 70^\circ$, find $x$.

Identify the value of $y$ in the figure where angles $2y$ and $3y$ form a right angle.

If $\angle x$ and $\angle y$ are vertically opposite and $x + y = 160^\circ$, find $x$.

Four angles at a point are in the ratio $1:2:3:4$. Find the largest angle.

Find the value of $a$ and $b$ if $a-b = 40^\circ$ and they are supplementary.

At a traffic junction, the angles between four roads meeting at a point are $x^{\circ}$, $(2x+10)^{\circ}$, $(3x-20)^{\circ}$, and $(4x+30)^{\circ}$ in clockwise order. What is the measure of the smallest angle?

If the complement of an angle is equal to one-fourth of its supplement, calculate the measure of the angle.

In a sundial design, $\angle PQR$ and $\angle RQS$ are adjacent angles forming a linear pair. If $\angle PQR$ is five times $\angle RQS$, find the measure of $\angle PQR$.

In the figure, $m \parallel n$ and $l$ is a transversal. If $\angle 4 = 100^{\circ}$, what is $\angle 5$ (interior angle on the same side)?

When a transversal cuts two parallel lines, how many pairs of corresponding angles are formed?

Find the measure of $\angle ABC$ if $L \parallel M$ and the given angle is $55^{\circ}$ as shown.

If two lines are intersected by a transversal such that the alternate interior angles are equal, then the lines are:

In the given figure, line $l$ is parallel to line $m$ ($l \parallel m$) and line $t$ is a transversal. If one of the interior angles on the same side of the transversal is $(2x + 10)^{\circ}$ and the other is $(3x - 15)^{\circ}$, find the value of $x$.

Find the value of $k$ if the interior angles on the same side of a transversal are $(4k + 10)^{\circ}$ and $(5k - 10)^{\circ}$ for parallel lines.

In the figure, line $a \parallel b$. If $\angle 1 = 3x$ and $\angle 2 = 2x + 20$ are corresponding angles, find the measure of $\angle 1$.

A city planning engineer is designing a residential layout where two main roads, $L_1$ and $L_2$, run parallel to each other. A diagonal walkway $T$ crosses both roads. On road $L_1$, the engineer measures an angle formed with the walkway as $(3x - 15)^{\circ}$. On road $L_2$, the interior angle on the same side of the walkway is measured as $(2x + 45)^{\circ}$. To ensure the roads are perfectly parallel, what must be the value of $x$, and what is the measure of the larger of these two angles?

A shelf bracket consists of two parallel supports and a brace. If the interior angles on the same side of the brace are $(2x + 10)^{\circ}$ and $(3x + 20)^{\circ}$, calculate the value of $x$.

In a geometric design, three lines $l, m,$ and $n$ are such that $l \parallel m$ and $m \parallel n$. A transversal $t$ cuts them. If $\angle 1 = 65^{\circ}$, find the measure of $\angle 2$ which is the alternate interior angle to $\angle 1$ with respect to lines $l$ and $n$.

Find the measure of angle $x$ in the following figure.

In a right-angled triangle, if one of the acute angles is $35^\circ$, find the other acute angle.

In the figure, find the value of $\angle ACD$.

Each angle of an equilateral triangle measures:

In the given figure, triangle $ABC$ has side $BC$ extended to point $D$. If $\angle ABC = 45^\circ$ and the exterior angle $\angle ACD = 115^\circ$, find the value of interior angle $\angle BAC$.

The three angles of a triangle are $x + 10^\circ$, $x - 20^\circ$, and $x + 40^\circ$. Find the largest angle.

In the given figure, $PQ \parallel RS$. Find the value of $x$ using the triangle property.

In the figure provided, $AB$ is parallel to $CD$. If $\angle ABE = 120^\circ$ and $\angle DCE = 140^\circ$, find the measure of $\angle BEC$. (Hint: $E$ is a point such that $ABEC$ forms two triangles or a quadrilateral; here consider the reflex angle or use exterior angle properties by drawing a line through $E$ parallel to $AB$).

In $\triangle PQR$, the angle $P$ is half of angle $Q$, and angle $R$ is $20^\circ$ more than angle $Q$. What is the measure of the exterior angle at vertex $P$?

A star-shaped polygon is created. Consider one of the triangles forming a point of the star. If the exterior angle at the base of this triangle is $140^\circ$ and the triangle is isosceles (with the two interior angles at the base being equal), find the vertex angle of the point.

A triangle has three medians that all intersect at a single point inside the triangle. What is the special name for this point of intersection?

In triangle $PQR$, the line segment $PS$ is drawn from vertex $P$ such that it is perpendicular to the opposite side $QR$. What is $PS$ called in relation to the triangle?

In the given triangle $ABC$, $AD$ is a line segment that connects the vertex $A$ to the midpoint $D$ of the opposite side $BC$. What is the specific name given to the segment $AD$?

Which of the following always lies inside the triangle regardless of its type?

If a triangle has sides $6$ cm, $8$ cm, and $10$ cm, what is the length of the altitude drawn to the $10$ cm side?

In $\triangle ABC$, $AD$ is the median to $BC$. If the coordinates of $B$ are $(2, 0)$ and $C$ are $(6, 0)$, what are the coordinates of $D$?

Which statement is true about the altitudes of an acute-angled triangle?

If the centroid $G$ of a triangle $ABC$ is located at $(2, 4)$ and two of its vertices are $A(1, 2)$ and $B(3, 5)$, what are the coordinates of the third vertex $C$?

In triangle $PQR$, $PX$ is the altitude to $QR$ and $PY$ is the median to $QR$. If $QR = 14 \text{ cm}$ and the area of $\triangle PQR$ is $56 \text{ cm}^2$, find the length of the altitude $PX$.

A median of a triangle divides it into two triangles of:

In a right triangle, the side opposite to the $90^{\circ}$ angle is called the:

The sides of a triangle are $8$ cm, $15$ cm and $17$ cm. Is this a right-angled triangle?

In triangle $XYZ$, $\angle Y = 90^{\circ}$. If $XY = 9$ and $YZ = 12$, find $XZ$.

If the hypotenuse of a right triangle is $10$ cm and one side is $6$ cm, find the other side.

Find the length of the side of a square whose diagonal is $10$ cm.

In a right triangle $ABC$ (right-angled at $C$), if $AC = 2.4$ cm and $BC = 0.7$ cm, find $AB$.

The diagonal of a square is $12\sqrt{2}$ cm. Find its area.

A square field has an area of 400 sq. m. A person walks along the diagonal from one corner to the opposite corner. Approximately how much distance does he cover?

A rhombus has diagonals of length 16 cm and 12 cm. What is the length of each side of the rhombus?

A ladder $15$ m long reaches a window which is $9$ m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window $12$ m high. What is the width of the street?

In the given figure, triangle $ABC$ and triangle $PQR$ are such that $AB = PQ$, $BC = QR$, and $AC = PR$. Which congruence criterion proves that $\triangle ABC \cong \triangle PQR$?

Which congruence criterion uses two angles and the side between them?

In $\triangle ABC$ and $\triangle PQR$, if $AB = PQ$, $\angle B = \angle Q$, and $BC = QR$, then $\triangle ABC \cong \triangle PQR$. This is the ___ criterion.

In a right-angled $\triangle ABC$ where $\angle B = 90^{\circ}$, and $\triangle DEF$ where $\angle E = 90^{\circ}$, if $AC = DF$ and $AB = DE$, then the triangles are congruent by:

Two isosceles triangles have equal vertical angles and equal areas. Are they necessarily congruent?

In the figure, $O$ is the center of the circle and $AB$ is a chord. $M$ is the midpoint of $AB$. $\triangle OMA \cong \triangle OMB$ by which criterion?

If $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC$ is congruent to $\triangle GHI$. This property is called:

In a circle with center $O$, two chords $AB$ and $CD$ are equal in length. To prove that the triangles $\triangle OAB$ and $\triangle OCD$ are congruent, which criterion is most directly applicable using radius properties?

In $\triangle DEF$ and $\triangle PQR$, if $DE = PQ$, $EF = QR$ and $\angle E = \angle Q$, then $\triangle DEF \cong \triangle PQR$. If $\angle D = 70^\circ$ and $\angle E = 50^\circ$, calculate the value of $\angle R$.

A ladder $10$ m long reaches a window $8$ m high on one side of a street. On the other side, another ladder of the same length $10$ m reaches a window $8$ m high. If the ground is level, the triangles formed by the ladders and walls are congruent by:

The diagram shows an equilateral triangle. What is the order of rotational symmetry for this figure about its center?

A rectangle is shown in the diagram. How many lines of symmetry does a standard rectangle have?

If a figure is rotated $180^\circ$ and looks exactly the same, what is its minimum order of rotational symmetry?

Which of these shapes has only 1 line of symmetry?

In the diagram, a square is inscribed in a circle. If the figure is rotated about its center, what is the smallest non-zero angle of rotation that maps the figure onto itself, and how many lines of symmetry does the combined figure have?

The diagram displays a regular hexagon. What is the angle of rotation for its rotational symmetry?

Which of these quadrilaterals has the same number of lines of symmetry as its order of rotational symmetry?

A rectangle has dimensions 12 cm by 8 cm. Four identical quarter-circles are removed from each of its four corners, each with a radius of 2 cm. For this new shape, what is the order of rotational symmetry and how many lines of symmetry does it have?

A star is formed by overlapping two identical equilateral triangles, one pointing up and one pointing down, sharing the same center (a hexagram). If a small circle is removed from the very center of the star, how many lines of symmetry does the resulting shape have?

A paper cutout is made in the shape of a letter 'H'. If the height of the 'H' is 10 cm, the width is 8 cm, and the thickness of all bars is 2 cm, find the number of lines of symmetry and the angle of rotational symmetry.