The Triangle and its Properties

Does every triangle have exactly 3 altitudes?

The point of intersection of the altitudes of a triangle is called the:

In a triangle, can a median and an altitude be the same line segment?

The altitude of a triangle always forms an angle of ____ with the base.

In a triangle $ABC$, the altitude $AM$ is $8$ cm and the median $AN$ is $10$ cm. Which of the following is true about $M$ and $N$?

In an equilateral triangle, what is the relationship between the centroid and the orthocentre?

Which point of concurrency is equidistant from the vertices of a triangle? (Note: This relates to medians/altitudes as special cases)

In an acute-angled triangle, the point where all three altitudes intersect is called the Orthocentre. Where does this point lie?

A point $K$ lies inside $\triangle ABC$ such that it is the intersection of the medians. If the distance from $K$ to the midpoint of $BC$ is $5$ units, what is the distance from $K$ to the vertex $A$?

In $\triangle ABC$, $BE$ and $CF$ are altitudes. If $BE = CF$, which of the following must be true about $\triangle ABC$?

Find the value of $x$ in the figure below.

If two interior opposite angles are $35^\circ$ and $45^\circ$, find the exterior angle.

In an equilateral triangle, what is the measure of each exterior angle?

Which of the following can never be the measure of an exterior angle of a triangle?

In $\triangle ABC$, the exterior angle at $C$ is $k$. If the interior angle at $A$ is $50^\circ$ and the interior angle at $C$ is $70^\circ$, find $k$.

Two exterior angles of a triangle are $120^\circ$ and $130^\circ$. Find the third exterior angle.

The exterior angle at vertex $A$ of $\triangle ABC$ is $125^\circ$. If $\angle B = 2x$ and $\angle C = 3x$, find the value of $x$.

A triangular plot has an exterior angle of $150^{\circ}$ at one vertex. The difference between the two interior opposite angles is $30^{\circ}$. Find the measure of the larger interior opposite angle.

An architect extended the base of a pyramid-shaped facade. The exterior angle formed is $(4x + 20)^{\circ}$. The two interior opposite angles are $(2x + 10)^{\circ}$ and $80^{\circ}$. Determine the measure of the exterior angle.

For a specialized folding bracket, the exterior angle is $110^{\circ}$ and one of the interior opposite angles is $40^{\circ}$. If the interior angle adjacent to the exterior angle is $x^{\circ}$, what is the value of $x +$ (the other interior opposite angle)?

Which of the following sets of angles can form a triangle?

Find the value of $x$ and $y$ in the following figure if the triangle is right-angled at $B$ and $AB = BC$.

In an isosceles triangle, the vertex angle is $40^{\circ}$. Find the measure of each base angle.

In the given triangle, find the value of $z$.

In the figure, find the value of $x + y$.

In a triangle, one angle is $30^{\circ}$ more than the smallest angle, and the third angle is $60^{\circ}$ more than the smallest angle. Find the smallest angle.

Find the value of $x$ if the angles of a triangle are $2x-5^{\circ}$, $x-10^{\circ}$, and $x+15^{\circ}$.

In $\triangle ABC$, if $2\angle A = 3\angle B = 6\angle C$, what is the measure of $\angle B$?

For a school project, a student draws a triangle where one angle is the average of the other two. If the difference between the largest and the smallest angle is $50^\circ$, find the measure of the largest angle.

A triangular park $XYZ$ has $\angle X - \angle Y = 40^\circ$ and $\angle Y - \angle Z = 10^\circ$. Calculate the measure of the smallest angle $\angle Z$ in the park.

In the figure, $O$ is a point in the interior of $\triangle ABC$. Which of the following is always true?

If the sides of a triangle are $x, x+1, x+2$, what is the smallest integer value of $x$ for which the triangle exists?

Can a triangle have sides of $2.5\text{ cm}$, $2.5\text{ cm}$, and $5.0\text{ cm}$?

Which inequality represents the Triangle Inequality Theorem for sides $a, b, c$?

In the figure, $O$ is an interior point. Which relation is true for the perimeter $P$ of $\triangle ABC$?

Which set of lengths can be the sides of a triangle?

If the sides of a triangle are $5\text{ cm}$, $12\text{ cm}$, and $n\text{ cm}$, where $n$ is an integer, how many possible values can $n$ take?

If the sides of a triangle are in the ratio $3:4:5$, and the shortest side is $9$ cm, what is the sum of the lengths of the two shorter sides, and is it greater than the third side?

A fence is to be built around a triangular plot. If two sides are $25$ m and $30$ m, and the perimeter must be less than $100$ m, which of the following could be the length of the third side?

In triangle $LMN$, if the difference between lengths of sides $LM$ and $MN$ is $5$ units, and the sum of lengths $LM$ and $MN$ is $15$ units, which of the following could be the length of side $LN$?

If the hypotenuse of a right-angled triangle is $25\text{ cm}$ and one side is $7\text{ cm}$, find the other side.

Which set of numbers cannot be the sides of a right-angled triangle?

The property that 'In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the legs' is named after which mathematician?

A man goes $24\text{ m}$ due west and then $10\text{ m}$ due north. How far is he from the starting point?

In the given right-angled triangle $PQR$, the side $QR$ has a length of $24$ cm and the side $PQ$ has a length of $7$ cm. Calculate the length of the hypotenuse $PR$.

A $15$ m long ladder is placed against a wall to reach a window at a height of $12$ m from the ground. Find the distance of the foot of the ladder from the base of the wall.

In the figure, if $AD \perp BC$, $AB=25$, $AD=24$, and $AC=26$, find the length of $BC$.

A square has a diagonal of length $10\sqrt{2}$ cm. What is the area of the square?

In triangle $PQR$, the side $PQ = 13$ cm, $QR = 5$ cm, and $PR = 12$ cm. Which angle of the triangle is a right angle?

A wire is attached to the top of a 15 m high flag pole and to a hook on the ground. If the wire is 17 m long, how far is the hook from the base of the pole?