Practice Questions

Describe the Angle Sum Property of a triangle.

Critique the following statement and justify your conclusion: 'It is possible to construct an equilateral triangle that is also an obtuse-angled triangle.'

Examine if a triangle can be formed with side lengths of $6$ cm, $9$ cm, and $16$ cm. Justify your answer.

In $\triangle ABC$, if $\angle A = 55^\circ$ and $\angle B = 45^\circ$, calculate the measure of $\angle C$.

What is the sum of the three angles in any triangle?

Name the three types of triangles based on the lengths of their sides.

Create a set of three side lengths that would form an isosceles triangle but not an equilateral triangle. Justify that your chosen lengths can indeed form a triangle by using the triangle inequality theorem.

The three angles of a triangle are in the ratio $1:2:3$. Calculate the measure of each angle and classify the triangle based on its angles.

Define an equilateral triangle.

Design a set of measurements (two angles and an included side) for which it is impossible to construct a triangle. Justify why your chosen values make the construction impossible.

Explain why it is more efficient to use a compass and a ruler to construct an equilateral triangle than just a ruler.

In an isosceles triangle, one of the equal base angles is $50^\circ$. Calculate the vertex angle.

Analyze why a triangle cannot have two obtuse angles.

In $\triangle XYZ$, the measure of $\angle X$ is $30^\circ$ more than $\angle Y$, and $\angle Z$ is a right angle. Calculate the measures of $\angle X$ and $\angle Y$.

Two sides of a triangular garden measure $10$ m and $15$ m. Analyze and determine the possible range of lengths for the third side.

A triangle has side lengths of $7$ cm, $7$ cm, and $10$ cm. Examine if this triangle is possible and, if so, classify it based on its sides.

List the three cases of given measurements for which the construction of a unique triangle is described in the text.

Formulate a logical argument to prove that the exterior angle of a triangle is equal to the sum of its two interior opposite angles. You may use the fact that the sum of angles in a triangle is $180^\circ$.

Identify the property that must be satisfied by the side lengths for a triangle to be possible.

What is an altitude of a triangle?

In an isosceles triangle, the vertex angle is $80^\circ$. An exterior angle is formed by extending the base at one of the vertices. Calculate the measure of this exterior angle.

Explain what happens if you try to construct a triangle with side lengths 4 cm, 5 cm, and 10 cm. Relate this to the triangle inequality.

Summarize the steps to construct a triangle when two sides and the included angle are given.

Describe the difference between an acute-angled triangle, a right-angled triangle, and an obtuse-angled triangle.

Summarize the Triangle Inequality property. Explain with an example how to use this property to check if a triangle can be formed with side lengths 12 cm, 7 cm, and 4 cm.

Describe the complete process for constructing a triangle when the lengths of its three sides are given, for example, 5 cm, 6 cm, and 7 cm. List the tools required and explain the role of each step.

An exterior angle of a triangle is $125^\circ$. If one of the interior opposite angles is $70^\circ$, calculate the other interior opposite angle.

A plot of land is shaped like a triangle $\triangle PQR$. Solve the following problems related to it. (a) The owner claims the side lengths are $PQ = 45$ m, $QR = 105$ m, and $RP = 58$ m. Analyze if a triangular plot with these dimensions is possible. (b) In a different valid triangular plot, the angle at vertex $P$ is $80^\circ$ and the angle at vertex $Q$ is $42^\circ$. Calculate the angle at vertex $R$. (c) In a third plot, the side $QR$ is extended to a point $S$. If $\angle P = 60^\circ$ and $\angle Q = 50^\circ$, calculate the exterior angle $\angle PRS$.

Consider an isosceles triangle $\triangle ABC$ where $AB = AC$. (a) If the vertex angle $\angle A = 40^\circ$, calculate the measure of the base angles $\angle B$ and $\angle C$. (b) The side $BC$ is extended to a point $D$. Calculate the exterior angle $\angle ACD$. (c) Using your results, demonstrate that the exterior angle $\angle ACD$ is equal to the sum of the interior opposite angles.

Justify why, when checking the triangle inequality for sides $a, b, c$, it is sufficient to only check if the sum of the two shorter sides is greater than the longest side.

Propose a logical reason, based on the angle sum property, why the sum of any two angles in a triangle must always be less than $180^\circ$.

Evaluate the statement: 'If you are given the lengths of three sides that satisfy the triangle inequality, you can construct only one unique triangle.' Justify your answer.

Rohan claims he has constructed a triangle with angles $40^\circ$, $60^\circ$, and $80^\circ$. Anjali also claims she has constructed a triangle with the same angles, but her triangle is visibly larger than Rohan's. Critique this situation and justify whether it is possible for both of them to be correct.

A student attempts to construct a triangle with side lengths $4$ cm, $6$ cm, and $11$ cm. They draw the $11$ cm base, then use a compass to draw arcs from each endpoint. Justify geometrically why the two arcs will never intersect.

Evaluate whether it is possible for a triangle to be both right-angled and isosceles. If it is possible, design the measures for its three angles and justify your design using the angle sum property.

Create a formal proof, complete with a labeled diagram, to demonstrate that the sum of the interior angles of any triangle is always $180^\circ$. Justify your steps using the properties of parallel lines and transversals.

In $\triangle ABC$, a line segment $DE$ is drawn with $D$ on side $AB$ and $E$ on side $AC$, such that $DE$ is parallel to $BC$. If $\angle ABC = 55^\circ$ and $\angle ACB = 45^\circ$, calculate all three angles of $\triangle ADE$.

One angle of a triangle is equal to the sum of the other two angles. Calculate the measure of this specific angle and demonstrate what type of triangle it is.

Explain the condition related to the sum of two angles for a triangle to be possible when two angles and an included side are given.

A student correctly constructs a triangle with side lengths $6$ cm, $8$ cm, and $10$ cm and observes that it appears to be a right-angled triangle. Critique this observation and justify mathematically why this specific set of side lengths must form a right-angled triangle.

Design a step-by-step geometric construction method using a compass and a straightedge to find the altitude from vertex P for an obtuse-angled triangle $\triangle PQR$ where $\angle Q$ is the obtuse angle. Justify why your method works, explaining how it ensures the constructed line is perpendicular to the line containing side $\overline{QR}$.

Formulate a general rule, in the form of a compound inequality, for the possible length of the third side of a triangle, $c$, given the lengths of the other two sides, $a$ and $b$.

Explain how the concept of parallel lines is used to prove that the sum of the angles in a triangle is $180^\circ$. You can describe the steps with reference to a general triangle $\triangle ABC$.

A designer is creating a logo based on a triangle $\triangle XYZ$. Solve the following design problems. (a) The first proposal has side lengths $8$ cm, $15$ cm, and $6$ cm. Analyze if a triangle can be constructed with these side lengths and provide a reason. (b) In a second design, the triangle is an isosceles right-angled triangle with $\angle X = 90^\circ$ and side $XY = XZ$. Calculate the measures of angles $\angle Y$ and $\angle Z$. (c) In a third design, the exterior angle at vertex $Z$ is $140^\circ$, and the interior angle $\angle Y = 85^\circ$. Calculate the interior angle $\angle X$.

An architect is designing a triangular garden plot with two sides measuring $8$ m and $13$ m. Formulate a compound inequality that describes all possible lengths, $x$, for the third side. Justify why a side length of $4$ m is impossible, and then propose two different valid lengths for the third side that would make the garden an isosceles triangle.