Side and Angle Conditions for a Triangle

To determine if these can form a triangle, apply the triangle inequality theorem. Add the lengths of any two sides and compare it to the third: $7$ cm $+ 10$ cm = $17$ cm, which is less than $18$ cm. Therefore, these sides do not meet the triangle inequality theorem, and they cannot form a triangle.

To check if these sides can form a triangle, apply the triangle inequality theorem:

• $8$ cm + $15$ cm = $23$ cm, which is greater than $22$ cm,
• $15$ cm + $22$ cm = $37$ cm, which is greater than $8$ cm,
• $8$ cm + $22$ cm = $30$ cm, which is greater than $15$ cm.

All conditions satisfy the triangle inequality theorem, so these sides can form a triangle.

To find the measure of the third angle, use the angle sum theorem, which states that the sum of the interior angles of a triangle is always $180$ degrees. Add the known angles and subtract from 180 degrees:

• $65^\circ + 45^\circ = 110^\circ$,
• $180^\circ - 110^\circ = 70^\circ$.

Therefore, the third angle measures $70^\circ$.

Using the angle sum theorem:

• The sum of the angles in any triangle is $180^\circ$,
• With a right angle ($90^\circ$) and another angle of $30^\circ$, add these angles: $90^\circ + 30^\circ = 120^\circ$,
• Subtract this sum from $180^\circ$ to find the third angle: $180^\circ - 120^\circ = 60^\circ$.

Thus, the third angle measures $60^\circ$.

Using the angle sum theorem, calculate the third angle: $180^\circ - (50^\circ + 60^\circ) = 70^\circ$.

The sum of the angles in a triangle is $180^\circ$. Thus, the remaining angle is $180^\circ - (35^\circ + 95^\circ) = 50^\circ$.

Calculate the smallest angle: $180^\circ - (75^\circ + 55^\circ) = 50^\circ$.

No, because $2 \text{ cm} + 3 \text{ cm} < 5 \text{ cm}$, which violates the triangle inequality theorem.

No, $7 \text{ cm} + 8 \text{ cm} = 15 \text{ cm}$, which means these sides lie on a straight line, not forming a triangle.

No, since $6 \text{ cm} + 7 \text{ cm} = 13 \text{ cm}$, which is not greater than $12 \text{ cm}$, they cannot form a triangle.

Let the third angle be $2x$ and the second angle be $x$. Solve $120^\circ + x + 2x = 180^\circ$, giving $x = 20^\circ$. So, the angles are $20^\circ$ and $40^\circ$.

No, because $90^\circ + 50^\circ + 60^\circ = 200^\circ$, which exceeds $180^\circ$, violating the angle sum theorem.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This ensures that the sides can meet to form a triangle.

The sum of the interior angles in a triangle always equals $180$ degrees due to the geometric properties of flat, two-dimensional spaces where triangles exist. This is a fundamental rule in Euclidean geometry.

Yes, a triangle can have a 90-degree angle, known as a right triangle, and still adhere to the angle sum theorem. The other two angles must sum to $90$ degrees, maintaining the total of 180 degrees.

If the sum of two sides equals the third side, the figure cannot form a triangle and is instead considered a straight line. This situation does not meet the conditions of the triangle inequality theorem.

To find an unknown angle in a triangle, add the measures of the known angles and subtract from $180$ degrees. The result is the measure of the unknown angle.

Yes, a triangle with all angles of equal measure is called an equilateral triangle, and each angle is $60$ degrees, satisfying the angle sum theorem.

Yes, by applying the triangle inequality theorem (checking if the sum of any two sides is greater than the third), you can determine if three lengths can form a triangle without needing to draw it.

Understanding triangle side and angle conditions is crucial in fields such as architecture, engineering, and construction, where the structural integrity often relies on the properties of triangles.

Yes, a triangle can have one angle greater than $90$ degrees, known as an obtuse triangle, as long as the sum of all three angles remains $180$ degrees.

Knowing these conditions helps ensure that the sides and angles chosen can form a valid triangle, which is essential in geometric constructions and various design tasks where specific triangle types are needed.

Heron's formula is a way to find the area of a triangle if you know the lengths of all three sides. It can be a very useful way of finding the area of a triangle if you’re given the sides and not the perpendicular height of the triangle.