Triangle Angles Practice Problems - Sum Theorem Exercises

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

In order to determine if segment CB is a side of one of the triangles, let's start by identifying the triangles and their corresponding vertices from the given diagram:

• Triangle 1 has vertices labeled as A, B, C.
• Triangle 2 has vertices labeled as D, E, F.

Now, to decide if CB is a side, we need to check if a line segment exists between points C and B in any of these triangles.

Upon examining the points:

• Point C is present in triangle 1.
• Point B is also present in triangle 1.
• The line segment connecting B and C is visible, forming the base of triangle 1.

Therefore, segment CB is indeed a side of triangle ABC, confirming that the answer is Yes.

Thus, the solution to the problem is $\text{Yes}$.

In order to solve this problem, we need to understand the basic properties of an isosceles triangle.

An isosceles triangle has two sides that are equal in length, often referred to as the "legs" of the triangle. The angle formed between these two equal sides, which are sometimes referred to as the "sides", is called the "vertex angle" or sometimes more colloquially as the "main angle".

When considering the vocabulary of the given multiple-choice answers, choice 2: $sides, main$ accurately fills the blanks, as the angle formed between the two equal sides can indeed be referred to as the "main angle".

Therefore, the correct answer to the problem is: $sides, main$.

Let's solve the problem step-by-step.

• We first consider the two triangles given in the diagram. The vertices of the first triangle are labeled $A$, $B$, and $C$. The vertices of the second triangle are labeled $D$, $E$, and $F$.
• Identify the sides of the first triangle: Since the vertices are $A$, $B$, and $C$, the sides of the triangle are $AB$, $BC$, and $CA$.
• Identify the sides of the second triangle: With vertices $D$, $E$, and $F$, the sides are $DE$, $EF$, and $FD$.
• Now, we ascertain whether $BC$ is a side. Upon inspection, $BC$ is clearly the side connecting vertex $B$ and vertex $C$ in the first triangle.

Thus, we conclude that Yes, $BC$ is indeed a side of one of the triangles.

The solution to the problem is: Yes

The task is to determine if the segment $AD$ is a side of any of the given triangles. Based on the diagram, we have two distinct triangles:

• $\triangle ABC$: Formed by the points $A, B, C$.
• $\triangle DEF$: Formed by the points $D, E, F$.

For $\triangle ABC$, the sides are $AB, BC,$ and $CA$.

For $\triangle DEF$, the sides are $DE, EF,$ and $FD$.

In analyzing both triangles, we observe that:

• The side $AD$ is not listed as one of the sides of either triangle.

Thus, the conclusion is clear: AD is not a side of either triangle.

Therefore, the answer is No.