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A) Which statement can verify the Triangle Inequality Theorem? (1 point)

The three angles of a triangle will always amount to 180 degrees.
The sum of any two sides of a triangle is greater than the length of the third side.
An exterior angle is equal to its two remote interior angles.
The formula a + = will find the length of the third side of any triangle.

9 answers

The statement that verifies the Triangle Inequality Theorem is:

"The sum of any two sides of a triangle is greater than the length of the third side."

This is the fundamental principle of the Triangle Inequality Theorem.

A) If a triangle has sides of 31 in. and 28 in., which is a possible length for the third side?
(1 point)
59 in.
62 in.
33 in.
3 in.

To determine a possible length for the third side of a triangle with sides of 31 in. and 28 in., we can use the Triangle Inequality Theorem. According to this theorem:

  1. The sum of any two sides must be greater than the length of the third side.
  2. Specifically, for sides \(a\), \(b\), and \(c\), the following must hold true:
    • \(a + b > c\)
    • \(a + c > b\)
    • \(b + c > a\)

Let’s denote the sides we have as:

  • \(a = 31\) in.
  • \(b = 28\) in.
  • \(c\) = length of the third side.

Using the Triangle Inequality Theorem, we can establish the following inequalities:

  1. \(31 + 28 > c\) → \(59 > c\) → \(c < 59\)
  2. \(31 + c > 28\) → \(c > -3\) (which is always true for positive lengths)
  3. \(28 + c > 31\) → \(c > 3\)

From these inequalities, we conclude:

  • The length of the third side \(c\) must be greater than 3 inches and less than 59 inches.

Now, evaluating the options given:

  • 59 in. (not possible, since \(c\) must be less than 59)
  • 62 in. (not possible, since it exceeds the upper limit of 59)
  • 33 in. (possible, since it is greater than 3 and less than 59)
  • 3 in. (not possible, since \(c\) must be greater than 3)

Therefore, the possible length for the third side is:

33 in.

A) A ruler measuring 1 foot is bisected. Using your knowledge of constructions, determine the
length of each segment of the bisected ruler. (1 point)
0.5 inch
6 inches
4 inches
3 inches

A ruler measuring 1 foot is equal to 12 inches. When you bisect the ruler, you divide the total length in half.

So, the length of each segment after bisecting will be:

\[ \frac{12 \text{ inches}}{2} = 6 \text{ inches} \]

Thus, the length of each segment of the bisected ruler is:

6 inches.

What is the next step in this construction?
(1 point)
connecting point R to point S to create ∠SRQ
connecting point Q to point S to create ∠SQR
creating a reference line with a straightedge
placing the compass point on the vertex and drawing an arc

To properly determine the next step in your construction, it would be helpful to know the context of the construction you are referencing. However, based on common geometric constructions, particularly those involving angles or triangles, the following is often the sequence of steps:

If you have a vertex point \( S \) and you are working on constructing an angle, the next logical step after establishing points \( R \) and \( Q \) around point \( S \) might likely be:

Placing the compass point on the vertex and drawing an arc.

This step is commonly used to create equal segments or to mark points where the angle will be formed. If you provide more context about the specific construction you are working on, I can give a more tailored response!

A) If an angle can be divided in half by constructing a bisector, then what method doubles an
angle? (1 point)
bisecting the angle twice
constructing a copy of the bisected angle attached to the original angle
constructing two of the bisected angles
constructing a copy attached to the original angle

To double an angle, the correct method is:

Constructing a copy of the bisected angle attached to the original angle.

Here's how it works: If you bisect an angle, you create two equal angles. To double the original angle, you take one of the bisected angles and attach it to the original angle, effectively creating an angle that is twice the size of the original.