Maria constructed the angle bisector of angle ABC by constructing ray BD.

Question

Maria then constructed the angle bisector of DBC by constructing ray BE.
If the measurement of angle ABC is 68 degrees, what is the measurement of angle EBC?
112degrees
34degrees
17degrees
51degrees

Answer 1

To find the measure of angle EBC, we will follow these steps:

1. Determine the measure of angle DBC:

   Since ray BD is the angle bisector of angle ABC, it divides angle ABC into two equal parts. Thus,

   \[ \text{Angle ABD} = \text{Angle DBC} = \frac{\text{Angle ABC}}{2} = \frac{68^\circ}{2} = 34^\circ. \]

2. Determine the measure of angle EBC:

   Next, ray BE is the angle bisector of angle DBC. Since angle DBC measures 34 degrees, we can find angle EBC by again dividing angle DBC by 2:

   \[ \text{Angle EBC} = \frac{\text{Angle DBC}}{2} = \frac{34^\circ}{2} = 17^\circ. \]

3. Conclusion:

   Therefore, the measurement of angle EBC is

   \[ \boxed{17 \text{ degrees}}. \]

Answer 2

Which set of side lengths do NOT form a triangle?
2,3,5
3,5,7
6,7,10
6,14,18

Answer 3

To determine whether a set of side lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be true:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

We will check each set of side lengths:

1. Set: \( 2, 3, 5 \)

   • \( 2 + 3 = 5 \) (not greater)
   • \( 2 + 5 = 7 > 3 \) (true)
   • \( 3 + 5 = 8 > 2 \) (true)
   • Since \( 2 + 3 \) is not greater than \( 5 \), this set does NOT form a triangle.

2. Set: \( 3, 5, 7 \)

   • \( 3 + 5 = 8 > 7 \) (true)
   • \( 3 + 7 = 10 > 5 \) (true)
   • \( 5 + 7 = 12 > 3 \) (true)
   • This set forms a triangle.

3. Set: \( 6, 7, 10 \)

   • \( 6 + 7 = 13 > 10 \) (true)
   • \( 6 + 10 = 16 > 7 \) (true)
   • \( 7 + 10 = 17 > 6 \) (true)
   • This set forms a triangle.

4. Set: \( 6, 14, 18 \)

   • \( 6 + 14 = 20 > 18 \) (true)
   • \( 6 + 18 = 24 > 14 \) (true)
   • \( 14 + 18 = 32 > 6 \) (true)
   • This set forms a triangle.

Conclusion: The set of side lengths that do NOT form a triangle is

\[ \boxed{2, 3, 5}. \]