Below are the answers to the questions:
Question 1:
To find angle \( x \):
The angles in a triangle sum to 180 degrees. For the lower triangle:
60° + 50° + angle \( y \) = 180°
Angle \( y = 180° - 60° - 50° = 70° \)
Angle \( x \) (the vertex angle of the inverted triangle) would be:
x + y = 180° (corresponding angles)
x + 70° = 180°
Thus, \( x = 180° - 70° = 110° \)
Answer: 110°
Question 2:
To find angle \( x \):
The top vertex angle \( y \) and the angles adjacent to it (57° and 61°) sum to 180°:
x + 57° + 61° = 180°
x + 118° = 180°
Thus, \( x = 180° - 118° = 62° \)
Answer: 62°
Question 3:
In triangle \( MNO \) with segment \( LM \) forming a straight angle with segment \( MO \). Given that \( \angle NOP = 104° \) and because segments \( MN \) and \( NO \) are congruent, this might lead us to utilize the Exterior Angle Theorem or consider \( \angle N + \angle M + \angle O = 180° \).
Since segment \( NO \) is congruent to segment \( MN \), \( \angle LMN \) would thus equal:
52°
Answer: 52°
Question 4:
To determine if segment \( GE \) can measure 6 units:
By triangle inequality, the sum of any two sides must be greater than the third side:
- 3 + 5 > 6 (True)
- 5 + 6 > 3 (True)
- 6 + 3 > 5 (True)
Thus, it is possible.
Answer: Yes, because 3 + 5 > 6, 5 + 6 > 3, and 6 + 3 > 5
Question 5:
To check whether a triangle can be formed with side lengths 4 in, 5 in, and 8 in:
Verify the triangle inequality:
- 4 + 5 = 9 (which is > 8)
Therefore, the student is wrong because:
Answer: No, because 4 + 5 = 9 > 8
Question 6:
We check the triangle inequality for each set of side lengths:
- 15 + 6 > 8 (True);
- 6 + 8 > 15 (False);
- 15 + 6 > 9 (True);
- 9 + 8 > 16 (False);
- 16 + 9 > 6 (True).
Check set 1 works:
Answer: Side = 15 cm, side = 6 cm, side = 8 cm
Question 7:
The missing statement relates to segments. Since segments \( BD \) must be reflexive, it leads to:
Answer: line segment BD ≅ line segment BD
Question 8:
To find angle \( x \):
The vertex angle sums with the base angles (which are equal) so:
x + 75° + 75° = 180°
x + 150° = 180°
Thus, \( x = 180° - 150° = 30° \)
Answer: 30°
Question 9:
Given that \( \angle BCA = 44° \) and since triangle \( ABC \) is isosceles where \( AB \cong BC \), the angles opposite these sides are equal. Thus:
m∠BCA + m∠BAC + m∠CBA = 180°
Let \( m∠BAC = m∠CBA \):
44° + 2x = 180°
2x = 136°
Thus, \( x = 68° \)
Answer: 68°
Question 10:
The flowchart details parallel lines help demonstrate that certain angles are equal through properties. The key reason involves angles that are congruent, leading to:
Answer: Alternate Interior Angles Theorem
These answers summarize and resolve the questions using geometrical principles related to triangles and their properties, ensuring clarity in the methodologies used.