Sure, here are the answers to your questions:
Question 1
Since the triangle is isosceles, the two base angles \(x\) are equal. According to the Triangle Angle Sum Theorem, the sum of the angles in a triangle is \(180^\circ\). Therefore:
\[ x + x + 18^\circ = 180^\circ \] \[ 2x = 180^\circ - 18^\circ = 162^\circ \] \[ x = \frac{162^\circ}{2} = 81^\circ \]
Response: 81°
Question 2
In a triangle, there can be at most one obtuse angle (an angle greater than 90°), because the sum of the angles must equal 180°.
Response: 1
Question 3
Using the Triangle Angle Sum Theorem:
\[ 39.5^\circ + 61.7^\circ + \text{third angle} = 180^\circ \] \[ \text{third angle} = 180^\circ - (39.5^\circ + 61.7^\circ) = 180^\circ - 101.2^\circ = 78.8^\circ \]
Response: 78.8°
Question 4
The Triangle Angle Sum Theorem states that the interior angles of a triangle sum to \(180^\circ\).
Response: 180°.
Question 5
Let the angles in the triangle be:
- Angle x = \(3y\)
- Angle w = \(4y - 4\)
- Angle y = \(y\)
Using the Triangle Angle Sum Theorem: \[ 3y + (4y - 4) + y = 180 \] \[ 8y - 4 = 180 \] \[ 8y = 184 \implies y = 23 \]
Substituting \(y\) back to find angle w: \[ w = 4(23) - 4 = 92 - 4 = 88° \]
Response: This seems to contradict the options given. Can you please specify if you want the angle w or an error in the problem statement, or need an adjustment?
Question 6
Let angle A = \(4x - 6\) and angles B and C be \(3x - 2\).
Using the Triangle Angle Sum Theorem: \[ (4x - 6) + (3x - 2) + (3x - 2) = 180 \] \[ 10x - 10 = 180 \] \[ 10x = 190 \implies x = 19 \]
Now substituting back to find angle A: \[ A = 4(19) - 6 = 76 - 6 = 70° \]
Response: 70°
Question 7
To ensure the three side lengths \(a\), \(b\), and \(c\) can form a triangle, they must satisfy the conditions of the Triangle Inequality Theorem. If \(a < b + c\), another valid condition would be \(b > a - c\).
Response: b>a−c
Question 8
For side lengths 11 and 3, we use the Triangle Inequality Theorem:
For the third side \(x\):
- \(11 + 3 > x\) which leads to \(x < 14\)
- \(11 + x > 3\) which leads to \(x > -8\) (not limiting since lengths are positive)
- \(3 + x > 11\) which leads to \(x > 8\)
Thus the possible third side should satisfy \(8 < x < 14\).
Therefore, the side measure that would not create a triangle is 13.
Response: 13
Question 9
Three congruent sides will always form a triangle, as they satisfy the Triangle Inequality Theorem.
Response: Three congruent sides will always form a triangle.
Question 10
Using the Triangle Inequality Theorem, for the side that measures \(s\):
- \(612 + 10 > s\) → \(622 > s\)
- \(612 + s > 10\) → \(s > -602\)
- \(10 + s > 612\) → \(s > 602\)
Thus, the measure of the third side must satisfy \(602 < s < 622\).
Assuming 16 cm is within the range of the Triangle Inequality Theorem.
Response: The response options given do not seem correct based on data given.
Question 11
To form a triangle, the following must be satisfied:
- \(4 + 4 > 5\) (True)
- \(4 + 5 > 4\) (True)
- \(4 + 5 > 4\) (True)
For \(4, 4, 7\):
- \(4 + 4 > 7\) (False)
Response: 4, 4, 7
Question 12
Given sides 24 and 30:
Using the Triangle Inequality Theorem, the third side \(s\) must satisfy:
- \(24 + 30 > s \implies s < 54\)
- \(30 + s > 24 \implies s > -6\)
- \(24 + s > 30 \implies s > 6\)
Thus, combining the conditions: \(6 < s < 54\).
Response: 6 < s < 54
Hope this helps! If there's anything more you would like assistance with, let me know!
1 answer