Use the image to answer the question.
A number line with arrows at both the ends ranges from negative 5 to 5 in unit increments. The number line shows closed points on the following numbers along with letters labeled above the points: A for negative 4, B for negative 1, C for 3, and D for 4.
A triangle has no congruent sides. The lengths of two sides of the triangle are AB and BD . Use the number line above to determine the lengths of AB and BD . Then, use the Triangle Inequality Theorem to determine which two segments have lengths that could be used as the third side of the triangle. Select a response containing 2 possible answers.
(1 point)
Responses
BC and AD
cap b cap c and cap A cap d
BC and CD
cap b cap c and cap c cap d
AC and BC
cap A cap c and cap b cap c
CD and AC
cap c cap d and cap A cap c
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The lengths of AB and BD can be determined from the number line. AB has a length of 3 units, and BD has a length of 1 unit.
According to the Triangle Inequality Theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Using this theorem, the two possible combinations are:
1. BC and AD
2. BC and CD
Therefore, the correct response is: BC and AD
A right triangle has acute angles measuring 2x+6 degrees and 3x−26 degrees. Use the Triangle Angle Sum Theorem to find the measures of the missing angles of the triangle.(4 points)
The Triangle Angle Sum Theorem states that the sum of the angles in any triangle is always 180 degrees.
In this right triangle, one angle measures 90 degrees since it is a right angle. Let's denote this angle as A.
We are given that the other two angles measure 2x+6 degrees and 3x-26 degrees. Let's denote these angles as B and C, respectively.
According to the Triangle Angle Sum Theorem, the sum of A, B, and C must be equal to 180 degrees.
So, we can set up the equation:
A + B + C = 180
Substituting the values we know:
90 + (2x+6) + (3x-26) = 180
Simplifying the equation:
90 + 2x + 6 + 3x - 26 = 180
5x + 70 = 180
Subtracting 70 from both sides:
5x = 110
Dividing both sides by 5:
x = 22
Now that we have the value of x, we can substitute it back into the expressions for the angles to find their values:
Angle B = 2x + 6 = 2(22) + 6 = 44 + 6 = 50 degrees
Angle C = 3x - 26 = 3(22) - 26 = 66 - 26 = 40 degrees
Therefore, the measures of the missing angles in the right triangle are:
Angle A = 90 degrees (right angle)
Angle B = 50 degrees
Angle C = 40 degrees
shorter
Sure, here's a shorter version:
We are given that one angle of the right triangle is 90 degrees (A), as it is a right angle.
Using the Triangle Angle Sum Theorem, we know that the sum of the three angles in a triangle is always 180 degrees. Therefore,
A + B + C = 180
Substituting the given measurements into the equation, we have:
90 + (2x+6) + (3x-26) = 180
Simplifying the equation:
5x - 30 = 90
Adding 30 to both sides:
5x = 120
Dividing both sides by 5:
x = 24
Substituting the value of x back into the expressions for the angles, we find:
Angle B = 2x + 6 = 2(24) + 6 = 48 + 6 = 54 degrees
Angle C = 3x - 26 = 3(24) - 26 = 72 - 26 = 46 degrees
Therefore, the measures of the missing angles in the right triangle are:
Angle A = 90 degrees (right angle)
Angle B = 54 degrees
Angle C = 46 degrees
even shorter
Sure, here's an even shorter version:
One angle of the right triangle is 90 degrees (A), as it is a right angle.
Using the Triangle Angle Sum Theorem, we know that the sum of the three angles in a triangle is always 180 degrees.
Given the angles 2x+6 and 3x-26, we can set up the equation:
90 + (2x+6) + (3x-26) = 180
Simplifying the equation:
5x - 14 = 90
Adding 14 to both sides:
5x = 104
Dividing both sides by 5:
x = 20.8
Substituting the value of x back into the expressions for the angles, we find:
Angle B = 2x + 6 = 2(20.8) + 6 = 41.6 + 6 = 47.6 degrees
Angle C = 3x - 26 = 3(20.8) - 26 = 62.4 - 26 = 36.4 degrees
Therefore, the measures of the missing angles in the right triangle are:
Angle A = 90 degrees (right angle)
Angle B = 47.6 degrees
Angle C = 36.4 degrees
very short
Certainly! Here's a very short version:
Using the Triangle Angle Sum Theorem, we know that the sum of the angles in a triangle is always 180 degrees.
In the right triangle, one angle is 90 degrees (A), as it is a right angle.
The other two angles are given as 2x + 6 and 3x - 26 degrees.
Setting up the equation:
90 + (2x + 6) + (3x - 26) = 180
Simplifying:
5x - 14 = 90
Solving for x:
x = 20
Substituting the value of x back into the expressions for the angles:
Angle B = 2(20) + 6 = 46 degrees
Angle C = 3(20) - 26 = 34 degrees
Therefore, the measures of the missing angles in the right triangle are:
Angle A = 90 degrees (right angle)
Angle B = 46 degrees
Angle C = 34 degrees